FACULTY OF SCIENCE - Pavol Jozef Šafárik University€¦ · Electronic academic textbook for Faculty of Science on P. J. Šafárik ... The text consists of four chapters that form

PAVOL JOZEF ŠAFAacuteRIK UNIVERSITY IN KOŠICE

FACULTY OF SCIENCE

DEPARTMENT OF PHYSICAL SCIENCES

MICHAL JAŠČUR

QUANTUM THEORY OF MAGNETISM

Academic textbooks

Košice 2013


copy 2013 doc RNDr Michal Jaščur CSc

Reviewers Prof RNDr Vladimiacuter Lisyacute DrSc

Prof RNDr Peter Kollaacuter CSc

Number of pages 65

Number of authorsʹ sheets 3

Electronic academic textbook for Faculty of Science on P J Šafaacuterik University in Košice

The author of this textbook is accountable for the professional level and language correctness

The language and arrangement of the manuscript have not been revised

Publisher Pavol Jozef Šafaacuterik University in Košice

Location httpwwwupjsskpracoviskauniverzitna-kniznicae-publikaciapf

Available 10 10 2013

Edition second

ISBN 978-80-8152-049-5

Contents

Preface 2

1 Many-Body Systems 711 Systems of Identical Particles 712 Heitler-London Theory of Direct Exchange 11

2 Bogolyubov Inequality and Its Applications 1921 General Remarks on Bogolyubov Inequality 1922 Mean-Field Theory of the Spin-12

Anisotropic Heisenberg Model 20221 General Formulation 20222 Thermodynamic Properties of the Heisenberg

Model 27

3 Spin-Wave Theory 3131 Bloch Spin-Wave Theory of Ferromagnets 3132 Holstein-Primakoff theory of ferromagnets 36

321 Internal Energy and Specific heat 42322 Magnetization and Critical Temperature 44

33 Holstein-Primakoff theory of antiferromagnets 45331 Ground-State Energy 49332 Sublattice Magnetization 51

3

Contents

4 Jordan-Wigner Transformation for the XY Model 55

4

Preface

Preface

This is a supporting text for the undergraduate students interested in quan-tum theory of magnetism The text represents an introduction to varioustheoretical approaches used in this field It is an open material in the sensethat it will be supported by appropriate problems to be solved by studentat seminars and also as a homework

The text consists of four chapters that form closed themes of the in-troductory quantum theory of magnetism In the first part we explain thequantum origin of the magnetism illustrating the appearance of the ex-change interaction for the case of hydrogen molecule The second chapter isdevoted to the application of the Bogolyubov inequality to the Heisenbergmodel and it represents the part which is usually missing in the textbooksof magnetism The third chapter deals with the standard spin wave-theoryincluding the Bloch and Holstein-Primakoff a approach Finally we discussthe application of the Jordan-Wigner transformation to one-dimensionalspin-12 XY model

The primary ambition of this supportive text is to explain diverse math-ematical methods and their applications in the quantum theory of mag-netism The text is written in the form which should be sufficient to de-velop the skills of the students up to the very high level In fact it isexpected that students following this course will able to make their ownoriginal applications of the methods presented in this book

The reading and understanding this text requires relevant knowledgesfrom the mathematics quantum and statistical mechanics and also fromthe theory of phase transitions It is also expected that students are familiarwith computational physics programming and numerical mathematics thatare necessary to solve the problems that will be included in the MOODLEeducational environment of the PJ Safarik University in Kosice

Finally one should note that the text is mathematically and physicallyrather advanced and its understanding assumes supplementary study of the

5

Preface

books listed in the list of references

Kosice September 2013 Michal Jascur

6

Chapter 1

Many-Body Systems

11 Systems of Identical Particles

Magnetism is a many-body phenomenon and its origin can only be explainedwithin the quantum physics In this part we will investigate some impor-tant properties of the wavefunctions of quantum systems consisting of manyidentical particles that are important to understand magnetic properties inthe solid state In general the Hamiltonian of the many-body system de-pends on the time coordinates and spin variables of all particles Howeverin this part we will mainly investigate the symmetry properties of the wave-function that are independent on the time and therefore we can excludethe time from our discussion For simplicity we neglect also spin-degrees offreedom

Let us consider a system of N identical particles described by the many-body Hamiltonian H(r1 r2 rN ) which remains unchanged with respectto the transposition of arbitrary two particles in the system This fact canbe mathematically expressed as

H(r1 r2 rj rk rN) = H(r1 r2 rk rj rN) (11)

7


or

PjkH(r1 r2 rj rk rN) = H(r1 r2 rk rj rN)Pjk

(12)

The previous equation defines the transposition operator Pjk of the jth andkth particle Since a mutual interchange of arbitrary particles in the systemdoes not change its state then the total wavefunction of the system obeysthe following relation

PjkΨ(r1 r2 rj rk rN) = eiαΨ(r1 r2 rk rj rN)

(13)

and

P2jkΨ(r1 r2 rj rk rN t) = e2iαΨ(r1 r2 rj rk rN)

(14)

where the real parameter α apparently satisfies equation

e2iα = 1 or eiα = plusmn1 (15)

The last equation implies that the total wavefunction either remains thesame or changes its sign when two arbitrary particles are interchanged inthe system In the first case the total wave function is called as a symmetricfunction and in the later one as an anti-symmetric function The symme-try of the wavefunction of an ensemble of identical particles is exclusivelygiven by the type of particles and does not depend on external conditionsIn fact it has been found that all particles obeying the Pauli exclusionprinciple have the antisymmetric wavefunction and all other particles havethe symmetric wavefunction Unfortunately the direct theoretical proof ofthis statement is impossible for strongly interacting particles since in real

8


many-body systems is not possible to calculate exactly the total wavefunc-tion On the other hand the situation becomes tractable for the case ofnon-interacting or weakly interacting particle Therefore we at first will in-vestigate the system of spinless non-interacting identical particles and thenwe clarify the role of the interaction and spin

In the case of noninteracting particles the total Hamiltonian of the sys-tem can be expressed as a sum of single-particle Hamiltonians ie

H =Nsumj

Hj (16)

where N denotes the number if particles It is also well known that thetotal wavefunction of non-interacting particles can be written as a productof the single-particle wavefunctions namely

Ψ(r1 r2 rj rk rN) =Nprodj=1

ϕnj(rj) (17)

where nj represents the set of all quantum numbers characterizing the rele-vant quantum state of the jth particle The single-particle functions ϕnj

(rj)are of course the solutions of the stationary Schrodinger equation

Hjϕnj(rj) = εnj

ϕnj(rj) (18)

Here εnjdenote the eigenvalues of the single-particle Hamiltonian Hj so

that the eigenvalue of the total Hamiltonian is given by En1n2nN=

sumj εnj

The symmetric wavefunction describing whole system can be written in theform

Ψ(r1 r2 rj rk rN) = AsumP

ϕn1(r1)ϕn2(r2) ϕnN(rN) (19)

9

12 Heitler-London Theory of Direct Exchange

where A denotes the normalization constant and summation is performedover all possible permutation of the particles Similarly for the antisymmet-ric case we obtain the form

Ψ(r1 r2 rj rk rN) =1radicN

∣∣∣∣∣∣∣∣∣ϕn1(r1) ϕn1(r2) ϕn1(rN)ϕn2(r1) ϕn2(r2) ϕn2(rN)

ϕnN(r1) ϕnN

(r2) ϕnN(rN)

∣∣∣∣∣∣∣∣∣(110)

which is usually called as the Slater determinant It is clear that inter-changing of two particles in the system corresponds to the interchange ofrelevant rows in the determinant (110) which naturally leads to the signchange of the wavefunction Moreover if two or more particles occupy thesame state then two or more rows in the determinant (110) are equal andconsequently the resulting wavefunction is equal to zero in agreement withthe Pauli exclusion principle

Until now we have considered a simplified system of non-interacting par-ticles and we have completely neglected spin degrees of freedom Howeverthe situation in realistic experimental systems is much more complicatedsince each particle has a spin and moreover the interactions between parti-cles often also substantially influence the behavior of the system Howeverif we assume the case of weak interactions and if we neglect the spin-orbitinteractions then the main findings discussed above remain valid and more-over the existence of the exchange interaction can be clearly demonstratedInstead of developing an abstract and general theory for such a case itis much more useful to study a typical realistic example of the hydrogenmolecule which illustrates principal physical mechanisms leading to theappearance of magnetism

10



In the previous part we have found that the total wavefunction of themany-electron system must be anti-symmetric In this subsection we willdemonstrate that the use of anti-symmetric wave functions leads to thepurely quantum contribution to the energy of the system which is calledthe exchange energy In fact this energy initiates a certain ordering ofthe spins ie it may lead to the magnetic order in the system A similareffect we also obtain using the single-product wavefunctions if we explicitlyinclude the exchange-interaction term into Hamiltonian This effect wasindependently found by Heisenberg and Dirac in 1926 and it represent themodern quantum-mechanical basis for understanding magnetic propertiesin many real systems

a b

2

1

r

rrab

r r

ra

a

1

2 2b

21

1b

Figure 11 Schematic geometry of the H2 molecule The larger circles denotethe protons and the smaller ones represent two electrons The relevant distancesbetween different pairs of particles are denoted as rij where i j = a b 1 2

We start our analysis with one of the simplest possible system namely

11


H2 molecule In the hydrogen molecule two electrons interact with eachother and with the nuclei of the atoms The situation is schematicallydepicted in Fig11 and considering this geometry we can write the Hamil-tonian of the H2 molecule in the form

H = H1 + H2 + W + HLS (111)

where

H1 = minus ~2

2m∆1 minus

e20ra1

(112)

H2 = minus ~2

2m∆2 minus

e20rb2

(113)

W =e20rab

+e20r12

minus e20rb1

minus e20ra2

(114)

and we have used the abbreviation e0 = e(4πε0) The terms H1 and H2 in(111) describe the situation when the two hydrogen atoms are isolated Theoperator W describes the interaction between two cores electrons as wellas between electrons and relevant nuclei Finally the last term representsthe spin-orbit interaction which is assumed to be small so that we canseparate the orbital and spin degrees of freedom To solve the problem of thehydrogen molecule we apply the first-order perturbation theory neglectingthe spin-orbit coupling and taking the term W as a small perturbationAs it is usual in the perturbation theory we at first solve the unperturbedproblem ie the system of two non-interacting hydrogen atoms Thus wehave to solve the Schrodinger equation

(H1 + H2)Ψ = U0Ψ (115)

12


As we already have noted above we will assume that the total wave functioncan be written as a product of the orbital and spin functions ie Ψ =φ(r1 r2)χ(s

z1 s

z2) Another very important point to be emphasized here

is that our simplified Hamiltonian does not explicitly depend on the spinvariables Consequently we can at first evaluate only the problem withorbital functions and at the end of the calculation we can multiply theresult by an appropriate spin function to ensure the anti-symmetry of thetotal wavefunction In order to express this situation mathematically weintroduce the following notations

bull ϕα(ri)where α = a b i = 1 2 represents the orbital wave func-tion of the isolated hydrogen atom when the ith electron is localizedclosely to the αth nucleus

bull ξγ(i)where γ =uarr darr i = 1 2 is the spin function describing thespin up or spin down of the ith electron

Now let us proceed with the discussion of the orbital functions of the systemdescribed by Eq(115) Since the orbital functions ϕα(ri) are eigenfunctionsof the relevant one-atom Schrodinger equation with the same eigenvalueE0 = minus1355 eV it is clear that the ground state of the two noninteractinghydrogen atoms has the energy U0 = 2E0 and it is doubly degeneratedThis is so-called exchange degeneracy and two wavefunctions correspondingto this U0 can be expressed as φ1(r1 r2) = ϕa(r1)ϕb(r2) and φ2(r1 r2) =ϕa(r2)ϕb(r1)

If we include into Hamiltonian also the interaction term W then the sit-uation becomes much more complex and the eigenfunctions and eigenvaluescannot be found exactly In the spirit of the perturbation theory we willassume that the wavefunction of the interacting system can be expressed inthe form

φ(r1 r2) = c1φ1(r1 r2) + c2φ2(r1 r2)

= c1ϕa(r1)ϕb(r2) + c2ϕa(r2)ϕb(r1) (116)

13


where c1 and c2 are the constant that will be determined later The systemis now described by the following Schrodinger equation(H1 + H2 + W

)[c1φ1(r1 r2) + c2φ2(r1 r2

]= E

[c1φ1(r1 r2) + c2φ2(r1 r2)

](117)

Multiplying the previous equation by φlowast1(r1 r2) and integrating over the

space one obtains

2c1E0 + 2c2E0

intintφlowast1(r1 r2)φ2(r1 r2)dV1dV2

+ 2c1

intintφlowast1(r1 r2)Wφ1(r1 r2)dV1dV2 + c2

intintφlowast1(r1 r2)Wφ2(r1 r2)dV1dV2

= c1E + c2E

intintφlowast1(r1 r2)φ2(r1 r2)dV1dV2 (118)

Similarly by multiplying Eq(117) by φlowast2(r1 r2) and integrating over the

space one finds

c1E0

intintφlowast2(r1 r2)φ1(r1 r2)dV1dV2 + c1


+ 2c2E0 + c2


= c1E

intintφlowast2(r1 r2)φ1(r1 r2)dV1dV2 + c2E (119)

Here one should notice that in deriving Eqs(118) and (119) we have usednormalization conditions for φ1 and φ2

In order to express two previous equations in an abbreviated form oneintroduces the following quantities

1 The overlap integral

S0 =

intϕlowasta(ri)ϕb(ri)dVi i = 1 2 (120)

14


2 The Coulomb integral

K =

intintφlowast1(r1 r2)Wφ1(r1 r2)dV1dV2 =


(121)

3 The exchange integral

A =



(122)

Applying this notation we rewrite Eqs(118) and (119) in the form

c1(2E0 +K minus E) + c2(2E0S20 + Aminus ES2

0) = 0

c1(2E0S20 + Aminus ES2

0) + c2(2E0 +K minus E) = 0 (123)

This is a homogeneous set of equations which determines unknown coeffi-cients c1 and c2 Of course we search for non-trivial solutions that can benaturally determined from the equation∣∣∣∣ 2E0 +K minus E 2E0S

20 + Aminus ES2

0

2E0S20 + Aminus ES2

0 2E0 +K minus E

∣∣∣∣ = 0 (124)

From this equation we easily find two possible solutions for the energy Eand the coefficients ci namely

Ea = 2E0 +K minus A

1minus S20

c1 = minusc2 (125)

and

Es = 2E0 +K + A

1 + S20

c1 = c2 (126)

15


Finally from the normalization condition of the total wavefunction oneobtains

1 =

int intφlowast(r1 r2)φ(r1 r2)dV1dV2

= 2|c1|2(1∓ S2) (127)

or

c1 =1radic

2(1∓ S20) (128)

where the minus and + sign corresponds to Ea and Es respectively Using(128) we can express the orbital part of the total wavefunctions as

φ(r1 r2) =1radic

2(1minus S20)

[ϕa(r1)ϕb(r2)minus ϕa(r2)ϕb(r1)

](129)

which is clearly the anti-symmetric expression and it corresponds to EaSimilarly the symmetric orbital function which corresponds to Es is givenby

φ(r1 r2) =1radic

2(1 + S20)

[ϕa(r1)ϕb(r2) + ϕa(r2)ϕb(r1)

] (130)

Finally we also can account for the spins of the electrons and to con-struct the following anti-symmetric wavefunctions

Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξuarr(1)ξuarr(2) S = 1 Sz = 1 (131)

Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξdarr(1)ξdarr(2) S = 1 Sz = 0 (132)

16


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ 1radic2[ξuarr(1)ξdarr(2) + ξdarr(1)ξuarr(2)]

S = 1 Sz = minus1 (133)

and

Ψ =1radic

2(1 + S20)

[ϕa(r1)ϕb(r2) + ϕb(r1)ϕa(r2)

] 1radic2

∣∣∣∣ ξuarr(1) ξuarr(2)ξdarr(1) ξuarr(2)

∣∣∣∣S = 0 Sz = 0 (134)

Here the wavefunctions (131)-(133) form a triplet (threefold degeneratestate) with the energy Ea the total spin S = 1 and the total zth componentSz = plusmn1 0 Similarly the wavefunction (134) describes a singlet state withenergy Es the total spin S = 0 and Sz = 0 Of course both the triplet andsinglet states can be the ground state of the system depending on the signof the exchange integral In order to prove this statement we calculate theenergy difference between Ea and Es ie

∆E = Ea minus Es =2(KS2

0 minus A)

1minus S40

asymp minus2A (135)

where we have neglected the overlap integral which very small in comparisonwith other terms in the expression It is clear from the previous equationthat for A gt 0 the ground state of the system will be represented by thetriplet state with parallel alignment of spins and for A lt 0 the singlet statewith anti-parallel orientation of the spins Of course the parallel (anti-parallel) spin orientation implies also parallel (anti-parallel) alignment of themagnetic moments which is crucial for the observation of the ferromagneticor antiferromagnetic ordering in the real systems

The situation in the hydrogen molecule of course differs from the realsituation in ionic crystal in many respects For example the electrons in

17


the hydrogen molecule occupy 1s orbitals while in the ionic crystals theunpaired localized electrons usually occupy d or f orbitals In fact thecalculation of the exchange integral using 3d orbital functions leads to thepositive value of the A Therefore the direct exchange interactions leadsto the ferromagnetic ordering in such systems Later it was discovered byAnderson that the antiferromagnetic ordering is caused by so-called indi-rect exchange interaction via a non-magnetic bridging atom The situationin real materials is usually much more complicated and other mechanismsuch as the Dzyaloshinkii-Moriya or Ruderman-Kittel-Kasuya-Yosida inter-actions can play an important role

Although the calculation performed for the hydrogen molecule does notreflect the complex situation of the real magnetic systems it is extremelyimportant from the methodological point of view In fact it is clear from ourcalculation that the parallel or anti-parallel alignment of the magnetic mo-ments appears due to new contribution to the energy of the system whichoriginates from the indistinguishability of quantum particles The othermechanisms leading to the macroscopic magnetism have also quantum na-ture and cannot be included into the classical theory

Thus we can conclude that the physical origin of magnetism can becorrectly understood and described only within the quantum theory

18

Chapter 2

Bogolyubov Inequality and ItsApplications

21 General Remarks on Bogolyubov Inequal-

ity

The Bogolyubov inequality for the Gibbs free energy G of an interactingmany-body system described by the Hamiltonian H is usually written inthe form

G le G0 + ⟨H minus H0⟩0 = ϕ(λx λy λz ) (21)

In this equation the so-called trial Hamiltonian H0 = H0(λx λy λz )depends on some variational parameters λi that are naturally determinedin the process of calculation and the symbol ⟨ ⟩0 stands for the usualensemble average calculated with the trial Hamiltonian Finally G0 denotesthe Gibbs free energy of the trial system defined by

G0 = G0(λx λy λz ) = minus 1

βlnZ0 (22)

19

22 Mean-Field Theory of Anisotropic Heisenberg Model

where β = 1kBT and the the partition function Z0 is given by

Z0 = Tr eminusβH0 (23)

In general there is a remarkable freedom in defining the trial Hamilto-nian In fact the only limitations to be taken into account follow from thetwo obvious requirements

1 The trial Hamiltonian should naturally represent a simplified physicalmodel of the real system

2 The expression of G0 must be calculated exactly in order to obtain aclosed-form formula for the rhs of Eqs(21)

22 Mean-Field Theory of the Spin-12

Anisotropic Heisenberg Model

221 General Formulation

The accurate theoretical analysis of the Heisenberg model is extremely harddue to interaction terms in the Hamiltonian including non-commutative spinoperators Probably the simplest analytic theory applicable to the modelis the standard mean-field approach Although the mean-field theory canmathematically be formulated in many different ways we will develop in thistext an approach based on the Bogolyubov inequality for the free energy ofthe system The main advantage of this formulation is its completeness (wecan derive analytic formulas for all thermodynamic quantities of interest)and a possibility of further generalizations for example an extension to theOguchi approximation

20


In this part we will study the spin-12 anisotropic Heisenberg model ona crystal lattice described by the Hamiltonian

H = minussumij

(JxSxi S

xj + JyS

yi S

yj + JzS

zi S

zj )minus gmicroB

sumi

(HxSxi +HyS

yi +HzS

zi )

(24)where g is the Lande factor microB is the Bohr magneton Jα and Hα α =x y z denote the spatial components of the exchange interaction and exter-nal magnetic field respectively One should note here that Jα is assumedto be positive for the ferromagnetic systems and negative for the antifer-romagnetic ones Moreover it is well-known that the absolute value ofexchange integrals very rapidly decreases with the distance thus we canrestrict the summation in the first term of (24) to the nearest-neighboringpairs of atoms on the lattice Finally the spatial components of the spin-12operators are given by

Sxk =

1

2

(0 11 0

)k

Syk =

1

2

(0 minusii 0

)k

Szk =

1

2

(1 00 minus1

)k

(25)

To proceed further we choose the trial Hamiltonian in the form

H0 = minusNsumk

(λxS

xk + λyS

yk + λzS

zk

) (26)

where N denotes the total number of the lattice sites and λi are variationalparameters to be determined be minimizing of the rhs of the Bogolyubovinequality After substituting (25) into (26) we can apparently rewriteprevious equation as follows

H0 =Nsumk=1

H0k (27)

21


where the site Hamiltonian H0k is given by

H0k = minus1

2

(λz λx minus iλy

λx + iλy minusλz

)k

(28)

In order to evaluate the terms entering the Bogolyubov inequality we atfirst have to calculate the partition function Z0 Taking into account thecommutation relation [H0k H0ℓ] = 0 for k = ℓ we obtain the followingexpression

Z0 = Tr exp(minusβ

Nsumk=1

H0k

)=

Nprodk=1

Trk exp(minusβH0k

) (29)

where Trk denotes the trace of the relevant operator related to kth latticesite Now after setting Eq(28) into (29) one obtains

Z0 =Nprodk=1

Trk exp

(β2λz

β2(λx minus iλy)

β2(λx + iλy) minusβ

2λz

)k

=

[Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

]N

(210)

The crucial point for further calculation is the evaluation of exponentialfunction in previous equation Let us note that in performing this task wewill not follow the usual approach based on the diagonalization of modifiedmatrix (28) entering the argument of exponential Instead we will performall calculations in the real space by applying the following matrix form ofthe Cauchy integral formula

eminusβH0k =1

2πi

∮C

ezdz

zI minus (minusβH0k) (211)

22


After a straightforward calculation we obtain

eminusβH0k =

(h11 h12

h21 h22

)(212)

with

h11 = cosh

(β

2Λ

)+

λz

Λsinh

(β

2Λ

)(213)

h12 =λx minus iλy

Λsinh

(β

2Λ

)(214)

h21 =λx + iλy

Λsinh

(β

2Λ

)(215)

h22 = cosh

(β

2Λ

)minus λz

Λsinh

(β

2Λ

) (216)

where we have introduced the parameter

Λ =radicλ2x + λ2

y + λ2z (217)

Using previous equations we easily obtain the following expressions for Z0

and G0

Z0 =

[2 cosh

(β

2Λ

)]N(218)

G0 = minusN

βln

[2 cosh

(β2Λ

)] (219)

23


To obtain the complete expression for ϕ(λx λy λz ) we have to eval-

uate the terms ⟨H⟩0 and ⟨H0⟩0 that can be expressed as

⟨H⟩0 = minusNq

2

(Jx⟨Sx

i Sxj ⟩0 + Jy⟨Sy

i Syj ⟩0 + Jz⟨Sz

i Szj ⟩0

)minus N

(hx⟨Sx

j ⟩0 + hy⟨Syj ⟩0 + hz⟨Sz

j ⟩0) (220)

where q denotes the coordination number of the lattice and hα = gmicroBHα(α = x y z) Similarly we have

⟨H0⟩0 = minusN(λx⟨Sx

j ⟩0 + λy⟨Syj ⟩0 + λz⟨Sz

j ⟩0) (221)

The ensemble averages of ⟨Sαj ⟩0 entering two previous equations can be

expressed in the form

⟨Sαj ⟩0 =

1

Z0

TrSαj e

minusβH0

=TrjS

αj exp

(minusβH0j

)prodNminus1k =j Trk exp

(minusβH0k

)Trj exp

(minusβH0j


(minusβH0k

) (222)

which can be simplified as follows

⟨Sαj ⟩0 =

TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) (223)

Similarly one easily finds that

⟨Sαi S

αj ⟩0 =

TriSαi exp

(minusβH0i

)Tri exp

(minusβH0i

) TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) =(⟨Sα

j ⟩0)2

(224)

24


In order to complete our calculation we substitute Eqs (212)-(216) intoEq (223) and then we evaluate the trace of relevant matrices In this waywe obtain the following equation

⟨Sαj ⟩0 =

λα

2Λtanh

(β

2Λ

) (225)

and

⟨Sαi S

αj ⟩0 =

(λα

2Λ

)2

tanh2

(β

2Λ

) (226)

After introducing notation ⟨Sαj ⟩0 = mα α = x y z we can rewrite the rhs

of the Bogolyubov inequality in the form

ϕ(λx λy λz) = minusN

βln(2 cosh

β

2Λ)minus Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

)+ N

[(λx minus hx)mx + (λy minus hy)my + (λz minus hz)mz

] (227)

Since the quantity ϕ represents an upper bound for the exact Gibbs freeenergy of the system then the best possible approximation (with the trialHamiltonian H0) corresponds for those values of λx λy λz that minimizeϕ(λx λy λz) Relevant values of λx λy λz are obviously determined fromthe equations

partϕ

partλx

= 0partϕ

partλy

= 0partϕ

partλz

= 0 (228)

The above conditions lead to three independent equations that can berewritten in the following simple matrix form

partmx

partλx

partmy

partλx

partmz

partλxpartmx

partλy

partmy

partλy

partmz

partλy

partmx

partλz

partmy

partλz

partmz

partλz

λx minus qJxmx minus hx

λy minus qJymy minus hy

λz minus qJzmz minus hz

= 0 (229)

25


Since the first matrix in (229) is in general non-zero then the only possiblesolution of this equation is given by

λx = qJxmx + hx λy = qJymy + hy λz = qJzmz + hz (230)

and the cooresponding equilibrium Gibbs free energy can be after a smallmanipulation expressed in the form

G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)

where we have denoted

hef =radic

(qJxmx + hx)2 + (qJymy + hy)2 + (qJzmz + hz)2 (232)

Having obtained the last equation we are able to identify the physicalmeaning of the parameters mxmymx and consequently also the meaningof variational parameters λx λy λz

For this purpose we at first recall that the spatial component of thetotal magnetization Mα is defined as

Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)

After a straightforward calculation we obtain the following very simple ex-pressions

Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)

which clearly indicate that mα represents the spatial components of themagnetization Consequently the parameters λα have the meaning of ofthe molecular-field components acting on one atom in the lattice Thusour formulation is nothing but the standard mean-field theory The mainadvantage of our approach is that this formulation provides a close-form an-alytical expression for the Gibbs free energy which is of principal importancefor finding stability conditions of various physical quantities

26


222 Thermodynamic Properties of the HeisenbergModel

Now let us proceed with the calculation of several interesting physical quan-tities for the system under investigation Using Eqs (222) (230) and(234) one simply obtains for the components of reduced magnetization aset of three coupled equations namely

mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)

where we have denotedIn order to investigate the long-range ordering the systems we set hx =

hy = hz = 0 and obtain

mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with

h0 =radic(Jxmx)2 + (Jymy)2 + (Jzmz)2 (241)

These three nonlinear equations have always the trivial solution mx =my = mz = 0 corresponding to the disordered paramagnetic phase (DP)Moreover depending on the temperature and the values of exchange pa-rameters there also exist non-trivial solutions corresponding to ordered fer-romagnetic phase (OP) The stability of the relevant solutions is then de-termined by the minimum of the Gibbs free energy given by Eq(231) Nowwe briefly analyze possible trivial and non-trivial solutions of (238)-(240)for some particular cases of the exchange interactions

1 Ising model Jx = Jy = 0 Jz = J = 0

mx = my = 0 and mz =1

2tanh

(qβJ2

mz

)for T lt Tc

mx = my = mz = 0 for T gt Tc (242)

2 Isotropic XY model Jx = Jy = J = 0 Jz = 0

mz = 0my = mx and mx =1

2radic2tanh

(radic2qβJ

2mx

)for T lt Tc


3 Isotropic Heisenberg model Jx = Jy = Jz = 0

my = mx = mz and mz =1

2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28


Here one should note that non-trivial results for isotropic XY and Heisen-berg model model represents valid solution for the models when the axis ofquantization is given by the equation y = x = t t isin R for the XY modeland y = x = z = t t isin R for the Heisenberg model On the other handif choose as a quantization axis for example the x axis then we obtain forboth models identical result as that for the Ising one Thus one can concludethat mean-field theory is not able to describe correctly quantum featuresof localized spin models In order to illustrate the numerical accuracy ofthe method we have compared in Fig 23 the temperature variation of themean-field magnetization with exact result on the square lattice with q = 4

00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT

Figure 21 Comparisom of the temperature dependence of magnetization forthe Ising model on a square lattice

The critical (or Curie) temperature is defined as a temperature at whichthe long-range order (or magnetization) vanishes In order to determine the

29


Curie temperature we take into account that tanh(x) asymp x for x ltlt 1 andobtain an universal solution

Tc =qJα

4kB α = x y z (245)

which valid for all three cases mentioned above Of course the critical tem-perature should be different depending on the model considered Thus thepresent method is not able to distinguish specific features of the models aslong as concerns the critical behavior Moreover it is also clear that thetopology of the lattice is taken into account only through its coordinationnumber that is also weak point of the present method It is well knownthat this approach significantly fails for the low-dimensional systems nev-ertheless it gives rather good qualitative picture for the three-dimensionalsystems and therefore it is frequently used to obtain analytical results par-ticularly for complex systems

Finally one should note that calculations of other thermodynamic quan-tities is easy and straightforward since we have analytical solution for theGibbs free energy of the system from which other quantities are easily ob-tained

This calculation including numerical analysis and discussion is left forthe readers as an appropriate exercise

Let us conclude this part by noticing that the present approach can beextended to case of the the so-called Oguchi approximation which alreadyaccount much better for the quantum effects as well as for the spin-spincorrelations Possible extensions are rather involved and require lengthycalculations that are beyond the scope of the present book

30

Chapter 3

Spin-Wave Theory

31 Bloch Spin-Wave Theory of Ferromag-

nets

In this part we will discuss the concept of spin waves which enables to studythe some physical properties of magnetic systems at low temperatures Thistheory was originally introduced by F Bloch [1] and its main assumptionis that the ground state of the system is ordered and the excited states aredescribed as a collection of spin waves At low temperatures it is reasonableto expect the small amplitudes of the spin waves thus the interaction amongspin waves can be neglected We will consider in this part an isotropicHeisenberg model defined by the Hamiltonian

H = minusJsumij

SiSj minus gmicroBHsumi

Szi (31)

Here Si represents a vector spin-operator at the ith site of the lattice Thisoperator is naturally defined as Si = (Sx

i Syi S

zi ) where Sα

i (α = x y z)represent spatial components of the standard spin operators g is the Lande

31

31 Bloch Spin-Wave Theory of Ferromagnets

factor microB is the Bohr magneton and H denotes the external magnetic fieldwhich is applied along z axis J is the exchange integral which is assumedto be positive since we are going to investigate the ferromagnetic systems

Adopting the basic concept of lattice vibrations we can write the fol-lowing equation of motion of the spin Sℓ on the j th lattice point

dSℓ

dt=

i

~[H Sℓ] (32)

In what follows we will use the following commutation relations for spinoperators

[Sai S

bj ] = iεabcδijS

cj a b c = x y or z (33)

where εabc represents Levi-Civita tensor and δij is the Kronecker symbolNow substituting the Hamiltonian (31) into (32) and employing relations(33) one obtains

dSℓ

dt= minus1

~(Hℓ times Sℓ) (34)

where the local magnetic field Hℓ acting on the j th spin is given by

Hℓ =sumj

JjℓSj + gmicroBHe3 (35)

with e3 = (0 0 1) Using two previous equations one easily obtains thefollowing equations of motion of the x y and z components of the spinoperator

~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz

j Syℓ ) + gmicroBHSy

ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx

j Szℓ )minus gmicroBHSx

ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)

If we restrict ourselves to the case of low-energy excitation then the compo-nents Sx

ℓ Syℓ can be regarded as small quantities of first order Consequently

the right-hand side of (38) is a small quantity of the second order whichcan be neglected and it follows from Eq (38) that the zth component be-comes a constant Moreover it is reasonable to put explicitly Sz

ℓ = S sincethe external magnetic field is assumed to be applied parallel to the zth axisUnder these assumptions Eqs (36) and (37) can be rewritten in the form

~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy

ℓ ) + gmicroBHSyℓ (39)

~dSy

ℓ

dt= minusS

sumj

Jjℓ(Sxℓ minus Sx

j )minus gmicroBHSxℓ (310)

or alternatively

~dS+

ℓ

dt= minusiS

sumj

Jjℓ(S+ℓ minus S+

j )minus gmicroBHS+ℓ (311)

~dSminus

ℓ

dt= minusiS

sumj

Jjℓ(Sminusℓ minus Sminus

j )minus gmicroBHSminusℓ (312)

where we have introduced the spin-raising and spin-lowering operators

S+ℓ = Sx

ℓ + iSyℓ Sminus

ℓ = Sxℓ minus iSy

ℓ (313)

that obey the following commutation rules

[Szk S

+ℓ ] = S+

k δkℓ [Szk S

minusℓ ] = minusSminus

k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33


The above relations are directly obtained with the help of well-known com-mutation relations of the spin operators (33)

To proceed further we now express the spin-raising and spin-loweringoperators with the help of the Fourier transform ie

S+q =

1radicN

sumj

eiqmiddotRj S+j Sminus

q =1radicN

sumj

eminusiqmiddotRj Sminusj (315)

S+j =

1radicN

sumq

eminusiqmiddotRj Sq Sminusj =

1radicN

sumq

eiqmiddotRj Sq (316)

Substituting Eq(316) into Eq(312) and applying the following relationsumℓ

eminusi(q1minusq2)middotRℓ = Nδq1q2 (317)

one gets equation

minusi~dSminus

q

dt=

[Ssumℓ

Jjℓ(1minus eminusiqmiddot(RjminusRℓ)) + gmicroBH]Sminusq (318)

If we now introduce a real quantity

J(q) =sumj

J(Rj)eminusiqmiddotRj (319)

then we rewrite (318) as

minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)

It is straightforward to obtain the solution of this equation in the form

Sq = δSqei(ωqt+α) (321)

34


where the eigenfrequencies of the spin waves are given by

~ωq = S[J(0)minus J(q)

]+ gmicroBH (322)

One should notice here that in the limit of small q we obtain ~ωq asymp q2Now using Eq(321) we can express the spin operators in the real space

as follows

Sminusj =

1radicNδSqe

i(qmiddotRj+ωqt+α) (323)

Sxj =

1radicNδSq cos

[i(q middotRj + ωqt+ α)

](324)

and

Syj = minus 1radic

NδSq sin


] (325)

Finally it is straightforward to prove that the energy difference betweenan excited state and ground state is given by

Eq minus E0 = ~ωq (326)

where ~ωq represents the excitation energy of the spin wave with the wavevec-tor q The proof of the last statement is left as an appropriate exercise forreaders

35

32 Holstein-Primakoff theory of ferromagnets

32 Holstein-Primakoff theory of ferromag-

nets

The Blochrsquos spin-wave theory discussed in previous chapter represents a verysimple approach in which we completely neglect the interaction among spinwaves This deficiency can be eliminated using an alternative formulationbased on magnon variables (or creation-annihilation operators) developedby Holstein and Primakoff [2] In this part we apply The Holstein-Primakofftheory to case of an anisotropic quantum Heisenberg model described bythe Hamiltonian

H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz

i Szj )minus gmicroBH

sumi

Szi (327)

The meaning of all symbols in previous Eq (327) is the same as in previoustext

In order to apply a spin-wave picture to analyze magnetic properties ofthe model under investigation one has to perform three subsequent math-ematical transformation At first we express the components of the spinoperators trough spin-lowering and spin raising operators then we intro-duce so-called creation and annihilation operators and finally we performa Fourier transform of the relevant all creation and annihilation operatorsentering the Hamiltonian The eigenvalues of the final Hamiltonian then en-able to determine relevant physical quantities applying standard relationsof statistical mechanics For further manipulations it is more convenient torewrite (327) as follows

H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz

ℓ Szℓ+δ)minus gmicroBH

sumℓ

Szℓ (328)

where ℓ labels the lattice sites and δ denotes nearest neighbors of the relevantsite and we assume that J gt 0 The integrals of motion for Hamiltonian

36


(328) are the total z th component Sz =sum

k Szk and the total spin of the

systemThe ground state of the system is ordered and for H gt 0 the magnetic

moment is paralell to the z axis If we consider the lattice consisting of Natoms with spin S then the ground-state vector in the spin basis reads |0⟩= |NSNS⟩ Of course we work in a real space basis in which the spinoperators Sz

ℓ are diagonal We recall that the eigenvalues MS = minusS minusS +1 S and eigenvectors |MS⟩ℓ of the operator Sz

ℓ satisfy the followingequation

Szℓ |MS⟩ℓ = MS|MS⟩ℓ (329)

Now applying operators (313) to the eigenvectors |MS⟩ℓ one obtains

S+ℓ |MS⟩ℓ =

radic(S minusMS)(S minusMS minus 1)|MS + 1⟩ℓ (330)

and

Sminusℓ |MS⟩ℓ =

radic(S +MS)(S minusMS minus 1)|MS minus 1⟩ℓ (331)

In addition to the spin-raising and spin-lowering operators we will alsouse in our analysis the operator describing a deviation of the zth spin-component from its maximum value namely

nℓ = S minus Szℓ (332)

The eigenvalues of this operator are integers nl = 0 1 2 2S (or nl =S minusMS) thus we can rewrite Eqs(330) in the form

S+ℓ |nℓ⟩ℓ =

radic2S

radic1minus (nℓ minus 1)

2S

radicnℓ|nℓ minus 1⟩ℓ (333)

Le us note that performing a similar procedure is also possible for Eq(331)and it will be used later

37


Now with the help of (313) we can rewrite the Hamiltonian (328) inthe form

H = minusJsumℓδ

[12(S+

ℓ Sminusℓ+δ + Sminus

ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)

which is suitable for introducing the Holstein-Primakoff magnon represen-tation In order to define the relevant transformation we at first introducethe standard creation and annihilation operators that are defined as

adaggerℓΨnℓ=

radicnℓ + 1Ψnℓ+1 aℓΨnℓ

=radicnℓΨnℓminus1 (335)

The central idea of the Holstein-Primakoff approach is based on the intro-duction of the following transformation

S+ℓ = (2S)12

(1minus adaggerℓaℓ

2S

)12

aℓ (336)

Sminusℓ = (2S)12adaggerℓ


2S

)12

(337)

Szℓ = S minus adaggerℓaℓ (338)

As already noted above the eigenvalues of adaggerℓaℓ are arbitrary integers whilethat of nℓ are limited to the range 0 le nℓ le 2S Fortunately this discrep-ancy does not play any important role in the calculation since the transitionfrom the state with nℓ le 2S to states with nℓ gt 2S never occur eg

Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)

Now we transform the operators aℓ adaggerℓ from the real space to the reciprocal

space by the following Fourier tansform

aℓ =1radicN

sumq

eminusiqmiddotRℓbq (340)

38


adaggerℓ =1radicN

sumq

eiqmiddotRℓbdaggerq (341)

where N denotes the total number of atoms q represents a vector of thereciprocal space and Rℓ represents a vector specifying the position of ℓthatom in the lattice

The magnon-annihilation and magnon-creation operators are given by

bq =1radicN

sumℓ

eiqmiddotRℓaℓ (342)

bdaggerq =1radicN

sumℓ

eminusiqmiddotRℓadaggerℓ (343)

and they obey the usual magnon commutation relations

[bq1 bdaggerq2] = δq1q2 [bq1 bq2 ] = [bdaggerq1

bdaggerq2] = 0 (344)

It is useful to note here that the operator bdaggerq creates a magnon with thewave vector q and the operator bq annihilates it Moreover it follows fromthe transformation (340) and (341) that any change of the spin state atarbitrary lattice site is described as a superposition of an infinite number ofthe spin waves

At this stage it is clear that further exact treatment of the problem isimpossible due to the complexity of applied transformations The progressis however still possible if we restrict ourselves to the case when the numberof excited states is small in comparison with 2S Under this assumptionwe can neglect the higher order terms in expansion of the square root andrewrite Eqs (336) and (337) in the form

S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)

Finally substituting (340) and (341) into previous equation one obtains

S+ℓ =

(2S

N

)12[sumq1

eminusiq1middotRℓbq1 minus1

4SN

sumq1q2q3

ei(q1minusq2minusq3)middotRℓbdaggerq1bq2bq3 +

](347)

Sminusℓ =

(2S

N

)12[sumq1

eiq1middotRℓbdaggerq1minus 1

4SN

sumq1q2q3

ei(q1+q2minusq3)middotRℓbdaggerq1bdaggerq2

bq3 +

](348)

Similarly the operator Szℓ is given by

Szℓ = S minus 1

N

sumq1q2

ei(q1minusq2)middotRℓbdaggerq1bq2 (349)

If we assume that each lattice site has z nearest neighbors then we canspecify their positions on the lattice by introducing the vector

δ = Rℓ+δ minusRℓ (350)

Substituting (347)-(350) into Eq(334) and performing a rearrange-ment of the terms one can rewrite the Hamiltonian as

H = minusJNzS2 minus gmicroBNHS + H0 + H1 (351)

where H0 is a bilinear form in the magnon variables and it is given by

H0 = minus JS

N

sumℓδ

sumq1q2

[eminusi(q1minusq2)middotRℓeiq2middotδbq1b

daggerq2

+ ei(q1minusq2)middotRℓeminusiq2middotδbdaggerq1bq2

]+

JS

N

sumℓδ

sumq1q2

[ei(q1minusq2)middotRℓbq1b

daggerq2

+ eminusi(q1minusq2)middotRℓei(q1minusq2)middotδbdaggerq1bq2

]+

gmicroBH

N

sumℓq1q2


40


The term H1 includes higher-order terms and for the sake of simplicity willbe neglected Now taking into account the relation (317) and performingthe summation over ℓ and q2 we can simplify Eq(352) as follows

H0 = minusJzSsumq

(γqbqbdaggerq + γminusqb

daggerqbq minus 2bdaggerqbq) + gmicroBH

sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)

and the summation in (354) runs over all nearest neighbors For furthermanipulation it useful noticing that

sumq γq = 0 and moreover for the crys-

tals with the symmetry center we have γq = γminusk Employing the magnoncommutation rules [bdaggerqbq] = 1 one obtains

H0 =sumq

[2JzS(1minus γq) + gmicroBH

]bdaggerqbq (355)

The last equation can be rewritten in the following very simple form

H0 =sumq

~ωqnq (356)

where nq is the magnon occupation operator and

~ωq = 2JzS(1minus γq) + gmicroBH (357)

Since the quantity γq can be expressed as follows

γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ

(eiqmiddotδ + eminusiqmiddotδ) =1

2z

sumδ

cos(q middot δ) (358)

then the dispersion relation becomes

~ωq = 2JzS[1minus 1

2z

sumδ

cos(q middot δ)]+ gmicroBH (359)

41


321 Internal Energy and Specific heat

In this subsection we will show how one can calculate the internal energyand specific heat of Heisenberg model within the Holstein-Primakoff theoryBefore performing any calculation it is useful noticing that due to neglectinghigher order terms in Eq (345) and (346) we treat the ensemble of non-interacting magnons In further calculatiuon we also will assume H = 0and |q middot δ| ≪ 1

Under these assumptions the internal energy of the magnon gas in ther-modynamic equilibrium at temperature T can be expressed as

U =sumq

~ωq⟨nq⟩ (360)

where ⟨nq⟩ is given by the well-known Bose-Einstein relation

⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)

Substituting (361) into (360) one gets

U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)

where we have introduced the lattice stiffnes as D = 2SJa2 If we denotex = Dq2kBT then we can rewrite (362) as

U =(kBT )

52

4π2D32

int xm

0

x32

exp(x)minus 1dx (363)

42


The last expression can be further simplified by considering the regionkBT ≪ ωmax Then the upper limit in the integral (363) can be approxi-mated by infinity and one obtains for internal energy the following analyticalexpression for internal energy of the system

U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)

Here Γ and ζ denote the Gamma- and Riemann zeta-functions respectively

and their numerical values of relevant arguments are known to be Γ(

52

)=

3radicπ

4and ζ

(52 1)= 1341

Thus we finally obtain

U ≃ 045(kBT )52

π2D32 (365)

Consequently for the magnetic part of specific heat at constant volume isgiven by

CV = 0113kB

(kBTD

) 32 (366)

In this work we consider strictly insulating materials thus there is no elec-tron contribution to the total specific heat Consequently taking into ac-count the phonon and magnon contribution we can express the total specificheat at low temperatures by the formula

C = aT 3 + bT 32 (367)

It is clear from this equation that the magnon contribution can experimen-tally easily determined if we plot the dependence of reduced specific heatCT 32 versus T 32 The situation is schematically illustrated in Fig 31

43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution

Figure 31 The re-scaled specific heat vs T 32

322 Magnetization and Critical Temperature

The magnetic moment of the system can be calculated from the equation

MS = gmicroB

⟨(NS minus

sumq

bdaggerqbq)⟩ (368)

The deviation of the magnetization from its saturation value can be ex-pressed as

∆M = M(0)minusM(T ) = gmicroB

sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44

33 Holstein-Primakoff theory of antiferromagnets

At low temperatures (Dq2 ≫ kBTJ) we obtain

∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)

The finite-temperature magnetization can be then expressed as

M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)

where the critical temperature Tc is given by

Tc =( M(0)

0117gmicroB

) 23 D

kB (372)

33 Holstein-Primakoff theory of antiferro-

magnets

The Holstein-Primakoff theory can be extended also to the case of antifero-magnetic Heisenberg model which can describe many real materials [7] Onthe other hand the generalization of the Holstein-Primakoff for quantumantiferromagnets is very interesting because such systems exhibit very inter-esting magnetic properties that differ from quantum ferromagnets in manyrespects Our aim in this subsection is to investigate the antiferromagneticHeisenberg model spin system consisting of two interpenetrating sublatticesa and b which is described by the Hamiltonian

H = minusJsumij

SaiSbj minus gmicroBHa

sumi

Szai + gmicroBHa

sumj

Szbj (373)

45


where J lt 0 is the exchange interaction which couples the nearest neighborson lattice and the quantity Ha gt 0 represents an external magnetic fieldwhich is parallel to the z axis In what follows we will treat the problem ap-plying the Holstein-Primakoff transformation for each sublattice separatelyAt first we express the spin rising and lowering sublattice operators withthe help of creation-annihilation operators as follows

S+aj = (2S)12

(1minus

adaggerjaj

2S

)12

aj Sminusaj = (2S)12adaggerj

(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12

(1minus bdaggerℓbℓ

2S

)12

bℓ Sminusbℓ = (2S)12bdaggerℓ


2S

)12

(375)

Szaj = S minus adaggerjaj minusSz

bℓ = S minus bdaggerℓbℓ (376)

Next we introduce the sublattice spin-wave variables

cq =2radicN

sumj

eminusiqmiddotRjaj cdaggerq =2radicN

sumj

eiqmiddotRjadaggerj (377)

dq =2radicN

sumℓ

eiqmiddotRℓbℓ ddaggerq =2radicN

sumℓ

eminusiqmiddotRℓbdaggerℓ (378)

In previous equations summations run over all N atoms of the a and b sub-lattice respectively Moreover the following magnon commutation relationshold

[cq1 cdaggerq2] = [dq1 d

daggerq2] = δq1q2

[cq1 cq2 ] = [cdaggerq1 cdaggerq2

] = 0 [dq1 dq2 ] = [ddaggerq1 ddaggerq2

] = 0 (379)

46


Now we expand (374)-(375) as

S+aj =

(4S

N

)12[sumq1

eminusiq1middotRjcq1 minus1

8SN

sumq1q2q3

ei(q1minusq2minusq3)middotRjcdaggerq1cq2cq3 +

](380)

Sminusaj =

(4S

N

)12[sumq1

eiq1middotRjcdaggerq1minus 1

8SN

sumq1q2q3

ei(q1+q2minusq3)middotRjcdaggerq1cdaggerq2

cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1

eminusiq1middotRℓdq1 minus1

8SN

sumq1q2q3

ei(q1minusq2minusq3)middotRℓddaggerq1dq2dq3 +

](382)

Sminusbℓ =

(4S

N

)12[sumq1

eiq1middotRℓddaggerq1minus 1

8SN

sumq1q2q3

ei(q1+q2minusq3)middotRℓddaggerq1ddaggerq2

dq3 +

](383)

Szaj = S minus 1

N

sumq1q2

ei(q1minusq2)Rjcdaggerq1cq2 (384)

Szbℓ = minusS +

1

N

sumq1q2

eminusi(q1minusq2)Rℓddaggerq1dq2 (385)

Substituting these expansions into (373) and using (317) we obtain

H = NzJS2 + H0 + H1 (386)

47


where H0 is a part of the Hamiltonian which is bilinear in magnon variablesie

H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq

)+(cdaggerqcq + ddaggerqdq

)]+ gmicroBH

sumq

(cdaggerqcq minus ddaggerqdq

)(387)

where γq is given by (354) and the existence of symmetry center of the

lattice has been assumed The quantity H1 includes higher-order termsand will be for the simplicity neglected However even after neglectingH1 the energy spectrum of system cannot be simply obtained since theHamiltonian is not yet diagonal Thus the main problem to be solved is thediagonalization of H0 In order to make a progress we introduce here newset of the creation and annihilation operators αdagger

q αq βdaggerq βq

αq = uqcq minus vqddaggerq αdagger

q = uqcdaggerq minus vqdq (388)

βq = uqdq minus vqcdaggerq βdagger

q = uqddaggerq minus vqcq (389)

obeying the following commutation rules

[αq αdaggerq] = 1 [βk β

daggerq] = 1 [αq βq] = 0 (390)

The real quantities entering previous equations satisfy relation

u2q minus v2q = 1 (391)

and the inverse transformation are given by

cq = uqαq minus vqβdaggerq cdaggerq = uqα

daggerq minus vqβq (392)

dq = uqβq minus vqαdaggerq ddaggerq = uqβ

daggerq minus vqαq (393)

48


The parameters uq and vq in transformations (392) and (393) can be de-termined explicitly in the form

uq = cosh θq vq = sinh θq (394)

where θq is given by

tanh(2θq) = γq (395)

The explicit forms of the transformation given by (394) and (395) is usuallycalled in the literature as Bogolyubov transformation Substituting (392)and (393) into (387) one finally obtains for H the folloving simple relation

H = NJzS(S + 1) +sumq

[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)

where we have defined the eigenfrequencies of the system as

~ωplusmnq = minus2JzS

radic1minus γ2

q plusmn gmicroBH (397)

331 Ground-State Energy

In this part we investigate the ground-state energy of the two sublatticeantiferromagnetic Heisenberg system

The lowest possible energy (ie the ground-state energy) can be straight-forwardly obtained after setting H = 0 and averaging Eq(396)

Eg equiv ⟨H⟩ = NJzS(S + 1) +sumq

[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)

Of course the ground state corresponds to the vacuum state in which no ex-cited magnons are present in the system and therefore the previous equation

49


reduces to

Eg = NJzS(S + 1)minus 2JzSsumq

(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)

The parameter γq depends on the lattice structure and it is given by

γq =1

d

dsumi=1

cos qi (3100)

where d = 1 for the one-dimensional chain d = 2 for two-dimensionalsquare and d = 3 for three-dimensional simple-cubic lattice respectivelyIn order to find numerical values of Eg we have to evaluate the sum over qin Eq (399) which must be taken over all N2 points in the first Brillouinzone of the sublattice The relevant sum can be expressed as

Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)

Evaluating the above expression one gets

I1= 0637 I2

= 0842 I3

= 0903 (3102)

and using these numerical values we obtain the ground-state energy in the

50


form

Eg = 2NJS2(1 +

0363

S

)for the linear chain

Eg = 4NJS2(1 +

0158

S

)for the square lattice (3103)

Eg = 6NJS2(1 +

0097

S

)for the cubic lattice

One should note here that all numerical values of the ground-state energysatisfy the Anderson inequality

1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)

It is also useful to note that for the spin-12 linear chain Eq(3104) givesthe value Eg(NJ) = 0863 which is in agreement with the exact Bethe-Hulten [3 4] value Eg(NJ) = 05(4 ln 2 minus 1) = 0886 On tha basis ofthis excellent agreement one should assume that the spin-wave theory givesacceptable quantitative results also for other lattices and spin values

The most important qualitative finding in this part is the fact that theenergy corresponding to the saturated antiparallel ordering is not the groundstate energy of the two-sublattice antiferromagnetic system This finding isa direct consequence of the fact that the total sublattice magnetization arenot the constants of motion

332 Sublattice Magnetization

At first let us examine in this part the ground-state sublattice magnetizationwhich is defined as

Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51


Using the Bogolyubov transformation (394) we obtain

Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq

⟩minus sinh(θq) cosh(θq)(

⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)

At zero temperature there are no excited states and therefore Eq(3106)simplifies as follows

Ma =1

2NS minus

sumq

sinh2(θq) (3107)

After substituting the relevant expression for sinh(θq) one obtains expression

Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)

which can finally be rewritten as

Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)

In order to proceed further we express the sum Sd =2N

sumq

1radic1minusγ2

q

appearing

in (3109) in the following integral form

Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ

middot middot middotint π

minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52


Evaluating this integral for the simple cubic lattice (d = 3) and plane squarelattice (d = 2) one obtains values S3 = 1156 and S2 = 1393 from whichthe sublattice magnetization are given by

Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)

Thus we can conclude that the spin-wave theory predicts a long-range or-der in three- and two dimensional antiferromagnetic Heisenberg systemshowever as it is clear from (3110)-(3112) the sublattice magnetizationsdo not take their saturation values at zero temperature This behavior ap-pears due to quantum fluctuations at the ground state that are stronger inlower dimensions

In the case of linear chain (d = 1) the integral S1 diverges logarith-mically indicating that the sublattice magnetization has zero expectationvalue This result is in agreement with exact calculations and represents asignificant improvement over the standard mean-field theory which predictsthe existence of long-range order also for one dimensional Heisenberg model

Finally let us comment on the calculation of the temperature depen-dence of the sublattice magnetization for the antiferromagnetic Heisenbergmodel The deviation of the sublattice magnetization from its ground-statevalue is calculated from

Ma(0)minusMa(T ) =sumq

⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)

where the mean number of bosons is obviously given by

⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)

However the explicit calculation of sublattice magnetizations at arbitrarytemperature represents hard and non-trivial mathematical task Neverthe-less at the temperatures that are significantly lower then the Neel transitiontemperature TN one finds that the sublattice magnetization is proportionalto (TTN)

2 The proof of this behavior is left for the reader

54

Chapter 4

Jordan-Wigner Transformationfor the XY Model

In previous parts of this text we have discussed how to apply the bosoniza-tion technique within the Holstein-Primakoff approach to the ferromagneticand antiferromagnetic spin systems In the present chapter we will investi-gate a similar approach which is known as a method of fermioniozation Inparticular we will discuss application of the Wigner-Jordan transformationto the isotropic quantum XY model in order to clarify important points ofthis method

The main idea of this approach is to transform the Hamiltonian de-scribing a spin system by the use of new operators obeying the fermionanticommutation rules

Before we start our analysis it is useful to note that the spectrum of theXY model was exactly found by HBethe [3] in 1931 The Bethersquos approachis of great importance however it is quite involved and rather abstractthus it is difficult to understand even such basic properties as long-rangeorder A much more natural approach to the problem of interacting spin-12 systems was originally introduced in 1928 by Jordan and Wigner [5]

55

41 Jordan-Wigner Transformation for the XY Model

who invented simple mathematical transformations converting spin-12 sys-tems into problems of interacting (and in some cases even non-interacting)spinless fermions In fact the XY model which is a special case of theHeisenberg Hamiltonian reduces to a free theory of spinless fermions un-der the Jordan-Wigner transformations Another reason for choosing theXY model is that the low-energy properties of the full anti-ferromagneticHeisenberg chain such as the presence of gapless excitations and absenceof a long range order are very similar to those of the XY model (see [6] andreferences therein)

We will study a linear chain of N spin-12 atoms interacting antiferro-magnetically with their nearest neighbors This system is described by theHamiltonian

H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)

where J gt 0 represents the exchange interaction ans Sαi are spin-12 opera-

tors obeying the usual commutation relations (33) As usually we assumecyclic boundary conditions with Sα

N+1 = Sα1 α = x y Of course for

J lt 0 the ferromagnetic case of (41) is obtained Thus if we solve theantiferromagnetic problem exactly then we can immediately obtain alsothe solution of the ferromagnetic XY chain Similarly as in the case ofbozonization we at first introduce spin-raising and spin-lowering operators(313) and rewrite the Hamiltonian as a bilinear form in these operatorsie

H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)

It clear that without loss of generality we can set J = 1 in all subsequentcalculations We know that the Splusmn operators belonging to the same siteobey anticommutation relations while that ones on different sites satisfy

56


usual commutation relations and this mathematical property disables a di-rect diagonalization of (42) The key to the solution of this problem is theJordan-Wigner transformation which enables to obtain the Hamiltonian interms of pure fermion operators

The JordanndashWigner transformation explicitly reads

cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)

and the inverse transformation is given by

S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)

Here it is worth noticing that the signs in the exponents and the order ofthe multipliers in (43) and (44) are not important

The operators cdaggerj ck obey the canonical fermion algebra ie

cj cdaggerk = δjk cdaggerj cdaggerk = 0 cj ck = 0 (45)

To illustrate the mathematics behind we will explicitly prove the first rela-tion while the other anti-commutators in (45) can be computed in a similarfashion

At first we recall the validity of the following relations [S+j Sj S

+k Sk] = 0

[S+j Sj Sk] = minusδjkSk [S

+j Sj S

+k ] = δjkS

+k (S

+j Sj)

2 = S+j Sj since the spin-

raising and spin-lowering operators on different sites commute and on thesame site behave like fermions Therefore

exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57


and the exponential function on the rhs of previous equation can be easilyevaluated as follows

exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj

= 1 + (eplusmniπ minus 1)S+j S

minusj = 1minus 2S+

j Sminusj (47)

Similarly using the following anticommutators

Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)

one obtains relations[exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)

Although all the above calculations have a preparatory character they are ofcrucial importance to demonstrate the fermionic nature of cj and cdaggerk Reallyusing (46)- (410) we can now straightforwardly calculate the following

58


anti-commutators

cj cdaggerk = Sminusj exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)

In the same way we derive relation

cj cdaggerk = ck cdaggerkdagger = 0 k lt j (412)

ck cdaggerk = Sminusk S

+k + S+

k Sminusk = 1 (413)

Thus we have successfully proved the fermionic commutation relations andnow we can transform the relevant terms in the Hamiltonian

For 1 le j le N minus 1 one obtains

Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1

= cj exp(iπcdaggerjcj)c

daggerj+1 = cj(1minus 2cdaggerjcj)c

daggerj+1

= minus(1minus 2cdaggerjcj)cjcdaggerj+1) = minuscjc

daggerj+1 = cdaggerj+1cj (414)

and similarly we have

Sminusj+1S

+j = (Sminus

j S+j+1)

dagger = cdaggerjcj+1 (415)

59


Here one should notice that in deriving previous equations we have usedthe following relations

[cdaggerjcj cdaggerkck] = 0 (cdaggerjcj)

2 = cdaggerjcj

[cdaggerjcj ck] = minusδjkck [cdaggerjcj cdaggerk] = minusδjkc

daggerk

1minus 2cdaggerjcj cj = 0 1minus 2cdaggerjcj cdaggerj = 0 (416)

[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)

After expressing the special cyclic boundary term (SminusN S

+1 + Sminus

1 S+N) in

terms of crsquos and substituting Eqs(414) and (415) into Eq(42) we canrewrite the Hamiltonian of the system as

H =1

2

sumj

(cdaggerj+1cj + cdaggerjcj+1) + Hb (419)

where

Hb = minus1

2

(cdagger1cN + cdaggerNc1

)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)

represents the boundary term which gives an O(Nminus1) contributions to themacroscopic physical quantities and for this reason will be neglected infurther calculations

60


Applying this simplification we are left with the problem of diagonal-ization of the following bilinear form

H =1

2

sumj

(cdaggerj+1cj + cdaggerjcj+1) (421)

which describes free spinless fermions on a cyclic chain with nearest neighborhopping

In order to diagonalize the Hamiltonian (421) we rewrite it in the form

H =sumjk

cdaggerjAjkck (422)

where the elements of the matrix A are given by

Ajk =1

2(δjk+1 + δkj+1) (423)

Taking into account the translational invariance of the lattice we find theeigenvectors and eigenvalues of A to be

ϕqj =1radicNeiqj Λq = cos(q) (424)

with q = 2πnN

and N2 le n le N2 minus 1 The eigenfunctions ϕqk form anorthonormal and complete set thus we can introduce new operators obeyingthe canonical anticommutation rule

ηq =sumj

ϕlowastqjcj ηdaggerq =

sumj

ϕqjcdaggerj (425)

Using the following inverse transformations

cj =sumq

ϕqjηq cdaggerj =sumq

ϕqjηdaggerq (426)

61


the Hamiltonian (422) can be rewritten in the diagonal form

H =sumq

Λqηdaggerqηq (427)

Since some of the eigenvalues of A will be negative we perform an additionaltransformation

ξq = ηq Λq ge 0 (428)

ξq = ηdaggerq Λq lt 0 (429)

and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq

|Λq|ξdaggerqξq minussum

qΛqlt0

|Λq|

=sumqΛq

|Λq|(ξdaggerqξq minus

1

2

) (430)

Here the ξ operators again obey the canonical anticommutation rulesAs usually the ground state |0⟩ satisfies equation

ξq|0⟩ = 0 forallq (431)

and the operator ξdaggerq generates an elementary fermion excitation with energy|Λq| above the ground state

The dispersion relation for the model under investigation is shown in Fig41 As one can see there always exist gapless excitations near q = plusmnπ2The ground state energy reduced per one spin is given by

U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq

4π| cos(q)| = minus 1

π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20

Figure 41 Dispersion relation of the XY model

Of course the method of fermionization can been extended to manyother systems including the systems with higher spin and higher spatialdimensions However the analysis of this systems requires complex andtedious calculations and sometimes also the applications additional theo-retical tools methods (for example the Green function technique) that arebeyond the scope of this introductory text

63

Conclusion

Conclusion

In this work we have discussed some theoretical tools that are frequentlyused to investigate the localized quantum spin models

The first approach is the standard mean-field theory which representsthe simplest possible technique usually applied to understand basic quan-titative features of various theoretical models of quantum magnetism Al-though this method is included in many textbooks the formulation is usu-ally very simplified and frequently the authors do not derive the Gibbsfree energy within this method To avoid this problem we have used andelegant formulation based on the Bogolyubov inequality which enables toextend our approach to formulate more accurate theories (for example theOguchi approximation or Constant Coupling Method) One should also em-phasize that in deriving the partition function we have applied the Cauchyintegral formula which enables to calculate exponential function with a ma-trix argument This is an unknown trick which can be used in calculationsin nay branch of theoretical physics

In the Chapter 3 we have discussed the standard spin-wave theory sep-arately for ferromagnetic and antiferromagnetic materials We have shownin detail how to introduce the boson operators that enable to calculate sev-eral physical quantities such as magnetization internal energy and magneticcontribution to the specific heat

The last part is dedicated to the method of fermionization Here we havediscussed the application o Jordan-Wigner transformation to the quantumXY model with spin 12 This approach represent another standard theo-retical approach which is used to investigate quantum magnetism

It is clear from the content of this supporting text that we have coveredjust one special part of quantum theory of magnetism which is howeverdiscussed elementary enough In order to understand further interesting fea-tures of the quantum magnetism it is necessary to study further textbooksfor example [8] [9] [10] [11] )

64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)

[2] T Holstein and H Primakoff Phys Rev 58 1908 (1940)

[3] H Bethe Z Physik 71 205 (1931)

[4] L Hulten Ark Met Asron Fysik 26 A 11 (1938)

[5] P Jordan and E Wigner Z Phys 47 631 (1928)

[6] A Tsvelik Quantum Field Theory in Condensed Matter PhysicsCambridge Cambridge University Press 1995

[7] LJ de Jongh AR Miedema Adv Phys 23 2 (1974)

[8] C Kittel Quantum Theory of Solids New York John Wiley andSons 1987

[9] K Yosida Theory of Magnetism Berlin Springer 1998

[10] P Mohn Magnetism in the Solid State Berlin Springer 2003

[11] N Majlis The Quantum Theory of Magnetism Singapore World Sci-entific 2007

65


copy 2013 doc RNDr Michal Jaščur CSc

Reviewers Prof RNDr Vladimiacuter Lisyacute DrSc

Prof RNDr Peter Kollaacuter CSc

Number of pages 65

Number of authorsʹ sheets 3

Electronic academic textbook for Faculty of Science on P J Šafaacuterik University in Košice

The author of this textbook is accountable for the professional level and language correctness

The language and arrangement of the manuscript have not been revised

Publisher Pavol Jozef Šafaacuterik University in Košice

Location httpwwwupjsskpracoviskauniverzitna-kniznicae-publikaciapf

Available 10 10 2013

Edition second

ISBN 978-80-8152-049-5

Contents

Preface 2




Model 27




3

Contents


4

Preface

Preface






5

Preface



6

Chapter 1

Many-Body Systems





7


or


(12)



(13)

and


(14)




8




H =Nsumj

Hj (16)



ϕnj(rj) (17)



Hjϕnj(rj) = εnj

ϕnj(rj) (18)



sumj εnj




9





ϕnN(r1) ϕnN

(r2) ϕnN(rN)

∣∣∣∣∣∣∣∣∣(110)



10




a b

2

1

r

rrab

r r

ra

a

1

2 2b

21

1b



11



H = H1 + H2 + W + HLS (111)

where

H1 = minus ~2

2m∆1 minus

e20ra1

(112)

H2 = minus ~2

2m∆2 minus

e20rb2

(113)

W =e20rab

+e20r12

minus e20rb1

minus e20ra2

(114)


(H1 + H2)Ψ = U0Ψ (115)

12



z1 s







φ(r1 r2) = c1φ1(r1 r2) + c2φ2(r1 r2)

= c1ϕa(r1)ϕb(r2) + c2ϕa(r2)ϕb(r1) (116)

13



)[c1φ1(r1 r2) + c2φ2(r1 r2

]= E

[c1φ1(r1 r2) + c2φ2(r1 r2)

](117)


space one obtains

2c1E0 + 2c2E0


+ 2c1



= c1E + c2E



space one finds

c1E0



+ 2c2E0 + c2


= c1E





S0 =


14



K =



(121)


A =



(122)



0) = 0


0) + c2(2E0 +K minus E) = 0 (123)


20 + Aminus ES2

0

2E0S20 + Aminus ES2

0 2E0 +K minus E

∣∣∣∣ = 0 (124)


Ea = 2E0 +K minus A

1minus S20

c1 = minusc2 (125)

and

Es = 2E0 +K + A

1 + S20

c1 = c2 (126)

15



1 =


= 2|c1|2(1∓ S2) (127)

or

c1 =1radic

2(1∓ S20) (128)


φ(r1 r2) =1radic

2(1minus S20)


](129)


φ(r1 r2) =1radic

2(1 + S20)


] (130)


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξuarr(1)ξuarr(2) S = 1 Sz = 1 (131)

Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξdarr(1)ξdarr(2) S = 1 Sz = 0 (132)

16


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)


S = 1 Sz = minus1 (133)

and

Ψ =1radic

2(1 + S20)


] 1radic2


∣∣∣∣S = 0 Sz = 0 (134)



0 minus A)

1minus S40

asymp minus2A (135)



17





18

Chapter 2



ity





βlnZ0 (22)

19











20



H = minussumij

(JxSxi S

xj + JyS

yi S

yj + JzS

zi S

zj )minus gmicroB

sumi

(HxSxi +HyS

yi +HzS

zi )


Sxk =

1

2

(0 11 0

)k

Syk =

1

2

(0 minusii 0

)k

Szk =

1

2

(1 00 minus1

)k

(25)


H0 = minusNsumk

(λxS

xk + λyS

yk + λzS

zk

) (26)


H0 =Nsumk=1

H0k (27)

21



H0k = minus1

2

(λz λx minus iλy

λx + iλy minusλz

)k

(28)


Z0 = Tr exp(minusβ

Nsumk=1

H0k

)=

Nprodk=1

Trk exp(minusβH0k

) (29)


Z0 =Nprodk=1

Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

=

[Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

]N

(210)


eminusβH0k =1

2πi

∮C

ezdz


22



eminusβH0k =

(h11 h12

h21 h22

)(212)

with

h11 = cosh

(β

2Λ

)+

λz

Λsinh

(β

2Λ

)(213)

h12 =λx minus iλy

Λsinh

(β

2Λ

)(214)

h21 =λx + iλy

Λsinh

(β

2Λ

)(215)

h22 = cosh

(β

2Λ

)minus λz

Λsinh

(β

2Λ

) (216)


Λ =radicλ2x + λ2

y + λ2z (217)


and G0

Z0 =

[2 cosh

(β

2Λ

)]N(218)

G0 = minusN

βln

[2 cosh

(β2Λ

)] (219)

23




⟨H⟩0 = minusNq

2

(Jx⟨Sx



i Szj ⟩0

)minus N

(hx⟨Sx


j ⟩0) (220)




j ⟩0) (221)



⟨Sαj ⟩0 =

1

Z0

TrSαj e

minusβH0

=TrjS

αj exp

(minusβH0j


(minusβH0k

)Trj exp

(minusβH0j


(minusβH0k

) (222)


⟨Sαj ⟩0 =

TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) (223)


⟨Sαi S

αj ⟩0 =

TriSαi exp

(minusβH0i

)Tri exp

(minusβH0i

) TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) =(⟨Sα

j ⟩0)2

(224)

24



⟨Sαj ⟩0 =

λα

2Λtanh

(β

2Λ

) (225)

and

⟨Sαi S

αj ⟩0 =

(λα

2Λ

)2

tanh2

(β

2Λ

) (226)




βln(2 cosh

β

2Λ)minus Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

)+ N


] (227)


partϕ

partλx

= 0partϕ

partλy

= 0partϕ

partλz

= 0 (228)


partmx

partλx

partmy

partλx

partmz

partλxpartmx

partλy

partmy

partλy

partmz

partλy

partmx

partλz

partmy

partλz

partmz

partλz




= 0 (229)

25





G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)


hef =radic




Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)


Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)


26




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65

Contents

Preface 2




Model 27




3

Contents


4

Preface

Preface






5

Preface



6

Chapter 1

Many-Body Systems





7


or


(12)



(13)

and


(14)




8




H =Nsumj

Hj (16)



ϕnj(rj) (17)



Hjϕnj(rj) = εnj

ϕnj(rj) (18)



sumj εnj




9





ϕnN(r1) ϕnN

(r2) ϕnN(rN)

∣∣∣∣∣∣∣∣∣(110)



10




a b

2

1

r

rrab

r r

ra

a

1

2 2b

21

1b



11



H = H1 + H2 + W + HLS (111)

where

H1 = minus ~2

2m∆1 minus

e20ra1

(112)

H2 = minus ~2

2m∆2 minus

e20rb2

(113)

W =e20rab

+e20r12

minus e20rb1

minus e20ra2

(114)


(H1 + H2)Ψ = U0Ψ (115)

12



z1 s







φ(r1 r2) = c1φ1(r1 r2) + c2φ2(r1 r2)

= c1ϕa(r1)ϕb(r2) + c2ϕa(r2)ϕb(r1) (116)

13



)[c1φ1(r1 r2) + c2φ2(r1 r2

]= E

[c1φ1(r1 r2) + c2φ2(r1 r2)

](117)


space one obtains

2c1E0 + 2c2E0


+ 2c1



= c1E + c2E



space one finds

c1E0



+ 2c2E0 + c2


= c1E





S0 =


14



K =



(121)


A =



(122)



0) = 0


0) + c2(2E0 +K minus E) = 0 (123)


20 + Aminus ES2

0

2E0S20 + Aminus ES2

0 2E0 +K minus E

∣∣∣∣ = 0 (124)


Ea = 2E0 +K minus A

1minus S20

c1 = minusc2 (125)

and

Es = 2E0 +K + A

1 + S20

c1 = c2 (126)

15



1 =


= 2|c1|2(1∓ S2) (127)

or

c1 =1radic

2(1∓ S20) (128)


φ(r1 r2) =1radic

2(1minus S20)


](129)


φ(r1 r2) =1radic

2(1 + S20)


] (130)


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξuarr(1)ξuarr(2) S = 1 Sz = 1 (131)

Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξdarr(1)ξdarr(2) S = 1 Sz = 0 (132)

16


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)


S = 1 Sz = minus1 (133)

and

Ψ =1radic

2(1 + S20)


] 1radic2


∣∣∣∣S = 0 Sz = 0 (134)



0 minus A)

1minus S40

asymp minus2A (135)



17





18

Chapter 2



ity





βlnZ0 (22)

19











20



H = minussumij

(JxSxi S

xj + JyS

yi S

yj + JzS

zi S

zj )minus gmicroB

sumi

(HxSxi +HyS

yi +HzS

zi )


Sxk =

1

2

(0 11 0

)k

Syk =

1

2

(0 minusii 0

)k

Szk =

1

2

(1 00 minus1

)k

(25)


H0 = minusNsumk

(λxS

xk + λyS

yk + λzS

zk

) (26)


H0 =Nsumk=1

H0k (27)

21



H0k = minus1

2

(λz λx minus iλy

λx + iλy minusλz

)k

(28)


Z0 = Tr exp(minusβ

Nsumk=1

H0k

)=

Nprodk=1

Trk exp(minusβH0k

) (29)


Z0 =Nprodk=1

Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

=

[Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

]N

(210)


eminusβH0k =1

2πi

∮C

ezdz


22



eminusβH0k =

(h11 h12

h21 h22

)(212)

with

h11 = cosh

(β

2Λ

)+

λz

Λsinh

(β

2Λ

)(213)

h12 =λx minus iλy

Λsinh

(β

2Λ

)(214)

h21 =λx + iλy

Λsinh

(β

2Λ

)(215)

h22 = cosh

(β

2Λ

)minus λz

Λsinh

(β

2Λ

) (216)


Λ =radicλ2x + λ2

y + λ2z (217)


and G0

Z0 =

[2 cosh

(β

2Λ

)]N(218)

G0 = minusN

βln

[2 cosh

(β2Λ

)] (219)

23




⟨H⟩0 = minusNq

2

(Jx⟨Sx



i Szj ⟩0

)minus N

(hx⟨Sx


j ⟩0) (220)




j ⟩0) (221)



⟨Sαj ⟩0 =

1

Z0

TrSαj e

minusβH0

=TrjS

αj exp

(minusβH0j


(minusβH0k

)Trj exp

(minusβH0j


(minusβH0k

) (222)


⟨Sαj ⟩0 =

TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) (223)


⟨Sαi S

αj ⟩0 =

TriSαi exp

(minusβH0i

)Tri exp

(minusβH0i

) TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) =(⟨Sα

j ⟩0)2

(224)

24



⟨Sαj ⟩0 =

λα

2Λtanh

(β

2Λ

) (225)

and

⟨Sαi S

αj ⟩0 =

(λα

2Λ

)2

tanh2

(β

2Λ

) (226)




βln(2 cosh

β

2Λ)minus Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

)+ N


] (227)


partϕ

partλx

= 0partϕ

partλy

= 0partϕ

partλz

= 0 (228)


partmx

partλx

partmy

partλx

partmz

partλxpartmx

partλy

partmy

partλy

partmz

partλy

partmx

partλz

partmy

partλz

partmz

partλz




= 0 (229)

25





G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)


hef =radic




Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)


Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)


26




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65

Contents


4

Preface

Preface






5

Preface



6

Chapter 1

Many-Body Systems





7


or


(12)



(13)

and


(14)




8




H =Nsumj

Hj (16)



ϕnj(rj) (17)



Hjϕnj(rj) = εnj

ϕnj(rj) (18)



sumj εnj




9





ϕnN(r1) ϕnN

(r2) ϕnN(rN)

∣∣∣∣∣∣∣∣∣(110)



10




a b

2

1

r

rrab

r r

ra

a

1

2 2b

21

1b



11



H = H1 + H2 + W + HLS (111)

where

H1 = minus ~2

2m∆1 minus

e20ra1

(112)

H2 = minus ~2

2m∆2 minus

e20rb2

(113)

W =e20rab

+e20r12

minus e20rb1

minus e20ra2

(114)


(H1 + H2)Ψ = U0Ψ (115)

12



z1 s







φ(r1 r2) = c1φ1(r1 r2) + c2φ2(r1 r2)

= c1ϕa(r1)ϕb(r2) + c2ϕa(r2)ϕb(r1) (116)

13



)[c1φ1(r1 r2) + c2φ2(r1 r2

]= E

[c1φ1(r1 r2) + c2φ2(r1 r2)

](117)


space one obtains

2c1E0 + 2c2E0


+ 2c1



= c1E + c2E



space one finds

c1E0



+ 2c2E0 + c2


= c1E





S0 =


14



K =



(121)


A =



(122)



0) = 0


0) + c2(2E0 +K minus E) = 0 (123)


20 + Aminus ES2

0

2E0S20 + Aminus ES2

0 2E0 +K minus E

∣∣∣∣ = 0 (124)


Ea = 2E0 +K minus A

1minus S20

c1 = minusc2 (125)

and

Es = 2E0 +K + A

1 + S20

c1 = c2 (126)

15



1 =


= 2|c1|2(1∓ S2) (127)

or

c1 =1radic

2(1∓ S20) (128)


φ(r1 r2) =1radic

2(1minus S20)


](129)


φ(r1 r2) =1radic

2(1 + S20)


] (130)


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξuarr(1)ξuarr(2) S = 1 Sz = 1 (131)

Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξdarr(1)ξdarr(2) S = 1 Sz = 0 (132)

16


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)


S = 1 Sz = minus1 (133)

and

Ψ =1radic

2(1 + S20)


] 1radic2


∣∣∣∣S = 0 Sz = 0 (134)



0 minus A)

1minus S40

asymp minus2A (135)



17





18

Chapter 2



ity





βlnZ0 (22)

19











20



H = minussumij

(JxSxi S

xj + JyS

yi S

yj + JzS

zi S

zj )minus gmicroB

sumi

(HxSxi +HyS

yi +HzS

zi )


Sxk =

1

2

(0 11 0

)k

Syk =

1

2

(0 minusii 0

)k

Szk =

1

2

(1 00 minus1

)k

(25)


H0 = minusNsumk

(λxS

xk + λyS

yk + λzS

zk

) (26)


H0 =Nsumk=1

H0k (27)

21



H0k = minus1

2

(λz λx minus iλy

λx + iλy minusλz

)k

(28)


Z0 = Tr exp(minusβ

Nsumk=1

H0k

)=

Nprodk=1

Trk exp(minusβH0k

) (29)


Z0 =Nprodk=1

Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

=

[Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

]N

(210)


eminusβH0k =1

2πi

∮C

ezdz


22



eminusβH0k =

(h11 h12

h21 h22

)(212)

with

h11 = cosh

(β

2Λ

)+

λz

Λsinh

(β

2Λ

)(213)

h12 =λx minus iλy

Λsinh

(β

2Λ

)(214)

h21 =λx + iλy

Λsinh

(β

2Λ

)(215)

h22 = cosh

(β

2Λ

)minus λz

Λsinh

(β

2Λ

) (216)


Λ =radicλ2x + λ2

y + λ2z (217)


and G0

Z0 =

[2 cosh

(β

2Λ

)]N(218)

G0 = minusN

βln

[2 cosh

(β2Λ

)] (219)

23




⟨H⟩0 = minusNq

2

(Jx⟨Sx



i Szj ⟩0

)minus N

(hx⟨Sx


j ⟩0) (220)




j ⟩0) (221)



⟨Sαj ⟩0 =

1

Z0

TrSαj e

minusβH0

=TrjS

αj exp

(minusβH0j


(minusβH0k

)Trj exp

(minusβH0j


(minusβH0k

) (222)


⟨Sαj ⟩0 =

TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) (223)


⟨Sαi S

αj ⟩0 =

TriSαi exp

(minusβH0i

)Tri exp

(minusβH0i

) TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) =(⟨Sα

j ⟩0)2

(224)

24



⟨Sαj ⟩0 =

λα

2Λtanh

(β

2Λ

) (225)

and

⟨Sαi S

αj ⟩0 =

(λα

2Λ

)2

tanh2

(β

2Λ

) (226)




βln(2 cosh

β

2Λ)minus Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

)+ N


] (227)


partϕ

partλx

= 0partϕ

partλy

= 0partϕ

partλz

= 0 (228)


partmx

partλx

partmy

partλx

partmz

partλxpartmx

partλy

partmy

partλy

partmz

partλy

partmx

partλz

partmy

partλz

partmz

partλz




= 0 (229)

25





G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)


hef =radic




Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)


Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)


26




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65

Preface

Preface






5

Preface



6

Chapter 1

Many-Body Systems





7


or


(12)



(13)

and


(14)




8




H =Nsumj

Hj (16)



ϕnj(rj) (17)



Hjϕnj(rj) = εnj

ϕnj(rj) (18)



sumj εnj




9





ϕnN(r1) ϕnN

(r2) ϕnN(rN)

∣∣∣∣∣∣∣∣∣(110)



10




a b

2

1

r

rrab

r r

ra

a

1

2 2b

21

1b



11



H = H1 + H2 + W + HLS (111)

where

H1 = minus ~2

2m∆1 minus

e20ra1

(112)

H2 = minus ~2

2m∆2 minus

e20rb2

(113)

W =e20rab

+e20r12

minus e20rb1

minus e20ra2

(114)


(H1 + H2)Ψ = U0Ψ (115)

12



z1 s







φ(r1 r2) = c1φ1(r1 r2) + c2φ2(r1 r2)

= c1ϕa(r1)ϕb(r2) + c2ϕa(r2)ϕb(r1) (116)

13



)[c1φ1(r1 r2) + c2φ2(r1 r2

]= E

[c1φ1(r1 r2) + c2φ2(r1 r2)

](117)


space one obtains

2c1E0 + 2c2E0


+ 2c1



= c1E + c2E



space one finds

c1E0



+ 2c2E0 + c2


= c1E





S0 =


14



K =



(121)


A =



(122)



0) = 0


0) + c2(2E0 +K minus E) = 0 (123)


20 + Aminus ES2

0

2E0S20 + Aminus ES2

0 2E0 +K minus E

∣∣∣∣ = 0 (124)


Ea = 2E0 +K minus A

1minus S20

c1 = minusc2 (125)

and

Es = 2E0 +K + A

1 + S20

c1 = c2 (126)

15



1 =


= 2|c1|2(1∓ S2) (127)

or

c1 =1radic

2(1∓ S20) (128)


φ(r1 r2) =1radic

2(1minus S20)


](129)


φ(r1 r2) =1radic

2(1 + S20)


] (130)


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξuarr(1)ξuarr(2) S = 1 Sz = 1 (131)

Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξdarr(1)ξdarr(2) S = 1 Sz = 0 (132)

16


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)


S = 1 Sz = minus1 (133)

and

Ψ =1radic

2(1 + S20)


] 1radic2


∣∣∣∣S = 0 Sz = 0 (134)



0 minus A)

1minus S40

asymp minus2A (135)



17





18

Chapter 2



ity





βlnZ0 (22)

19











20



H = minussumij

(JxSxi S

xj + JyS

yi S

yj + JzS

zi S

zj )minus gmicroB

sumi

(HxSxi +HyS

yi +HzS

zi )


Sxk =

1

2

(0 11 0

)k

Syk =

1

2

(0 minusii 0

)k

Szk =

1

2

(1 00 minus1

)k

(25)


H0 = minusNsumk

(λxS

xk + λyS

yk + λzS

zk

) (26)


H0 =Nsumk=1

H0k (27)

21



H0k = minus1

2

(λz λx minus iλy

λx + iλy minusλz

)k

(28)


Z0 = Tr exp(minusβ

Nsumk=1

H0k

)=

Nprodk=1

Trk exp(minusβH0k

) (29)


Z0 =Nprodk=1

Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

=

[Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

]N

(210)


eminusβH0k =1

2πi

∮C

ezdz


22



eminusβH0k =

(h11 h12

h21 h22

)(212)

with

h11 = cosh

(β

2Λ

)+

λz

Λsinh

(β

2Λ

)(213)

h12 =λx minus iλy

Λsinh

(β

2Λ

)(214)

h21 =λx + iλy

Λsinh

(β

2Λ

)(215)

h22 = cosh

(β

2Λ

)minus λz

Λsinh

(β

2Λ

) (216)


Λ =radicλ2x + λ2

y + λ2z (217)


and G0

Z0 =

[2 cosh

(β

2Λ

)]N(218)

G0 = minusN

βln

[2 cosh

(β2Λ

)] (219)

23




⟨H⟩0 = minusNq

2

(Jx⟨Sx



i Szj ⟩0

)minus N

(hx⟨Sx


j ⟩0) (220)




j ⟩0) (221)



⟨Sαj ⟩0 =

1

Z0

TrSαj e

minusβH0

=TrjS

αj exp

(minusβH0j


(minusβH0k

)Trj exp

(minusβH0j


(minusβH0k

) (222)


⟨Sαj ⟩0 =

TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) (223)


⟨Sαi S

αj ⟩0 =

TriSαi exp

(minusβH0i

)Tri exp

(minusβH0i

) TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) =(⟨Sα

j ⟩0)2

(224)

24



⟨Sαj ⟩0 =

λα

2Λtanh

(β

2Λ

) (225)

and

⟨Sαi S

αj ⟩0 =

(λα

2Λ

)2

tanh2

(β

2Λ

) (226)




βln(2 cosh

β

2Λ)minus Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

)+ N


] (227)


partϕ

partλx

= 0partϕ

partλy

= 0partϕ

partλz

= 0 (228)


partmx

partλx

partmy

partλx

partmz

partλxpartmx

partλy

partmy

partλy

partmz

partλy

partmx

partλz

partmy

partλz

partmz

partλz




= 0 (229)

25





G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)


hef =radic




Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)


Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)


26




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65

Preface



6

Chapter 1

Many-Body Systems





7


or


(12)



(13)

and


(14)




8




H =Nsumj

Hj (16)



ϕnj(rj) (17)



Hjϕnj(rj) = εnj

ϕnj(rj) (18)



sumj εnj




9





ϕnN(r1) ϕnN

(r2) ϕnN(rN)

∣∣∣∣∣∣∣∣∣(110)



10




a b

2

1

r

rrab

r r

ra

a

1

2 2b

21

1b



11



H = H1 + H2 + W + HLS (111)

where

H1 = minus ~2

2m∆1 minus

e20ra1

(112)

H2 = minus ~2

2m∆2 minus

e20rb2

(113)

W =e20rab

+e20r12

minus e20rb1

minus e20ra2

(114)


(H1 + H2)Ψ = U0Ψ (115)

12



z1 s







φ(r1 r2) = c1φ1(r1 r2) + c2φ2(r1 r2)

= c1ϕa(r1)ϕb(r2) + c2ϕa(r2)ϕb(r1) (116)

13



)[c1φ1(r1 r2) + c2φ2(r1 r2

]= E

[c1φ1(r1 r2) + c2φ2(r1 r2)

](117)


space one obtains

2c1E0 + 2c2E0


+ 2c1



= c1E + c2E



space one finds

c1E0



+ 2c2E0 + c2


= c1E





S0 =


14



K =



(121)


A =



(122)



0) = 0


0) + c2(2E0 +K minus E) = 0 (123)


20 + Aminus ES2

0

2E0S20 + Aminus ES2

0 2E0 +K minus E

∣∣∣∣ = 0 (124)


Ea = 2E0 +K minus A

1minus S20

c1 = minusc2 (125)

and

Es = 2E0 +K + A

1 + S20

c1 = c2 (126)

15



1 =


= 2|c1|2(1∓ S2) (127)

or

c1 =1radic

2(1∓ S20) (128)


φ(r1 r2) =1radic

2(1minus S20)


](129)


φ(r1 r2) =1radic

2(1 + S20)


] (130)


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξuarr(1)ξuarr(2) S = 1 Sz = 1 (131)

Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξdarr(1)ξdarr(2) S = 1 Sz = 0 (132)

16


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)


S = 1 Sz = minus1 (133)

and

Ψ =1radic

2(1 + S20)


] 1radic2


∣∣∣∣S = 0 Sz = 0 (134)



0 minus A)

1minus S40

asymp minus2A (135)



17





18

Chapter 2



ity





βlnZ0 (22)

19











20



H = minussumij

(JxSxi S

xj + JyS

yi S

yj + JzS

zi S

zj )minus gmicroB

sumi

(HxSxi +HyS

yi +HzS

zi )


Sxk =

1

2

(0 11 0

)k

Syk =

1

2

(0 minusii 0

)k

Szk =

1

2

(1 00 minus1

)k

(25)


H0 = minusNsumk

(λxS

xk + λyS

yk + λzS

zk

) (26)


H0 =Nsumk=1

H0k (27)

21



H0k = minus1

2

(λz λx minus iλy

λx + iλy minusλz

)k

(28)


Z0 = Tr exp(minusβ

Nsumk=1

H0k

)=

Nprodk=1

Trk exp(minusβH0k

) (29)


Z0 =Nprodk=1

Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

=

[Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

]N

(210)


eminusβH0k =1

2πi

∮C

ezdz


22



eminusβH0k =

(h11 h12

h21 h22

)(212)

with

h11 = cosh

(β

2Λ

)+

λz

Λsinh

(β

2Λ

)(213)

h12 =λx minus iλy

Λsinh

(β

2Λ

)(214)

h21 =λx + iλy

Λsinh

(β

2Λ

)(215)

h22 = cosh

(β

2Λ

)minus λz

Λsinh

(β

2Λ

) (216)


Λ =radicλ2x + λ2

y + λ2z (217)


and G0

Z0 =

[2 cosh

(β

2Λ

)]N(218)

G0 = minusN

βln

[2 cosh

(β2Λ

)] (219)

23




⟨H⟩0 = minusNq

2

(Jx⟨Sx



i Szj ⟩0

)minus N

(hx⟨Sx


j ⟩0) (220)




j ⟩0) (221)



⟨Sαj ⟩0 =

1

Z0

TrSαj e

minusβH0

=TrjS

αj exp

(minusβH0j


(minusβH0k

)Trj exp

(minusβH0j


(minusβH0k

) (222)


⟨Sαj ⟩0 =

TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) (223)


⟨Sαi S

αj ⟩0 =

TriSαi exp

(minusβH0i

)Tri exp

(minusβH0i

) TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) =(⟨Sα

j ⟩0)2

(224)

24



⟨Sαj ⟩0 =

λα

2Λtanh

(β

2Λ

) (225)

and

⟨Sαi S

αj ⟩0 =

(λα

2Λ

)2

tanh2

(β

2Λ

) (226)




βln(2 cosh

β

2Λ)minus Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

)+ N


] (227)


partϕ

partλx

= 0partϕ

partλy

= 0partϕ

partλz

= 0 (228)


partmx

partλx

partmy

partλx

partmz

partλxpartmx

partλy

partmy

partλy

partmz

partλy

partmx

partλz

partmy

partλz

partmz

partλz




= 0 (229)

25





G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)


hef =radic




Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)


Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)


26




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65

Chapter 1

Many-Body Systems





7


or


(12)



(13)

and


(14)




8




H =Nsumj

Hj (16)



ϕnj(rj) (17)



Hjϕnj(rj) = εnj

ϕnj(rj) (18)



sumj εnj




9





ϕnN(r1) ϕnN

(r2) ϕnN(rN)

∣∣∣∣∣∣∣∣∣(110)



10




a b

2

1

r

rrab

r r

ra

a

1

2 2b

21

1b



11



H = H1 + H2 + W + HLS (111)

where

H1 = minus ~2

2m∆1 minus

e20ra1

(112)

H2 = minus ~2

2m∆2 minus

e20rb2

(113)

W =e20rab

+e20r12

minus e20rb1

minus e20ra2

(114)


(H1 + H2)Ψ = U0Ψ (115)

12



z1 s







φ(r1 r2) = c1φ1(r1 r2) + c2φ2(r1 r2)

= c1ϕa(r1)ϕb(r2) + c2ϕa(r2)ϕb(r1) (116)

13



)[c1φ1(r1 r2) + c2φ2(r1 r2

]= E

[c1φ1(r1 r2) + c2φ2(r1 r2)

](117)


space one obtains

2c1E0 + 2c2E0


+ 2c1



= c1E + c2E



space one finds

c1E0



+ 2c2E0 + c2


= c1E





S0 =


14



K =



(121)


A =



(122)



0) = 0


0) + c2(2E0 +K minus E) = 0 (123)


20 + Aminus ES2

0

2E0S20 + Aminus ES2

0 2E0 +K minus E

∣∣∣∣ = 0 (124)


Ea = 2E0 +K minus A

1minus S20

c1 = minusc2 (125)

and

Es = 2E0 +K + A

1 + S20

c1 = c2 (126)

15



1 =


= 2|c1|2(1∓ S2) (127)

or

c1 =1radic

2(1∓ S20) (128)


φ(r1 r2) =1radic

2(1minus S20)


](129)


φ(r1 r2) =1radic

2(1 + S20)


] (130)


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξuarr(1)ξuarr(2) S = 1 Sz = 1 (131)

Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξdarr(1)ξdarr(2) S = 1 Sz = 0 (132)

16


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)


S = 1 Sz = minus1 (133)

and

Ψ =1radic

2(1 + S20)


] 1radic2


∣∣∣∣S = 0 Sz = 0 (134)



0 minus A)

1minus S40

asymp minus2A (135)



17





18

Chapter 2



ity





βlnZ0 (22)

19











20



H = minussumij

(JxSxi S

xj + JyS

yi S

yj + JzS

zi S

zj )minus gmicroB

sumi

(HxSxi +HyS

yi +HzS

zi )


Sxk =

1

2

(0 11 0

)k

Syk =

1

2

(0 minusii 0

)k

Szk =

1

2

(1 00 minus1

)k

(25)


H0 = minusNsumk

(λxS

xk + λyS

yk + λzS

zk

) (26)


H0 =Nsumk=1

H0k (27)

21



H0k = minus1

2

(λz λx minus iλy

λx + iλy minusλz

)k

(28)


Z0 = Tr exp(minusβ

Nsumk=1

H0k

)=

Nprodk=1

Trk exp(minusβH0k

) (29)


Z0 =Nprodk=1

Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

=

[Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

]N

(210)


eminusβH0k =1

2πi

∮C

ezdz


22



eminusβH0k =

(h11 h12

h21 h22

)(212)

with

h11 = cosh

(β

2Λ

)+

λz

Λsinh

(β

2Λ

)(213)

h12 =λx minus iλy

Λsinh

(β

2Λ

)(214)

h21 =λx + iλy

Λsinh

(β

2Λ

)(215)

h22 = cosh

(β

2Λ

)minus λz

Λsinh

(β

2Λ

) (216)


Λ =radicλ2x + λ2

y + λ2z (217)


and G0

Z0 =

[2 cosh

(β

2Λ

)]N(218)

G0 = minusN

βln

[2 cosh

(β2Λ

)] (219)

23




⟨H⟩0 = minusNq

2

(Jx⟨Sx



i Szj ⟩0

)minus N

(hx⟨Sx


j ⟩0) (220)




j ⟩0) (221)



⟨Sαj ⟩0 =

1

Z0

TrSαj e

minusβH0

=TrjS

αj exp

(minusβH0j


(minusβH0k

)Trj exp

(minusβH0j


(minusβH0k

) (222)


⟨Sαj ⟩0 =

TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) (223)


⟨Sαi S

αj ⟩0 =

TriSαi exp

(minusβH0i

)Tri exp

(minusβH0i

) TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) =(⟨Sα

j ⟩0)2

(224)

24



⟨Sαj ⟩0 =

λα

2Λtanh

(β

2Λ

) (225)

and

⟨Sαi S

αj ⟩0 =

(λα

2Λ

)2

tanh2

(β

2Λ

) (226)




βln(2 cosh

β

2Λ)minus Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

)+ N


] (227)


partϕ

partλx

= 0partϕ

partλy

= 0partϕ

partλz

= 0 (228)


partmx

partλx

partmy

partλx

partmz

partλxpartmx

partλy

partmy

partλy

partmz

partλy

partmx

partλz

partmy

partλz

partmz

partλz




= 0 (229)

25





G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)


hef =radic




Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)


Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)


26




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65


or


(12)



(13)

and


(14)




8




H =Nsumj

Hj (16)



ϕnj(rj) (17)



Hjϕnj(rj) = εnj

ϕnj(rj) (18)



sumj εnj




9





ϕnN(r1) ϕnN

(r2) ϕnN(rN)

∣∣∣∣∣∣∣∣∣(110)



10




a b

2

1

r

rrab

r r

ra

a

1

2 2b

21

1b



11



H = H1 + H2 + W + HLS (111)

where

H1 = minus ~2

2m∆1 minus

e20ra1

(112)

H2 = minus ~2

2m∆2 minus

e20rb2

(113)

W =e20rab

+e20r12

minus e20rb1

minus e20ra2

(114)


(H1 + H2)Ψ = U0Ψ (115)

12



z1 s







φ(r1 r2) = c1φ1(r1 r2) + c2φ2(r1 r2)

= c1ϕa(r1)ϕb(r2) + c2ϕa(r2)ϕb(r1) (116)

13



)[c1φ1(r1 r2) + c2φ2(r1 r2

]= E

[c1φ1(r1 r2) + c2φ2(r1 r2)

](117)


space one obtains

2c1E0 + 2c2E0


+ 2c1



= c1E + c2E



space one finds

c1E0



+ 2c2E0 + c2


= c1E





S0 =


14



K =



(121)


A =



(122)



0) = 0


0) + c2(2E0 +K minus E) = 0 (123)


20 + Aminus ES2

0

2E0S20 + Aminus ES2

0 2E0 +K minus E

∣∣∣∣ = 0 (124)


Ea = 2E0 +K minus A

1minus S20

c1 = minusc2 (125)

and

Es = 2E0 +K + A

1 + S20

c1 = c2 (126)

15



1 =


= 2|c1|2(1∓ S2) (127)

or

c1 =1radic

2(1∓ S20) (128)


φ(r1 r2) =1radic

2(1minus S20)


](129)


φ(r1 r2) =1radic

2(1 + S20)


] (130)


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξuarr(1)ξuarr(2) S = 1 Sz = 1 (131)

Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξdarr(1)ξdarr(2) S = 1 Sz = 0 (132)

16


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)


S = 1 Sz = minus1 (133)

and

Ψ =1radic

2(1 + S20)


] 1radic2


∣∣∣∣S = 0 Sz = 0 (134)



0 minus A)

1minus S40

asymp minus2A (135)



17





18

Chapter 2



ity





βlnZ0 (22)

19











20



H = minussumij

(JxSxi S

xj + JyS

yi S

yj + JzS

zi S

zj )minus gmicroB

sumi

(HxSxi +HyS

yi +HzS

zi )


Sxk =

1

2

(0 11 0

)k

Syk =

1

2

(0 minusii 0

)k

Szk =

1

2

(1 00 minus1

)k

(25)


H0 = minusNsumk

(λxS

xk + λyS

yk + λzS

zk

) (26)


H0 =Nsumk=1

H0k (27)

21



H0k = minus1

2

(λz λx minus iλy

λx + iλy minusλz

)k

(28)


Z0 = Tr exp(minusβ

Nsumk=1

H0k

)=

Nprodk=1

Trk exp(minusβH0k

) (29)


Z0 =Nprodk=1

Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

=

[Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

]N

(210)


eminusβH0k =1

2πi

∮C

ezdz


22



eminusβH0k =

(h11 h12

h21 h22

)(212)

with

h11 = cosh

(β

2Λ

)+

λz

Λsinh

(β

2Λ

)(213)

h12 =λx minus iλy

Λsinh

(β

2Λ

)(214)

h21 =λx + iλy

Λsinh

(β

2Λ

)(215)

h22 = cosh

(β

2Λ

)minus λz

Λsinh

(β

2Λ

) (216)


Λ =radicλ2x + λ2

y + λ2z (217)


and G0

Z0 =

[2 cosh

(β

2Λ

)]N(218)

G0 = minusN

βln

[2 cosh

(β2Λ

)] (219)

23




⟨H⟩0 = minusNq

2

(Jx⟨Sx



i Szj ⟩0

)minus N

(hx⟨Sx


j ⟩0) (220)




j ⟩0) (221)



⟨Sαj ⟩0 =

1

Z0

TrSαj e

minusβH0

=TrjS

αj exp

(minusβH0j


(minusβH0k

)Trj exp

(minusβH0j


(minusβH0k

) (222)


⟨Sαj ⟩0 =

TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) (223)


⟨Sαi S

αj ⟩0 =

TriSαi exp

(minusβH0i

)Tri exp

(minusβH0i

) TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) =(⟨Sα

j ⟩0)2

(224)

24



⟨Sαj ⟩0 =

λα

2Λtanh

(β

2Λ

) (225)

and

⟨Sαi S

αj ⟩0 =

(λα

2Λ

)2

tanh2

(β

2Λ

) (226)




βln(2 cosh

β

2Λ)minus Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

)+ N


] (227)


partϕ

partλx

= 0partϕ

partλy

= 0partϕ

partλz

= 0 (228)


partmx

partλx

partmy

partλx

partmz

partλxpartmx

partλy

partmy

partλy

partmz

partλy

partmx

partλz

partmy

partλz

partmz

partλz




= 0 (229)

25





G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)


hef =radic




Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)


Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)


26




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65




H =Nsumj

Hj (16)



ϕnj(rj) (17)



Hjϕnj(rj) = εnj

ϕnj(rj) (18)



sumj εnj




9





ϕnN(r1) ϕnN

(r2) ϕnN(rN)

∣∣∣∣∣∣∣∣∣(110)



10




a b

2

1

r

rrab

r r

ra

a

1

2 2b

21

1b



11



H = H1 + H2 + W + HLS (111)

where

H1 = minus ~2

2m∆1 minus

e20ra1

(112)

H2 = minus ~2

2m∆2 minus

e20rb2

(113)

W =e20rab

+e20r12

minus e20rb1

minus e20ra2

(114)


(H1 + H2)Ψ = U0Ψ (115)

12



z1 s







φ(r1 r2) = c1φ1(r1 r2) + c2φ2(r1 r2)

= c1ϕa(r1)ϕb(r2) + c2ϕa(r2)ϕb(r1) (116)

13



)[c1φ1(r1 r2) + c2φ2(r1 r2

]= E

[c1φ1(r1 r2) + c2φ2(r1 r2)

](117)


space one obtains

2c1E0 + 2c2E0


+ 2c1



= c1E + c2E



space one finds

c1E0



+ 2c2E0 + c2


= c1E





S0 =


14



K =



(121)


A =



(122)



0) = 0


0) + c2(2E0 +K minus E) = 0 (123)


20 + Aminus ES2

0

2E0S20 + Aminus ES2

0 2E0 +K minus E

∣∣∣∣ = 0 (124)


Ea = 2E0 +K minus A

1minus S20

c1 = minusc2 (125)

and

Es = 2E0 +K + A

1 + S20

c1 = c2 (126)

15



1 =


= 2|c1|2(1∓ S2) (127)

or

c1 =1radic

2(1∓ S20) (128)


φ(r1 r2) =1radic

2(1minus S20)


](129)


φ(r1 r2) =1radic

2(1 + S20)


] (130)


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξuarr(1)ξuarr(2) S = 1 Sz = 1 (131)

Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξdarr(1)ξdarr(2) S = 1 Sz = 0 (132)

16


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)


S = 1 Sz = minus1 (133)

and

Ψ =1radic

2(1 + S20)


] 1radic2


∣∣∣∣S = 0 Sz = 0 (134)



0 minus A)

1minus S40

asymp minus2A (135)



17





18

Chapter 2



ity





βlnZ0 (22)

19











20



H = minussumij

(JxSxi S

xj + JyS

yi S

yj + JzS

zi S

zj )minus gmicroB

sumi

(HxSxi +HyS

yi +HzS

zi )


Sxk =

1

2

(0 11 0

)k

Syk =

1

2

(0 minusii 0

)k

Szk =

1

2

(1 00 minus1

)k

(25)


H0 = minusNsumk

(λxS

xk + λyS

yk + λzS

zk

) (26)


H0 =Nsumk=1

H0k (27)

21



H0k = minus1

2

(λz λx minus iλy

λx + iλy minusλz

)k

(28)


Z0 = Tr exp(minusβ

Nsumk=1

H0k

)=

Nprodk=1

Trk exp(minusβH0k

) (29)


Z0 =Nprodk=1

Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

=

[Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

]N

(210)


eminusβH0k =1

2πi

∮C

ezdz


22



eminusβH0k =

(h11 h12

h21 h22

)(212)

with

h11 = cosh

(β

2Λ

)+

λz

Λsinh

(β

2Λ

)(213)

h12 =λx minus iλy

Λsinh

(β

2Λ

)(214)

h21 =λx + iλy

Λsinh

(β

2Λ

)(215)

h22 = cosh

(β

2Λ

)minus λz

Λsinh

(β

2Λ

) (216)


Λ =radicλ2x + λ2

y + λ2z (217)


and G0

Z0 =

[2 cosh

(β

2Λ

)]N(218)

G0 = minusN

βln

[2 cosh

(β2Λ

)] (219)

23




⟨H⟩0 = minusNq

2

(Jx⟨Sx



i Szj ⟩0

)minus N

(hx⟨Sx


j ⟩0) (220)




j ⟩0) (221)



⟨Sαj ⟩0 =

1

Z0

TrSαj e

minusβH0

=TrjS

αj exp

(minusβH0j


(minusβH0k

)Trj exp

(minusβH0j


(minusβH0k

) (222)


⟨Sαj ⟩0 =

TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) (223)


⟨Sαi S

αj ⟩0 =

TriSαi exp

(minusβH0i

)Tri exp

(minusβH0i

) TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) =(⟨Sα

j ⟩0)2

(224)

24



⟨Sαj ⟩0 =

λα

2Λtanh

(β

2Λ

) (225)

and

⟨Sαi S

αj ⟩0 =

(λα

2Λ

)2

tanh2

(β

2Λ

) (226)




βln(2 cosh

β

2Λ)minus Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

)+ N


] (227)


partϕ

partλx

= 0partϕ

partλy

= 0partϕ

partλz

= 0 (228)


partmx

partλx

partmy

partλx

partmz

partλxpartmx

partλy

partmy

partλy

partmz

partλy

partmx

partλz

partmy

partλz

partmz

partλz




= 0 (229)

25





G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)


hef =radic




Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)


Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)


26




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65





ϕnN(r1) ϕnN

(r2) ϕnN(rN)

∣∣∣∣∣∣∣∣∣(110)



10




a b

2

1

r

rrab

r r

ra

a

1

2 2b

21

1b



11



H = H1 + H2 + W + HLS (111)

where

H1 = minus ~2

2m∆1 minus

e20ra1

(112)

H2 = minus ~2

2m∆2 minus

e20rb2

(113)

W =e20rab

+e20r12

minus e20rb1

minus e20ra2

(114)


(H1 + H2)Ψ = U0Ψ (115)

12



z1 s







φ(r1 r2) = c1φ1(r1 r2) + c2φ2(r1 r2)

= c1ϕa(r1)ϕb(r2) + c2ϕa(r2)ϕb(r1) (116)

13



)[c1φ1(r1 r2) + c2φ2(r1 r2

]= E

[c1φ1(r1 r2) + c2φ2(r1 r2)

](117)


space one obtains

2c1E0 + 2c2E0


+ 2c1



= c1E + c2E



space one finds

c1E0



+ 2c2E0 + c2


= c1E





S0 =


14



K =



(121)


A =



(122)



0) = 0


0) + c2(2E0 +K minus E) = 0 (123)


20 + Aminus ES2

0

2E0S20 + Aminus ES2

0 2E0 +K minus E

∣∣∣∣ = 0 (124)


Ea = 2E0 +K minus A

1minus S20

c1 = minusc2 (125)

and

Es = 2E0 +K + A

1 + S20

c1 = c2 (126)

15



1 =


= 2|c1|2(1∓ S2) (127)

or

c1 =1radic

2(1∓ S20) (128)


φ(r1 r2) =1radic

2(1minus S20)


](129)


φ(r1 r2) =1radic

2(1 + S20)


] (130)


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξuarr(1)ξuarr(2) S = 1 Sz = 1 (131)

Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξdarr(1)ξdarr(2) S = 1 Sz = 0 (132)

16


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)


S = 1 Sz = minus1 (133)

and

Ψ =1radic

2(1 + S20)


] 1radic2


∣∣∣∣S = 0 Sz = 0 (134)



0 minus A)

1minus S40

asymp minus2A (135)



17





18

Chapter 2



ity





βlnZ0 (22)

19











20



H = minussumij

(JxSxi S

xj + JyS

yi S

yj + JzS

zi S

zj )minus gmicroB

sumi

(HxSxi +HyS

yi +HzS

zi )


Sxk =

1

2

(0 11 0

)k

Syk =

1

2

(0 minusii 0

)k

Szk =

1

2

(1 00 minus1

)k

(25)


H0 = minusNsumk

(λxS

xk + λyS

yk + λzS

zk

) (26)


H0 =Nsumk=1

H0k (27)

21



H0k = minus1

2

(λz λx minus iλy

λx + iλy minusλz

)k

(28)


Z0 = Tr exp(minusβ

Nsumk=1

H0k

)=

Nprodk=1

Trk exp(minusβH0k

) (29)


Z0 =Nprodk=1

Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

=

[Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

]N

(210)


eminusβH0k =1

2πi

∮C

ezdz


22



eminusβH0k =

(h11 h12

h21 h22

)(212)

with

h11 = cosh

(β

2Λ

)+

λz

Λsinh

(β

2Λ

)(213)

h12 =λx minus iλy

Λsinh

(β

2Λ

)(214)

h21 =λx + iλy

Λsinh

(β

2Λ

)(215)

h22 = cosh

(β

2Λ

)minus λz

Λsinh

(β

2Λ

) (216)


Λ =radicλ2x + λ2

y + λ2z (217)


and G0

Z0 =

[2 cosh

(β

2Λ

)]N(218)

G0 = minusN

βln

[2 cosh

(β2Λ

)] (219)

23




⟨H⟩0 = minusNq

2

(Jx⟨Sx



i Szj ⟩0

)minus N

(hx⟨Sx


j ⟩0) (220)




j ⟩0) (221)



⟨Sαj ⟩0 =

1

Z0

TrSαj e

minusβH0

=TrjS

αj exp

(minusβH0j


(minusβH0k

)Trj exp

(minusβH0j


(minusβH0k

) (222)


⟨Sαj ⟩0 =

TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) (223)


⟨Sαi S

αj ⟩0 =

TriSαi exp

(minusβH0i

)Tri exp

(minusβH0i

) TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) =(⟨Sα

j ⟩0)2

(224)

24



⟨Sαj ⟩0 =

λα

2Λtanh

(β

2Λ

) (225)

and

⟨Sαi S

αj ⟩0 =

(λα

2Λ

)2

tanh2

(β

2Λ

) (226)




βln(2 cosh

β

2Λ)minus Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

)+ N


] (227)


partϕ

partλx

= 0partϕ

partλy

= 0partϕ

partλz

= 0 (228)


partmx

partλx

partmy

partλx

partmz

partλxpartmx

partλy

partmy

partλy

partmz

partλy

partmx

partλz

partmy

partλz

partmz

partλz




= 0 (229)

25





G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)


hef =radic




Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)


Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)


26




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65




a b

2

1

r

rrab

r r

ra

a

1

2 2b

21

1b



11



H = H1 + H2 + W + HLS (111)

where

H1 = minus ~2

2m∆1 minus

e20ra1

(112)

H2 = minus ~2

2m∆2 minus

e20rb2

(113)

W =e20rab

+e20r12

minus e20rb1

minus e20ra2

(114)


(H1 + H2)Ψ = U0Ψ (115)

12



z1 s







φ(r1 r2) = c1φ1(r1 r2) + c2φ2(r1 r2)

= c1ϕa(r1)ϕb(r2) + c2ϕa(r2)ϕb(r1) (116)

13



)[c1φ1(r1 r2) + c2φ2(r1 r2

]= E

[c1φ1(r1 r2) + c2φ2(r1 r2)

](117)


space one obtains

2c1E0 + 2c2E0


+ 2c1



= c1E + c2E



space one finds

c1E0



+ 2c2E0 + c2


= c1E





S0 =


14



K =



(121)


A =



(122)



0) = 0


0) + c2(2E0 +K minus E) = 0 (123)


20 + Aminus ES2

0

2E0S20 + Aminus ES2

0 2E0 +K minus E

∣∣∣∣ = 0 (124)


Ea = 2E0 +K minus A

1minus S20

c1 = minusc2 (125)

and

Es = 2E0 +K + A

1 + S20

c1 = c2 (126)

15



1 =


= 2|c1|2(1∓ S2) (127)

or

c1 =1radic

2(1∓ S20) (128)


φ(r1 r2) =1radic

2(1minus S20)


](129)


φ(r1 r2) =1radic

2(1 + S20)


] (130)


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξuarr(1)ξuarr(2) S = 1 Sz = 1 (131)

Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξdarr(1)ξdarr(2) S = 1 Sz = 0 (132)

16


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)


S = 1 Sz = minus1 (133)

and

Ψ =1radic

2(1 + S20)


] 1radic2


∣∣∣∣S = 0 Sz = 0 (134)



0 minus A)

1minus S40

asymp minus2A (135)



17





18

Chapter 2



ity





βlnZ0 (22)

19











20



H = minussumij

(JxSxi S

xj + JyS

yi S

yj + JzS

zi S

zj )minus gmicroB

sumi

(HxSxi +HyS

yi +HzS

zi )


Sxk =

1

2

(0 11 0

)k

Syk =

1

2

(0 minusii 0

)k

Szk =

1

2

(1 00 minus1

)k

(25)


H0 = minusNsumk

(λxS

xk + λyS

yk + λzS

zk

) (26)


H0 =Nsumk=1

H0k (27)

21



H0k = minus1

2

(λz λx minus iλy

λx + iλy minusλz

)k

(28)


Z0 = Tr exp(minusβ

Nsumk=1

H0k

)=

Nprodk=1

Trk exp(minusβH0k

) (29)


Z0 =Nprodk=1

Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

=

[Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

]N

(210)


eminusβH0k =1

2πi

∮C

ezdz


22



eminusβH0k =

(h11 h12

h21 h22

)(212)

with

h11 = cosh

(β

2Λ

)+

λz

Λsinh

(β

2Λ

)(213)

h12 =λx minus iλy

Λsinh

(β

2Λ

)(214)

h21 =λx + iλy

Λsinh

(β

2Λ

)(215)

h22 = cosh

(β

2Λ

)minus λz

Λsinh

(β

2Λ

) (216)


Λ =radicλ2x + λ2

y + λ2z (217)


and G0

Z0 =

[2 cosh

(β

2Λ

)]N(218)

G0 = minusN

βln

[2 cosh

(β2Λ

)] (219)

23




⟨H⟩0 = minusNq

2

(Jx⟨Sx



i Szj ⟩0

)minus N

(hx⟨Sx


j ⟩0) (220)




j ⟩0) (221)



⟨Sαj ⟩0 =

1

Z0

TrSαj e

minusβH0

=TrjS

αj exp

(minusβH0j


(minusβH0k

)Trj exp

(minusβH0j


(minusβH0k

) (222)


⟨Sαj ⟩0 =

TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) (223)


⟨Sαi S

αj ⟩0 =

TriSαi exp

(minusβH0i

)Tri exp

(minusβH0i

) TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) =(⟨Sα

j ⟩0)2

(224)

24



⟨Sαj ⟩0 =

λα

2Λtanh

(β

2Λ

) (225)

and

⟨Sαi S

αj ⟩0 =

(λα

2Λ

)2

tanh2

(β

2Λ

) (226)




βln(2 cosh

β

2Λ)minus Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

)+ N


] (227)


partϕ

partλx

= 0partϕ

partλy

= 0partϕ

partλz

= 0 (228)


partmx

partλx

partmy

partλx

partmz

partλxpartmx

partλy

partmy

partλy

partmz

partλy

partmx

partλz

partmy

partλz

partmz

partλz




= 0 (229)

25





G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)


hef =radic




Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)


Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)


26




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65



H = H1 + H2 + W + HLS (111)

where

H1 = minus ~2

2m∆1 minus

e20ra1

(112)

H2 = minus ~2

2m∆2 minus

e20rb2

(113)

W =e20rab

+e20r12

minus e20rb1

minus e20ra2

(114)


(H1 + H2)Ψ = U0Ψ (115)

12



z1 s







φ(r1 r2) = c1φ1(r1 r2) + c2φ2(r1 r2)

= c1ϕa(r1)ϕb(r2) + c2ϕa(r2)ϕb(r1) (116)

13



)[c1φ1(r1 r2) + c2φ2(r1 r2

]= E

[c1φ1(r1 r2) + c2φ2(r1 r2)

](117)


space one obtains

2c1E0 + 2c2E0


+ 2c1



= c1E + c2E



space one finds

c1E0



+ 2c2E0 + c2


= c1E





S0 =


14



K =



(121)


A =



(122)



0) = 0


0) + c2(2E0 +K minus E) = 0 (123)


20 + Aminus ES2

0

2E0S20 + Aminus ES2

0 2E0 +K minus E

∣∣∣∣ = 0 (124)


Ea = 2E0 +K minus A

1minus S20

c1 = minusc2 (125)

and

Es = 2E0 +K + A

1 + S20

c1 = c2 (126)

15



1 =


= 2|c1|2(1∓ S2) (127)

or

c1 =1radic

2(1∓ S20) (128)


φ(r1 r2) =1radic

2(1minus S20)


](129)


φ(r1 r2) =1radic

2(1 + S20)


] (130)


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξuarr(1)ξuarr(2) S = 1 Sz = 1 (131)

Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξdarr(1)ξdarr(2) S = 1 Sz = 0 (132)

16


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)


S = 1 Sz = minus1 (133)

and

Ψ =1radic

2(1 + S20)


] 1radic2


∣∣∣∣S = 0 Sz = 0 (134)



0 minus A)

1minus S40

asymp minus2A (135)



17





18

Chapter 2



ity





βlnZ0 (22)

19











20



H = minussumij

(JxSxi S

xj + JyS

yi S

yj + JzS

zi S

zj )minus gmicroB

sumi

(HxSxi +HyS

yi +HzS

zi )


Sxk =

1

2

(0 11 0

)k

Syk =

1

2

(0 minusii 0

)k

Szk =

1

2

(1 00 minus1

)k

(25)


H0 = minusNsumk

(λxS

xk + λyS

yk + λzS

zk

) (26)


H0 =Nsumk=1

H0k (27)

21



H0k = minus1

2

(λz λx minus iλy

λx + iλy minusλz

)k

(28)


Z0 = Tr exp(minusβ

Nsumk=1

H0k

)=

Nprodk=1

Trk exp(minusβH0k

) (29)


Z0 =Nprodk=1

Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

=

[Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

]N

(210)


eminusβH0k =1

2πi

∮C

ezdz


22



eminusβH0k =

(h11 h12

h21 h22

)(212)

with

h11 = cosh

(β

2Λ

)+

λz

Λsinh

(β

2Λ

)(213)

h12 =λx minus iλy

Λsinh

(β

2Λ

)(214)

h21 =λx + iλy

Λsinh

(β

2Λ

)(215)

h22 = cosh

(β

2Λ

)minus λz

Λsinh

(β

2Λ

) (216)


Λ =radicλ2x + λ2

y + λ2z (217)


and G0

Z0 =

[2 cosh

(β

2Λ

)]N(218)

G0 = minusN

βln

[2 cosh

(β2Λ

)] (219)

23




⟨H⟩0 = minusNq

2

(Jx⟨Sx



i Szj ⟩0

)minus N

(hx⟨Sx


j ⟩0) (220)




j ⟩0) (221)



⟨Sαj ⟩0 =

1

Z0

TrSαj e

minusβH0

=TrjS

αj exp

(minusβH0j


(minusβH0k

)Trj exp

(minusβH0j


(minusβH0k

) (222)


⟨Sαj ⟩0 =

TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) (223)


⟨Sαi S

αj ⟩0 =

TriSαi exp

(minusβH0i

)Tri exp

(minusβH0i

) TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) =(⟨Sα

j ⟩0)2

(224)

24



⟨Sαj ⟩0 =

λα

2Λtanh

(β

2Λ

) (225)

and

⟨Sαi S

αj ⟩0 =

(λα

2Λ

)2

tanh2

(β

2Λ

) (226)




βln(2 cosh

β

2Λ)minus Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

)+ N


] (227)


partϕ

partλx

= 0partϕ

partλy

= 0partϕ

partλz

= 0 (228)


partmx

partλx

partmy

partλx

partmz

partλxpartmx

partλy

partmy

partλy

partmz

partλy

partmx

partλz

partmy

partλz

partmz

partλz




= 0 (229)

25





G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)


hef =radic




Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)


Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)


26




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65



z1 s







φ(r1 r2) = c1φ1(r1 r2) + c2φ2(r1 r2)

= c1ϕa(r1)ϕb(r2) + c2ϕa(r2)ϕb(r1) (116)

13



)[c1φ1(r1 r2) + c2φ2(r1 r2

]= E

[c1φ1(r1 r2) + c2φ2(r1 r2)

](117)


space one obtains

2c1E0 + 2c2E0


+ 2c1



= c1E + c2E



space one finds

c1E0



+ 2c2E0 + c2


= c1E





S0 =


14



K =



(121)


A =



(122)



0) = 0


0) + c2(2E0 +K minus E) = 0 (123)


20 + Aminus ES2

0

2E0S20 + Aminus ES2

0 2E0 +K minus E

∣∣∣∣ = 0 (124)


Ea = 2E0 +K minus A

1minus S20

c1 = minusc2 (125)

and

Es = 2E0 +K + A

1 + S20

c1 = c2 (126)

15



1 =


= 2|c1|2(1∓ S2) (127)

or

c1 =1radic

2(1∓ S20) (128)


φ(r1 r2) =1radic

2(1minus S20)


](129)


φ(r1 r2) =1radic

2(1 + S20)


] (130)


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξuarr(1)ξuarr(2) S = 1 Sz = 1 (131)

Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξdarr(1)ξdarr(2) S = 1 Sz = 0 (132)

16


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)


S = 1 Sz = minus1 (133)

and

Ψ =1radic

2(1 + S20)


] 1radic2


∣∣∣∣S = 0 Sz = 0 (134)



0 minus A)

1minus S40

asymp minus2A (135)



17





18

Chapter 2



ity





βlnZ0 (22)

19











20



H = minussumij

(JxSxi S

xj + JyS

yi S

yj + JzS

zi S

zj )minus gmicroB

sumi

(HxSxi +HyS

yi +HzS

zi )


Sxk =

1

2

(0 11 0

)k

Syk =

1

2

(0 minusii 0

)k

Szk =

1

2

(1 00 minus1

)k

(25)


H0 = minusNsumk

(λxS

xk + λyS

yk + λzS

zk

) (26)


H0 =Nsumk=1

H0k (27)

21



H0k = minus1

2

(λz λx minus iλy

λx + iλy minusλz

)k

(28)


Z0 = Tr exp(minusβ

Nsumk=1

H0k

)=

Nprodk=1

Trk exp(minusβH0k

) (29)


Z0 =Nprodk=1

Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

=

[Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

]N

(210)


eminusβH0k =1

2πi

∮C

ezdz


22



eminusβH0k =

(h11 h12

h21 h22

)(212)

with

h11 = cosh

(β

2Λ

)+

λz

Λsinh

(β

2Λ

)(213)

h12 =λx minus iλy

Λsinh

(β

2Λ

)(214)

h21 =λx + iλy

Λsinh

(β

2Λ

)(215)

h22 = cosh

(β

2Λ

)minus λz

Λsinh

(β

2Λ

) (216)


Λ =radicλ2x + λ2

y + λ2z (217)


and G0

Z0 =

[2 cosh

(β

2Λ

)]N(218)

G0 = minusN

βln

[2 cosh

(β2Λ

)] (219)

23




⟨H⟩0 = minusNq

2

(Jx⟨Sx



i Szj ⟩0

)minus N

(hx⟨Sx


j ⟩0) (220)




j ⟩0) (221)



⟨Sαj ⟩0 =

1

Z0

TrSαj e

minusβH0

=TrjS

αj exp

(minusβH0j


(minusβH0k

)Trj exp

(minusβH0j


(minusβH0k

) (222)


⟨Sαj ⟩0 =

TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) (223)


⟨Sαi S

αj ⟩0 =

TriSαi exp

(minusβH0i

)Tri exp

(minusβH0i

) TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) =(⟨Sα

j ⟩0)2

(224)

24



⟨Sαj ⟩0 =

λα

2Λtanh

(β

2Λ

) (225)

and

⟨Sαi S

αj ⟩0 =

(λα

2Λ

)2

tanh2

(β

2Λ

) (226)




βln(2 cosh

β

2Λ)minus Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

)+ N


] (227)


partϕ

partλx

= 0partϕ

partλy

= 0partϕ

partλz

= 0 (228)


partmx

partλx

partmy

partλx

partmz

partλxpartmx

partλy

partmy

partλy

partmz

partλy

partmx

partλz

partmy

partλz

partmz

partλz




= 0 (229)

25





G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)


hef =radic




Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)


Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)


26




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65



)[c1φ1(r1 r2) + c2φ2(r1 r2

]= E

[c1φ1(r1 r2) + c2φ2(r1 r2)

](117)


space one obtains

2c1E0 + 2c2E0


+ 2c1



= c1E + c2E



space one finds

c1E0



+ 2c2E0 + c2


= c1E





S0 =


14



K =



(121)


A =



(122)



0) = 0


0) + c2(2E0 +K minus E) = 0 (123)


20 + Aminus ES2

0

2E0S20 + Aminus ES2

0 2E0 +K minus E

∣∣∣∣ = 0 (124)


Ea = 2E0 +K minus A

1minus S20

c1 = minusc2 (125)

and

Es = 2E0 +K + A

1 + S20

c1 = c2 (126)

15



1 =


= 2|c1|2(1∓ S2) (127)

or

c1 =1radic

2(1∓ S20) (128)


φ(r1 r2) =1radic

2(1minus S20)


](129)


φ(r1 r2) =1radic

2(1 + S20)


] (130)


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξuarr(1)ξuarr(2) S = 1 Sz = 1 (131)

Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξdarr(1)ξdarr(2) S = 1 Sz = 0 (132)

16


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)


S = 1 Sz = minus1 (133)

and

Ψ =1radic

2(1 + S20)


] 1radic2


∣∣∣∣S = 0 Sz = 0 (134)



0 minus A)

1minus S40

asymp minus2A (135)



17





18

Chapter 2



ity





βlnZ0 (22)

19











20



H = minussumij

(JxSxi S

xj + JyS

yi S

yj + JzS

zi S

zj )minus gmicroB

sumi

(HxSxi +HyS

yi +HzS

zi )


Sxk =

1

2

(0 11 0

)k

Syk =

1

2

(0 minusii 0

)k

Szk =

1

2

(1 00 minus1

)k

(25)


H0 = minusNsumk

(λxS

xk + λyS

yk + λzS

zk

) (26)


H0 =Nsumk=1

H0k (27)

21



H0k = minus1

2

(λz λx minus iλy

λx + iλy minusλz

)k

(28)


Z0 = Tr exp(minusβ

Nsumk=1

H0k

)=

Nprodk=1

Trk exp(minusβH0k

) (29)


Z0 =Nprodk=1

Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

=

[Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

]N

(210)


eminusβH0k =1

2πi

∮C

ezdz


22



eminusβH0k =

(h11 h12

h21 h22

)(212)

with

h11 = cosh

(β

2Λ

)+

λz

Λsinh

(β

2Λ

)(213)

h12 =λx minus iλy

Λsinh

(β

2Λ

)(214)

h21 =λx + iλy

Λsinh

(β

2Λ

)(215)

h22 = cosh

(β

2Λ

)minus λz

Λsinh

(β

2Λ

) (216)


Λ =radicλ2x + λ2

y + λ2z (217)


and G0

Z0 =

[2 cosh

(β

2Λ

)]N(218)

G0 = minusN

βln

[2 cosh

(β2Λ

)] (219)

23




⟨H⟩0 = minusNq

2

(Jx⟨Sx



i Szj ⟩0

)minus N

(hx⟨Sx


j ⟩0) (220)




j ⟩0) (221)



⟨Sαj ⟩0 =

1

Z0

TrSαj e

minusβH0

=TrjS

αj exp

(minusβH0j


(minusβH0k

)Trj exp

(minusβH0j


(minusβH0k

) (222)


⟨Sαj ⟩0 =

TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) (223)


⟨Sαi S

αj ⟩0 =

TriSαi exp

(minusβH0i

)Tri exp

(minusβH0i

) TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) =(⟨Sα

j ⟩0)2

(224)

24



⟨Sαj ⟩0 =

λα

2Λtanh

(β

2Λ

) (225)

and

⟨Sαi S

αj ⟩0 =

(λα

2Λ

)2

tanh2

(β

2Λ

) (226)




βln(2 cosh

β

2Λ)minus Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

)+ N


] (227)


partϕ

partλx

= 0partϕ

partλy

= 0partϕ

partλz

= 0 (228)


partmx

partλx

partmy

partλx

partmz

partλxpartmx

partλy

partmy

partλy

partmz

partλy

partmx

partλz

partmy

partλz

partmz

partλz




= 0 (229)

25





G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)


hef =radic




Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)


Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)


26




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65



K =



(121)


A =



(122)



0) = 0


0) + c2(2E0 +K minus E) = 0 (123)


20 + Aminus ES2

0

2E0S20 + Aminus ES2

0 2E0 +K minus E

∣∣∣∣ = 0 (124)


Ea = 2E0 +K minus A

1minus S20

c1 = minusc2 (125)

and

Es = 2E0 +K + A

1 + S20

c1 = c2 (126)

15



1 =


= 2|c1|2(1∓ S2) (127)

or

c1 =1radic

2(1∓ S20) (128)


φ(r1 r2) =1radic

2(1minus S20)


](129)


φ(r1 r2) =1radic

2(1 + S20)


] (130)


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξuarr(1)ξuarr(2) S = 1 Sz = 1 (131)

Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξdarr(1)ξdarr(2) S = 1 Sz = 0 (132)

16


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)


S = 1 Sz = minus1 (133)

and

Ψ =1radic

2(1 + S20)


] 1radic2


∣∣∣∣S = 0 Sz = 0 (134)



0 minus A)

1minus S40

asymp minus2A (135)



17





18

Chapter 2



ity





βlnZ0 (22)

19











20



H = minussumij

(JxSxi S

xj + JyS

yi S

yj + JzS

zi S

zj )minus gmicroB

sumi

(HxSxi +HyS

yi +HzS

zi )


Sxk =

1

2

(0 11 0

)k

Syk =

1

2

(0 minusii 0

)k

Szk =

1

2

(1 00 minus1

)k

(25)


H0 = minusNsumk

(λxS

xk + λyS

yk + λzS

zk

) (26)


H0 =Nsumk=1

H0k (27)

21



H0k = minus1

2

(λz λx minus iλy

λx + iλy minusλz

)k

(28)


Z0 = Tr exp(minusβ

Nsumk=1

H0k

)=

Nprodk=1

Trk exp(minusβH0k

) (29)


Z0 =Nprodk=1

Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

=

[Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

]N

(210)


eminusβH0k =1

2πi

∮C

ezdz


22



eminusβH0k =

(h11 h12

h21 h22

)(212)

with

h11 = cosh

(β

2Λ

)+

λz

Λsinh

(β

2Λ

)(213)

h12 =λx minus iλy

Λsinh

(β

2Λ

)(214)

h21 =λx + iλy

Λsinh

(β

2Λ

)(215)

h22 = cosh

(β

2Λ

)minus λz

Λsinh

(β

2Λ

) (216)


Λ =radicλ2x + λ2

y + λ2z (217)


and G0

Z0 =

[2 cosh

(β

2Λ

)]N(218)

G0 = minusN

βln

[2 cosh

(β2Λ

)] (219)

23




⟨H⟩0 = minusNq

2

(Jx⟨Sx



i Szj ⟩0

)minus N

(hx⟨Sx


j ⟩0) (220)




j ⟩0) (221)



⟨Sαj ⟩0 =

1

Z0

TrSαj e

minusβH0

=TrjS

αj exp

(minusβH0j


(minusβH0k

)Trj exp

(minusβH0j


(minusβH0k

) (222)


⟨Sαj ⟩0 =

TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) (223)


⟨Sαi S

αj ⟩0 =

TriSαi exp

(minusβH0i

)Tri exp

(minusβH0i

) TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) =(⟨Sα

j ⟩0)2

(224)

24



⟨Sαj ⟩0 =

λα

2Λtanh

(β

2Λ

) (225)

and

⟨Sαi S

αj ⟩0 =

(λα

2Λ

)2

tanh2

(β

2Λ

) (226)




βln(2 cosh

β

2Λ)minus Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

)+ N


] (227)


partϕ

partλx

= 0partϕ

partλy

= 0partϕ

partλz

= 0 (228)


partmx

partλx

partmy

partλx

partmz

partλxpartmx

partλy

partmy

partλy

partmz

partλy

partmx

partλz

partmy

partλz

partmz

partλz




= 0 (229)

25





G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)


hef =radic




Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)


Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)


26




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65



1 =


= 2|c1|2(1∓ S2) (127)

or

c1 =1radic

2(1∓ S20) (128)


φ(r1 r2) =1radic

2(1minus S20)


](129)


φ(r1 r2) =1radic

2(1 + S20)


] (130)


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξuarr(1)ξuarr(2) S = 1 Sz = 1 (131)

Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)

∣∣∣∣ ξdarr(1)ξdarr(2) S = 1 Sz = 0 (132)

16


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)


S = 1 Sz = minus1 (133)

and

Ψ =1radic

2(1 + S20)


] 1radic2


∣∣∣∣S = 0 Sz = 0 (134)



0 minus A)

1minus S40

asymp minus2A (135)



17





18

Chapter 2



ity





βlnZ0 (22)

19











20



H = minussumij

(JxSxi S

xj + JyS

yi S

yj + JzS

zi S

zj )minus gmicroB

sumi

(HxSxi +HyS

yi +HzS

zi )


Sxk =

1

2

(0 11 0

)k

Syk =

1

2

(0 minusii 0

)k

Szk =

1

2

(1 00 minus1

)k

(25)


H0 = minusNsumk

(λxS

xk + λyS

yk + λzS

zk

) (26)


H0 =Nsumk=1

H0k (27)

21



H0k = minus1

2

(λz λx minus iλy

λx + iλy minusλz

)k

(28)


Z0 = Tr exp(minusβ

Nsumk=1

H0k

)=

Nprodk=1

Trk exp(minusβH0k

) (29)


Z0 =Nprodk=1

Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

=

[Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

]N

(210)


eminusβH0k =1

2πi

∮C

ezdz


22



eminusβH0k =

(h11 h12

h21 h22

)(212)

with

h11 = cosh

(β

2Λ

)+

λz

Λsinh

(β

2Λ

)(213)

h12 =λx minus iλy

Λsinh

(β

2Λ

)(214)

h21 =λx + iλy

Λsinh

(β

2Λ

)(215)

h22 = cosh

(β

2Λ

)minus λz

Λsinh

(β

2Λ

) (216)


Λ =radicλ2x + λ2

y + λ2z (217)


and G0

Z0 =

[2 cosh

(β

2Λ

)]N(218)

G0 = minusN

βln

[2 cosh

(β2Λ

)] (219)

23




⟨H⟩0 = minusNq

2

(Jx⟨Sx



i Szj ⟩0

)minus N

(hx⟨Sx


j ⟩0) (220)




j ⟩0) (221)



⟨Sαj ⟩0 =

1

Z0

TrSαj e

minusβH0

=TrjS

αj exp

(minusβH0j


(minusβH0k

)Trj exp

(minusβH0j


(minusβH0k

) (222)


⟨Sαj ⟩0 =

TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) (223)


⟨Sαi S

αj ⟩0 =

TriSαi exp

(minusβH0i

)Tri exp

(minusβH0i

) TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) =(⟨Sα

j ⟩0)2

(224)

24



⟨Sαj ⟩0 =

λα

2Λtanh

(β

2Λ

) (225)

and

⟨Sαi S

αj ⟩0 =

(λα

2Λ

)2

tanh2

(β

2Λ

) (226)




βln(2 cosh

β

2Λ)minus Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

)+ N


] (227)


partϕ

partλx

= 0partϕ

partλy

= 0partϕ

partλz

= 0 (228)


partmx

partλx

partmy

partλx

partmz

partλxpartmx

partλy

partmy

partλy

partmz

partλy

partmx

partλz

partmy

partλz

partmz

partλz




= 0 (229)

25





G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)


hef =radic




Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)


Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)


26




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65


Ψ =1radic

2(1minus S20)

∣∣∣∣ ϕa(r1) ϕa(r2)ϕb(r1) ϕb(r2)


S = 1 Sz = minus1 (133)

and

Ψ =1radic

2(1 + S20)


] 1radic2


∣∣∣∣S = 0 Sz = 0 (134)



0 minus A)

1minus S40

asymp minus2A (135)



17





18

Chapter 2



ity





βlnZ0 (22)

19











20



H = minussumij

(JxSxi S

xj + JyS

yi S

yj + JzS

zi S

zj )minus gmicroB

sumi

(HxSxi +HyS

yi +HzS

zi )


Sxk =

1

2

(0 11 0

)k

Syk =

1

2

(0 minusii 0

)k

Szk =

1

2

(1 00 minus1

)k

(25)


H0 = minusNsumk

(λxS

xk + λyS

yk + λzS

zk

) (26)


H0 =Nsumk=1

H0k (27)

21



H0k = minus1

2

(λz λx minus iλy

λx + iλy minusλz

)k

(28)


Z0 = Tr exp(minusβ

Nsumk=1

H0k

)=

Nprodk=1

Trk exp(minusβH0k

) (29)


Z0 =Nprodk=1

Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

=

[Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

]N

(210)


eminusβH0k =1

2πi

∮C

ezdz


22



eminusβH0k =

(h11 h12

h21 h22

)(212)

with

h11 = cosh

(β

2Λ

)+

λz

Λsinh

(β

2Λ

)(213)

h12 =λx minus iλy

Λsinh

(β

2Λ

)(214)

h21 =λx + iλy

Λsinh

(β

2Λ

)(215)

h22 = cosh

(β

2Λ

)minus λz

Λsinh

(β

2Λ

) (216)


Λ =radicλ2x + λ2

y + λ2z (217)


and G0

Z0 =

[2 cosh

(β

2Λ

)]N(218)

G0 = minusN

βln

[2 cosh

(β2Λ

)] (219)

23




⟨H⟩0 = minusNq

2

(Jx⟨Sx



i Szj ⟩0

)minus N

(hx⟨Sx


j ⟩0) (220)




j ⟩0) (221)



⟨Sαj ⟩0 =

1

Z0

TrSαj e

minusβH0

=TrjS

αj exp

(minusβH0j


(minusβH0k

)Trj exp

(minusβH0j


(minusβH0k

) (222)


⟨Sαj ⟩0 =

TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) (223)


⟨Sαi S

αj ⟩0 =

TriSαi exp

(minusβH0i

)Tri exp

(minusβH0i

) TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) =(⟨Sα

j ⟩0)2

(224)

24



⟨Sαj ⟩0 =

λα

2Λtanh

(β

2Λ

) (225)

and

⟨Sαi S

αj ⟩0 =

(λα

2Λ

)2

tanh2

(β

2Λ

) (226)




βln(2 cosh

β

2Λ)minus Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

)+ N


] (227)


partϕ

partλx

= 0partϕ

partλy

= 0partϕ

partλz

= 0 (228)


partmx

partλx

partmy

partλx

partmz

partλxpartmx

partλy

partmy

partλy

partmz

partλy

partmx

partλz

partmy

partλz

partmz

partλz




= 0 (229)

25





G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)


hef =radic




Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)


Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)


26




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65





18

Chapter 2



ity





βlnZ0 (22)

19











20



H = minussumij

(JxSxi S

xj + JyS

yi S

yj + JzS

zi S

zj )minus gmicroB

sumi

(HxSxi +HyS

yi +HzS

zi )


Sxk =

1

2

(0 11 0

)k

Syk =

1

2

(0 minusii 0

)k

Szk =

1

2

(1 00 minus1

)k

(25)


H0 = minusNsumk

(λxS

xk + λyS

yk + λzS

zk

) (26)


H0 =Nsumk=1

H0k (27)

21



H0k = minus1

2

(λz λx minus iλy

λx + iλy minusλz

)k

(28)


Z0 = Tr exp(minusβ

Nsumk=1

H0k

)=

Nprodk=1

Trk exp(minusβH0k

) (29)


Z0 =Nprodk=1

Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

=

[Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

]N

(210)


eminusβH0k =1

2πi

∮C

ezdz


22



eminusβH0k =

(h11 h12

h21 h22

)(212)

with

h11 = cosh

(β

2Λ

)+

λz

Λsinh

(β

2Λ

)(213)

h12 =λx minus iλy

Λsinh

(β

2Λ

)(214)

h21 =λx + iλy

Λsinh

(β

2Λ

)(215)

h22 = cosh

(β

2Λ

)minus λz

Λsinh

(β

2Λ

) (216)


Λ =radicλ2x + λ2

y + λ2z (217)


and G0

Z0 =

[2 cosh

(β

2Λ

)]N(218)

G0 = minusN

βln

[2 cosh

(β2Λ

)] (219)

23




⟨H⟩0 = minusNq

2

(Jx⟨Sx



i Szj ⟩0

)minus N

(hx⟨Sx


j ⟩0) (220)




j ⟩0) (221)



⟨Sαj ⟩0 =

1

Z0

TrSαj e

minusβH0

=TrjS

αj exp

(minusβH0j


(minusβH0k

)Trj exp

(minusβH0j


(minusβH0k

) (222)


⟨Sαj ⟩0 =

TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) (223)


⟨Sαi S

αj ⟩0 =

TriSαi exp

(minusβH0i

)Tri exp

(minusβH0i

) TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) =(⟨Sα

j ⟩0)2

(224)

24



⟨Sαj ⟩0 =

λα

2Λtanh

(β

2Λ

) (225)

and

⟨Sαi S

αj ⟩0 =

(λα

2Λ

)2

tanh2

(β

2Λ

) (226)




βln(2 cosh

β

2Λ)minus Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

)+ N


] (227)


partϕ

partλx

= 0partϕ

partλy

= 0partϕ

partλz

= 0 (228)


partmx

partλx

partmy

partλx

partmz

partλxpartmx

partλy

partmy

partλy

partmz

partλy

partmx

partλz

partmy

partλz

partmz

partλz




= 0 (229)

25





G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)


hef =radic




Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)


Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)


26




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65

Chapter 2



ity





βlnZ0 (22)

19











20



H = minussumij

(JxSxi S

xj + JyS

yi S

yj + JzS

zi S

zj )minus gmicroB

sumi

(HxSxi +HyS

yi +HzS

zi )


Sxk =

1

2

(0 11 0

)k

Syk =

1

2

(0 minusii 0

)k

Szk =

1

2

(1 00 minus1

)k

(25)


H0 = minusNsumk

(λxS

xk + λyS

yk + λzS

zk

) (26)


H0 =Nsumk=1

H0k (27)

21



H0k = minus1

2

(λz λx minus iλy

λx + iλy minusλz

)k

(28)


Z0 = Tr exp(minusβ

Nsumk=1

H0k

)=

Nprodk=1

Trk exp(minusβH0k

) (29)


Z0 =Nprodk=1

Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

=

[Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

]N

(210)


eminusβH0k =1

2πi

∮C

ezdz


22



eminusβH0k =

(h11 h12

h21 h22

)(212)

with

h11 = cosh

(β

2Λ

)+

λz

Λsinh

(β

2Λ

)(213)

h12 =λx minus iλy

Λsinh

(β

2Λ

)(214)

h21 =λx + iλy

Λsinh

(β

2Λ

)(215)

h22 = cosh

(β

2Λ

)minus λz

Λsinh

(β

2Λ

) (216)


Λ =radicλ2x + λ2

y + λ2z (217)


and G0

Z0 =

[2 cosh

(β

2Λ

)]N(218)

G0 = minusN

βln

[2 cosh

(β2Λ

)] (219)

23




⟨H⟩0 = minusNq

2

(Jx⟨Sx



i Szj ⟩0

)minus N

(hx⟨Sx


j ⟩0) (220)




j ⟩0) (221)



⟨Sαj ⟩0 =

1

Z0

TrSαj e

minusβH0

=TrjS

αj exp

(minusβH0j


(minusβH0k

)Trj exp

(minusβH0j


(minusβH0k

) (222)


⟨Sαj ⟩0 =

TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) (223)


⟨Sαi S

αj ⟩0 =

TriSαi exp

(minusβH0i

)Tri exp

(minusβH0i

) TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) =(⟨Sα

j ⟩0)2

(224)

24



⟨Sαj ⟩0 =

λα

2Λtanh

(β

2Λ

) (225)

and

⟨Sαi S

αj ⟩0 =

(λα

2Λ

)2

tanh2

(β

2Λ

) (226)




βln(2 cosh

β

2Λ)minus Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

)+ N


] (227)


partϕ

partλx

= 0partϕ

partλy

= 0partϕ

partλz

= 0 (228)


partmx

partλx

partmy

partλx

partmz

partλxpartmx

partλy

partmy

partλy

partmz

partλy

partmx

partλz

partmy

partλz

partmz

partλz




= 0 (229)

25





G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)


hef =radic




Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)


Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)


26




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65











20



H = minussumij

(JxSxi S

xj + JyS

yi S

yj + JzS

zi S

zj )minus gmicroB

sumi

(HxSxi +HyS

yi +HzS

zi )


Sxk =

1

2

(0 11 0

)k

Syk =

1

2

(0 minusii 0

)k

Szk =

1

2

(1 00 minus1

)k

(25)


H0 = minusNsumk

(λxS

xk + λyS

yk + λzS

zk

) (26)


H0 =Nsumk=1

H0k (27)

21



H0k = minus1

2

(λz λx minus iλy

λx + iλy minusλz

)k

(28)


Z0 = Tr exp(minusβ

Nsumk=1

H0k

)=

Nprodk=1

Trk exp(minusβH0k

) (29)


Z0 =Nprodk=1

Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

=

[Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

]N

(210)


eminusβH0k =1

2πi

∮C

ezdz


22



eminusβH0k =

(h11 h12

h21 h22

)(212)

with

h11 = cosh

(β

2Λ

)+

λz

Λsinh

(β

2Λ

)(213)

h12 =λx minus iλy

Λsinh

(β

2Λ

)(214)

h21 =λx + iλy

Λsinh

(β

2Λ

)(215)

h22 = cosh

(β

2Λ

)minus λz

Λsinh

(β

2Λ

) (216)


Λ =radicλ2x + λ2

y + λ2z (217)


and G0

Z0 =

[2 cosh

(β

2Λ

)]N(218)

G0 = minusN

βln

[2 cosh

(β2Λ

)] (219)

23




⟨H⟩0 = minusNq

2

(Jx⟨Sx



i Szj ⟩0

)minus N

(hx⟨Sx


j ⟩0) (220)




j ⟩0) (221)



⟨Sαj ⟩0 =

1

Z0

TrSαj e

minusβH0

=TrjS

αj exp

(minusβH0j


(minusβH0k

)Trj exp

(minusβH0j


(minusβH0k

) (222)


⟨Sαj ⟩0 =

TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) (223)


⟨Sαi S

αj ⟩0 =

TriSαi exp

(minusβH0i

)Tri exp

(minusβH0i

) TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) =(⟨Sα

j ⟩0)2

(224)

24



⟨Sαj ⟩0 =

λα

2Λtanh

(β

2Λ

) (225)

and

⟨Sαi S

αj ⟩0 =

(λα

2Λ

)2

tanh2

(β

2Λ

) (226)




βln(2 cosh

β

2Λ)minus Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

)+ N


] (227)


partϕ

partλx

= 0partϕ

partλy

= 0partϕ

partλz

= 0 (228)


partmx

partλx

partmy

partλx

partmz

partλxpartmx

partλy

partmy

partλy

partmz

partλy

partmx

partλz

partmy

partλz

partmz

partλz




= 0 (229)

25





G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)


hef =radic




Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)


Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)


26




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65



H = minussumij

(JxSxi S

xj + JyS

yi S

yj + JzS

zi S

zj )minus gmicroB

sumi

(HxSxi +HyS

yi +HzS

zi )


Sxk =

1

2

(0 11 0

)k

Syk =

1

2

(0 minusii 0

)k

Szk =

1

2

(1 00 minus1

)k

(25)


H0 = minusNsumk

(λxS

xk + λyS

yk + λzS

zk

) (26)


H0 =Nsumk=1

H0k (27)

21



H0k = minus1

2

(λz λx minus iλy

λx + iλy minusλz

)k

(28)


Z0 = Tr exp(minusβ

Nsumk=1

H0k

)=

Nprodk=1

Trk exp(minusβH0k

) (29)


Z0 =Nprodk=1

Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

=

[Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

]N

(210)


eminusβH0k =1

2πi

∮C

ezdz


22



eminusβH0k =

(h11 h12

h21 h22

)(212)

with

h11 = cosh

(β

2Λ

)+

λz

Λsinh

(β

2Λ

)(213)

h12 =λx minus iλy

Λsinh

(β

2Λ

)(214)

h21 =λx + iλy

Λsinh

(β

2Λ

)(215)

h22 = cosh

(β

2Λ

)minus λz

Λsinh

(β

2Λ

) (216)


Λ =radicλ2x + λ2

y + λ2z (217)


and G0

Z0 =

[2 cosh

(β

2Λ

)]N(218)

G0 = minusN

βln

[2 cosh

(β2Λ

)] (219)

23




⟨H⟩0 = minusNq

2

(Jx⟨Sx



i Szj ⟩0

)minus N

(hx⟨Sx


j ⟩0) (220)




j ⟩0) (221)



⟨Sαj ⟩0 =

1

Z0

TrSαj e

minusβH0

=TrjS

αj exp

(minusβH0j


(minusβH0k

)Trj exp

(minusβH0j


(minusβH0k

) (222)


⟨Sαj ⟩0 =

TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) (223)


⟨Sαi S

αj ⟩0 =

TriSαi exp

(minusβH0i

)Tri exp

(minusβH0i

) TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) =(⟨Sα

j ⟩0)2

(224)

24



⟨Sαj ⟩0 =

λα

2Λtanh

(β

2Λ

) (225)

and

⟨Sαi S

αj ⟩0 =

(λα

2Λ

)2

tanh2

(β

2Λ

) (226)




βln(2 cosh

β

2Λ)minus Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

)+ N


] (227)


partϕ

partλx

= 0partϕ

partλy

= 0partϕ

partλz

= 0 (228)


partmx

partλx

partmy

partλx

partmz

partλxpartmx

partλy

partmy

partλy

partmz

partλy

partmx

partλz

partmy

partλz

partmz

partλz




= 0 (229)

25





G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)


hef =radic




Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)


Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)


26




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65



H0k = minus1

2

(λz λx minus iλy

λx + iλy minusλz

)k

(28)


Z0 = Tr exp(minusβ

Nsumk=1

H0k

)=

Nprodk=1

Trk exp(minusβH0k

) (29)


Z0 =Nprodk=1

Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

=

[Trk exp

(β2λz

β2(λx minus iλy)


2λz

)k

]N

(210)


eminusβH0k =1

2πi

∮C

ezdz


22



eminusβH0k =

(h11 h12

h21 h22

)(212)

with

h11 = cosh

(β

2Λ

)+

λz

Λsinh

(β

2Λ

)(213)

h12 =λx minus iλy

Λsinh

(β

2Λ

)(214)

h21 =λx + iλy

Λsinh

(β

2Λ

)(215)

h22 = cosh

(β

2Λ

)minus λz

Λsinh

(β

2Λ

) (216)


Λ =radicλ2x + λ2

y + λ2z (217)


and G0

Z0 =

[2 cosh

(β

2Λ

)]N(218)

G0 = minusN

βln

[2 cosh

(β2Λ

)] (219)

23




⟨H⟩0 = minusNq

2

(Jx⟨Sx



i Szj ⟩0

)minus N

(hx⟨Sx


j ⟩0) (220)




j ⟩0) (221)



⟨Sαj ⟩0 =

1

Z0

TrSαj e

minusβH0

=TrjS

αj exp

(minusβH0j


(minusβH0k

)Trj exp

(minusβH0j


(minusβH0k

) (222)


⟨Sαj ⟩0 =

TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) (223)


⟨Sαi S

αj ⟩0 =

TriSαi exp

(minusβH0i

)Tri exp

(minusβH0i

) TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) =(⟨Sα

j ⟩0)2

(224)

24



⟨Sαj ⟩0 =

λα

2Λtanh

(β

2Λ

) (225)

and

⟨Sαi S

αj ⟩0 =

(λα

2Λ

)2

tanh2

(β

2Λ

) (226)




βln(2 cosh

β

2Λ)minus Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

)+ N


] (227)


partϕ

partλx

= 0partϕ

partλy

= 0partϕ

partλz

= 0 (228)


partmx

partλx

partmy

partλx

partmz

partλxpartmx

partλy

partmy

partλy

partmz

partλy

partmx

partλz

partmy

partλz

partmz

partλz




= 0 (229)

25





G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)


hef =radic




Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)


Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)


26




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65



eminusβH0k =

(h11 h12

h21 h22

)(212)

with

h11 = cosh

(β

2Λ

)+

λz

Λsinh

(β

2Λ

)(213)

h12 =λx minus iλy

Λsinh

(β

2Λ

)(214)

h21 =λx + iλy

Λsinh

(β

2Λ

)(215)

h22 = cosh

(β

2Λ

)minus λz

Λsinh

(β

2Λ

) (216)


Λ =radicλ2x + λ2

y + λ2z (217)


and G0

Z0 =

[2 cosh

(β

2Λ

)]N(218)

G0 = minusN

βln

[2 cosh

(β2Λ

)] (219)

23




⟨H⟩0 = minusNq

2

(Jx⟨Sx



i Szj ⟩0

)minus N

(hx⟨Sx


j ⟩0) (220)




j ⟩0) (221)



⟨Sαj ⟩0 =

1

Z0

TrSαj e

minusβH0

=TrjS

αj exp

(minusβH0j


(minusβH0k

)Trj exp

(minusβH0j


(minusβH0k

) (222)


⟨Sαj ⟩0 =

TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) (223)


⟨Sαi S

αj ⟩0 =

TriSαi exp

(minusβH0i

)Tri exp

(minusβH0i

) TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) =(⟨Sα

j ⟩0)2

(224)

24



⟨Sαj ⟩0 =

λα

2Λtanh

(β

2Λ

) (225)

and

⟨Sαi S

αj ⟩0 =

(λα

2Λ

)2

tanh2

(β

2Λ

) (226)




βln(2 cosh

β

2Λ)minus Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

)+ N


] (227)


partϕ

partλx

= 0partϕ

partλy

= 0partϕ

partλz

= 0 (228)


partmx

partλx

partmy

partλx

partmz

partλxpartmx

partλy

partmy

partλy

partmz

partλy

partmx

partλz

partmy

partλz

partmz

partλz




= 0 (229)

25





G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)


hef =radic




Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)


Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)


26




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65




⟨H⟩0 = minusNq

2

(Jx⟨Sx



i Szj ⟩0

)minus N

(hx⟨Sx


j ⟩0) (220)




j ⟩0) (221)



⟨Sαj ⟩0 =

1

Z0

TrSαj e

minusβH0

=TrjS

αj exp

(minusβH0j


(minusβH0k

)Trj exp

(minusβH0j


(minusβH0k

) (222)


⟨Sαj ⟩0 =

TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) (223)


⟨Sαi S

αj ⟩0 =

TriSαi exp

(minusβH0i

)Tri exp

(minusβH0i

) TrjSαj exp

(minusβH0j

)Trj exp

(minusβH0j

) =(⟨Sα

j ⟩0)2

(224)

24



⟨Sαj ⟩0 =

λα

2Λtanh

(β

2Λ

) (225)

and

⟨Sαi S

αj ⟩0 =

(λα

2Λ

)2

tanh2

(β

2Λ

) (226)




βln(2 cosh

β

2Λ)minus Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

)+ N


] (227)


partϕ

partλx

= 0partϕ

partλy

= 0partϕ

partλz

= 0 (228)


partmx

partλx

partmy

partλx

partmz

partλxpartmx

partλy

partmy

partλy

partmz

partλy

partmx

partλz

partmy

partλz

partmz

partλz




= 0 (229)

25





G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)


hef =radic




Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)


Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)


26




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65



⟨Sαj ⟩0 =

λα

2Λtanh

(β

2Λ

) (225)

and

⟨Sαi S

αj ⟩0 =

(λα

2Λ

)2

tanh2

(β

2Λ

) (226)




βln(2 cosh

β

2Λ)minus Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

)+ N


] (227)


partϕ

partλx

= 0partϕ

partλy

= 0partϕ

partλz

= 0 (228)


partmx

partλx

partmy

partλx

partmz

partλxpartmx

partλy

partmy

partλy

partmz

partλy

partmx

partλz

partmy

partλz

partmz

partλz




= 0 (229)

25





G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)


hef =radic




Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)


Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)


26




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65





G = minusN

βln2 cosh

(βhef

2

)+

Nq

2

(Jxm

2x + Jym

2y + Jzm

2z

) (231)


hef =radic




Mα = minus(

partGpartHα

)β

= minusgmicroB

(partGparthα

)β

(233)


Mx

NgmicroB

= mx = ⟨Sxj ⟩0

My

NgmicroB

= my = ⟨Syj ⟩0

Mz

NgmicroB

= mz = ⟨Szj ⟩0

(234)


26




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65




mx =qJxmx + hx

2hef

tanh

(β

2hef

) (235)

my =qJymy + hy

2hef

tanh

(β

2hef

) (236)

mz =qJzmz + hz

2hef

tanh

(β

2hef

) (237)



mx =Jxmx

2h0

tanh

(βq

2h0

) (238)

my =Jymy

2h0

tanh

(βq

2h0

) (239)

mz =Jzmz

2h0

tanh

(βq

2h0

) (240)

27


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65


with





2tanh

(qβJ2

mz

)for T lt Tc




2radic2tanh

(radic2qβJ

2mx

)for T lt Tc




2radic3tanh

(radic3qβJx2

mx

)for T lt Tc


28



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65



00 05 10 15000

025

050q = 4

m

kBT J

Exact

MFT



29



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65



Tc =qJα

4kB α = x y z (245)





30

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65

Chapter 3

Spin-Wave Theory


nets


H = minusJsumij


Szi (31)


i Syi S

zi ) where Sα


31




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65




dSℓ

dt=

i

~[H Sℓ] (32)


[Sai S

bj ] = iεabcδijS



dSℓ

dt= minus1



Hℓ =sumj



~dSx

ℓ

dt= minus

sumj

Jjℓ(Syj S

zℓ minus Sz


ℓ (36)

~dSy

ℓ

dt= minus

sumj

Jjℓ(Szj S

xℓ minus Sx


ℓ (37)

32


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65


~dSz

ℓ

dt= minus

sumj

Jjℓ(Sxj S

yℓ minus Sy

j Sxℓ ) (38)





~dSx

ℓ

dt= minusS

sumj

Jjℓ(Syj minus Sy


~dSy

ℓ

dt= minusS

sumj



or alternatively

~dS+

ℓ

dt= minusiS

sumj



~dSminus

ℓ

dt= minusiS

sumj




S+ℓ = Sx

ℓ + iSyℓ Sminus


ℓ (313)


[Szk S

+ℓ ] = S+

k δkℓ [Szk S


k δkℓ [S+k S

+ℓ ] = 2Sz

kδkℓ (314)

33




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65




S+q =

1radicN

sumj


q =1radicN

sumj


S+j =

1radicN

sumq


1radicN

sumq




one gets equation

minusi~dSminus

q

dt=

[Ssumℓ



J(q) =sumj



minusi~dSminus

q

dt=

S[J(0)minus J(q)

]+ gmicroBH

Sminusq (320)



34




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65




]+ gmicroBH (322)


as follows

Sminusj =

1radicNδSqe


Sxj =

1radicNδSq cos


](324)

and

Syj = minus 1radic

NδSq sin


] (325)




35



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65



nets


H = minusJsumij

(Sxi S

xj + Sy

i Syj + Sz


sumi

Szi (327)



H = minusJsumℓδ

(Sxℓ S

xℓ+δ + Sy

ℓ Syℓ+δ + Sz


sumℓ

Szℓ (328)


36










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65










S+ℓ |MS⟩ℓ =


and






S+ℓ |nℓ⟩ℓ =

radic2S


2S



37



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65



H = minusJsumℓδ

[12(S+


ℓ S+ℓ+δ) + Sz

ℓ Szℓ+δ

]minus gmicroBH

sumℓ

Szℓ (334)


adaggerℓΨnℓ=




S+ℓ = (2S)12


2S

)12

aℓ (336)



2S

)12

(337)



Sminusℓ Ψ2S = (2S)12(2S + 1)12

(1minus 2S

2S

)12

Ψ2S+1 = 0 (339)



aℓ =1radicN

sumq


38


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65


adaggerℓ =1radicN

sumq




bq =1radicN

sumℓ


bdaggerq =1radicN

sumℓ







S+ℓ = (2S)12


4S

)aℓ + (345)

39




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65




4S

)+ (346)


S+ℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


](347)

Sminusℓ =

(2S

N

)12[sumq1


4SN

sumq1q2q3


bq3 +

](348)


Szℓ = S minus 1

N

sumq1q2







H0 = minus JS

N

sumℓδ

sumq1q2


daggerq2


]+

JS

N

sumℓδ

sumq1q2


daggerq2


]+

gmicroBH

N

sumℓq1q2


40



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65



H0 = minusJzSsumq



sumq

bdaggerqbq (353)

where

γq =1

z

sumδ

eiqmiddotδ (354)




H0 =sumq


]bdaggerqbq (355)


H0 =sumq

~ωqnq (356)




γq =1

z

sumδ

eiqmiddotδ =1

2z

sumδ


2z

sumδ




2z

sumδ


41





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65





U =sumq

~ωq⟨nq⟩ (360)


⟨nq⟩ =[exp

( ~ωq

kBT

)minus 1

]minus1

(361)


U =sumq

~ωq

[exp

( ~ωq

kBT

)minus 1

]minus1

=1

(2π)3

intDq2

[exp

(Dq2

kBT

)minus 1

]minus1

d3q

=1

2π2

intDq4

[exp

(Dq2

kBT

)minus 1

]minus1

dq (362)


U =(kBT )

52

4π2D32

int xm

0

x32


42



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65



U =(kBT )

52

4π2D32Γ(52

)ζ(52 1) (364)



52

)=

3radicπ

4and ζ

(52 1)= 1341


U ≃ 045(kBT )52

π2D32 (365)


CV = 0113kB

(kBTD

) 32 (366)


C = aT 3 + bT 32 (367)


43


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65


0 2 4 6 80

1

2

CT -32

T 32

magnoncontribution




MS = gmicroB

⟨(NS minus

sumq




sumq

⟨nq⟩

=gmicroB

(2π)3

int qm

0

[exp

(Dq2

kBT

)minus 1

]minus1

d3q (369)

44



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65



∆M =gmicroB

2π2

(kBTJD

)32int infin

0

x12

exp(x)minus 1dx

=gmicroB

2π2

(kBTJD

)32

Γ(52

)ζ(32 1)

= 0117gmicroB

(kBTD

)32

(370)


M(T ) = M(0)[1minus

( T

Tc

) 32]

(371)


Tc =( M(0)

0117gmicroB

) 23 D

kB (372)


magnets


H = minusJsumij


sumi

Szai + gmicroBHa

sumj

Szbj (373)

45



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65



S+aj = (2S)12

(1minus

adaggerjaj

2S

)12


(1minus

adaggerjaj

2S

)12

(374)

S+bℓ = (2S)12


2S

)12



2S

)12

(375)




cq =2radicN

sumj


sumj


dq =2radicN

sumℓ


sumℓ




daggerq2] = δq1q2



] = 0 (379)

46



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65



S+aj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](380)

Sminusaj =

(4S

N

)12[sumq1


8SN

sumq1q2q3


cq3 +

](381)

S+bℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


](382)

Sminusbℓ =

(4S

N

)12[sumq1


8SN

sumq1q2q3


dq3 +

](383)

Szaj = S minus 1

N

sumq1q2


Szbℓ = minusS +

1

N

sumq1q2



H = NzJS2 + H0 + H1 (386)

47



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65



H0 = minus2JzSsumq

[γq(cdaggerqd

daggerq + cqdq


)]+ gmicroBH

sumq


)(387)



q αq βdaggerq βq







daggerq] = 1 [αq βq] = 0 (390)


u2q minus v2q = 1 (391)






48








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65








[~ω+

q

(αdaggerqαq +

1

2

)+ ~ωminus

q

(βdaggerqβq +

1

2

)](396)



radic1minus γ2






[~ω+

q

(⟨αdagger

qαq⟩+1

2

)+ ~ωminus

q

(⟨βdagger

qβq⟩+1

2

)]

(398)


49


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65


reduces to


(1minus γ2

q

)12

= NJzS2[1 +

1

S

(1minus 2

N

sumq

radic1minus γ2

q

)] (399)


γq =1

d

dsumi=1

cos qi (3100)


Id =2

N

radic1minus γ2

q

=1

(2π)d

int π

minusπ

int π

minusπ

dq1 dqd

[1minus

(1d

dsumi=1

cos qi

)2]12 (3101)


I1= 0637 I2

= 0842 I3

= 0903 (3102)


50


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65


form

Eg = 2NJS2(1 +

0363

S


Eg = 4NJS2(1 +

0158

S


Eg = 6NJS2(1 +

0097

S



1

2NJzS2(1 +

1

zS) lt Eg lt

1

2NJzS2 (3104)





Ma = gmicroB

⟨sumj

Szaj

⟩=

1

2NS minus

sumq

⟨cdaggerqcq

⟩(3105)

51



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65



Ma =1

2NS minus

sumq

cosh2(θq)

⟨αdaggerqαq

⟩+sinh2(θq)

⟨βqβ

daggerq


⟨αdaggerqβ

daggerq

⟩+⟨αqβq

⟩)

(3106)


Ma =1

2NS minus

sumq

sinh2(θq) (3107)


Ma =1

2NS minus 1

4

sumq

[(1minus γq1 + γq

)12

+

(1 + γq1minus γq

)12

minus 2

] (3108)


Ma =NS

2

[1minus 1

2S

(2

N

sumq

1radic1minus γ2

q

minus 1

)] (3109)


sumq

1radic1minusγ2

q

appearing


Sd =2

N

sumq

1radic1minus γ2

q

=1

(2π)d

int π

minusπ


minusπ

dλ1 dλd

[1minus

(1

d

dsumj=1

cosλj

)2]minus12

(3110)

52



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65



Ma =NS

2

(1minus 0078

S

)for d = 3 (3111)

Ma =NS

2

(1minus 0197

S

)for d = 2 (3112)





⟨nq⟩ cosh(2θq)

=sumq

⟨nq⟩(1minus γ2

q

)minus12(3113)


⟨nq⟩ =1

exp( ~ωq

kBT

)minus 1

(3114)

53


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65


with

~ωq = (2JSz)2(1minus γ2q) (3115)



54

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65

Chapter 4





55




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65




H = JNsumj=1

(Sxj S

xj+1 + Sy

j Syj+1) (41)





H =J

2

Nsumj=1

(Sminusj S

+j+1 + Sminus

j+1S+j ) (42)


56




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65




cdaggern = S+n exp

(minusiπ

nminus1sumj=1

S+j S

minusj

) cn = Sminus

n exp(iπ

nminus1sumj=1

S+j S

minusj

) (43)


S+n = cdaggern exp

(iπ

nminus1sumj=1

cdaggerjcj

) Sminus

n = cn exp(iπ

nminus1sumj=1

cdaggerncn

) (44)






+k Sk] = 0


+j Sj S

+k ] = δjkS

+k (S

+j Sj)



exp(plusmniπ

msumj=n

S+j S

minusj

)=

mprodj=n

exp(plusmniπS+j S

minusj ) (46)

57



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65



exp(plusmniπS+j S

minusj ) =

infinsumk=0

1

k(plusmniπ)k(S+

j Sminusj )

k

= 1 +infinsumk=1

1

k(plusmniπ)kS+

j Sminusj


minusj = 1minus 2S+

j Sminusj (47)


Sminusj 1minus 2S+

j Sminusj = 0 S+

j 1minus 2S+j S

minusj = 0 (48)


(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

]=

[exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

]= 0 k isin [nm]

(49)

exp

(plusmniπ

msumj=n

S+j S

minusj

) Sminus

k

=

exp

(plusmniπ

msumj=n

S+j S

minusj

) S+

k

= 0 k isin [nm]

(410)


58


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65


anti-commutators


(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)S+k

+ S+k exp

(minusiπ

kminus1sumℓ=1

S+ℓ S

minusℓ

)exp

(iπ

jminus1sumℓ=1

S+ℓ S

minusℓ

)Sminusj

= Sminusj exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)S+k S

+k exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)Sminusk

= (Sminusj S

+j minus S+

j Sminusj ) exp

(iπ

kminus1sumℓ=j

S+ℓ S

minusℓ

)= 0 k gt j (411)




+k + S+

k Sminusk = 1 (413)



Sminusj S

+j+1 = exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)cjc

daggerj+1 exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)= cj exp

(minusiπ

jminus1sumℓ=1

cdaggerℓcℓ

)exp

(iπ

jsumℓ=1

cdaggerℓcℓ

)cdaggerj+1



daggerj+1




Sminusj+1S

+j = (Sminus

j S+j+1)


59




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65




2 = cdaggerjcj


daggerk


[exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

]=

[exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

]= 0

k isin [nm] (417)

exp

(plusmniπ

msumj=n

cdaggerjcj

) ck

=

exp

(plusmniπ

msumj=n

cdaggerjcj

) cdaggerk

= 0

k isin [nm] (418)


+1 + Sminus

1 S+N) in


H =1

2

sumj


where

Hb = minus1

2


)[1 + exp

(plusmniπ

Nsumj=1

cdaggerjcj

)](420)


60



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65



H =1

2

sumj




H =sumjk

cdaggerjAjkck (422)


Ajk =1

2(δjk+1 + δkj+1) (423)



with q = 2πnN


ηq =sumj


sumj

ϕqjcdaggerj (425)


cj =sumq


ϕqjηdaggerq (426)

61



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65



H =sumq



ξq = ηq Λq ge 0 (428)


and obtain

H =sum

qΛqge0

Λqξdaggerqξq +

sumqΛqlt0

Λqξdaggerqξq

=sumqΛq


qΛqlt0

|Λq|

=sumqΛq


1

2

) (430)


ξq|0⟩ = 0 forallq (431)



U0 =⟨H⟩N

= minus 1

N

sumq

1

2|Λq|

=

int π

π

dq


π

int π2

0

cos(q)dq = minus 1

π(432)

62


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65


ω(q)

0

02

04

06

08

1

q-20 -15 -10 -05 00 05 10 15 20



63

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65

Conclusion

Conclusion






64

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65

Bibliography

[1] F Bloch Z Physik 61 206 (1930)


[3] H Bethe Z Physik 71 205 (1931)









65

Documents

FACULTY OF SCIENCE - Pavol Jozef Šafárik University€¦ · Electronic academic textbook for Faculty of Science on P. J. Šafárik ... The text consists of four chapters that form