Review Article Composite Operator Method Analysis of …downloads.hindawi.com/journals/acmp/2014/515698.pdf · Review Article Composite Operator Method Analysis of ... that strong

Review ArticleComposite Operator Method Analysis ofthe Underdoped Cuprates Puzzle

Adolfo Avella123

1 Dipartimento di Fisica ldquoER Caianiellordquo Universita degli Studi di Salerno 84084 Fisciano Italy2 Unita CNISM di Salerno Universita degli Studi di Salerno 84084 Fisciano Italy3 CNR-SPIN UoS di Salerno 84084 Fisciano Italy

Correspondence should be addressed to Adolfo Avella avellaphysicsunisait

Received 3 June 2014 Accepted 30 September 2014 Published 10 November 2014

Academic Editor Jorg Fink

Copyright copy 2014 Adolfo AvellaThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The microscopical analysis of the unconventional and puzzling physics of the underdoped cuprates as carried out lately by meansof the composite operator method (COM) applied to the 2D Hubbard model is reviewed and systematized The 2D Hubbardmodel has been adopted as it has been considered the minimal model capable of describing the most peculiar features of cupratesheld responsible for their anomalous behavior COM is designed to endorse since its foundation the systematic emergence inany SCS of new elementary excitations described by composite operators obeying noncanonical algebras In this case (underdopedcupratesmdash2DHubbardmodel) the residual interactionsmdashbeyond a 2-pole approximationmdashbetween the new elementary electronicexcitations dictated by the strong local Coulomb repulsion and well described by the two Hubbard composite operators have beentreated within the noncrossing approximation Given this recipe and exploiting the few unknowns to enforce the Pauli principlecontent in the solution it is possible to qualitatively describe some of the anomalous features of high-Tc cuprate superconductorssuch as large versus small Fermi surface dichotomy Fermi surface deconstruction (appearance of Fermi arcs) nodal versusantinodal physics pseudogap(s) and kinks in the electronic dispersion The resulting scenario envisages a smooth crossoverbetween an ordinary weakly interactingmetal sustainingweak short-range antiferromagnetic correlations in the overdoped regimeto an unconventional poor metal characterized by very strong long-but-finite-range antiferromagnetic correlations leading tomomentum-selective non-Fermi liquid features as well as to the opening of a pseudogap and to the striking differences betweenthe nodal and the antinodal dynamics in the underdoped regime

1 Introduction

11 Composite Fields One of the most intriguing chal-lenges in modern condensed matter physics is the theoret-ical description of the anomalous behaviors experimentallyobserved in many novel materials By anomalous behaviorswe mean those not predicted by standard many-body theorythat is behaviors in contradiction with the Fermi-liquidframework and diagrammatic expansions The most relevantcharacteristic of such novel materials is the presence ofso strong correlations among the electrons that classicalschemes based on the band picture and the perturbationtheory are definitely inapplicable Accordingly it is necessaryto move from a single-electron physics to a many-electronphysics where the dominant contributions come from the

strong interactions among the electrons usual schemes aresimply inadequate and new concepts must be introduced

The classical techniques are based on the hypothesisthat the interactions among the electrons are weak enoughor sufficiently well screened to be properly taken intoaccount within the framework of perturbativediagrammaticmethods However as many and many experimental andtheoretical studies of highly correlated systems have shownwith more and more convincing evidence all these methodsare no more viable The main concept that breaks down isthe existence of the electrons as particles or quasiparticleswith quite-well-defined propertiesThe presence of the inter-actions radically modifies the properties of the particles andat a macroscopic level what are observed are new particles(actually they are the only observable ones) with new peculiar

Hindawi Publishing CorporationAdvances in Condensed Matter PhysicsVolume 2014 Article ID 515698 29 pageshttpdxdoiorg1011552014515698

2 Advances in Condensed Matter Physics

properties entirely determined by the dynamics and by theboundary conditions (ie the phase under study the externalfields ) These new objects appear as the final result of themodifications imposed by the interactions on the originalparticles and contain by the very beginning the effects ofcorrelations

On the basis of this evidence one is induced to move theattention from the original fields to the new fields generatedby the interactions The operators describing these excita-tions once they have been identified can be written in termsof the original ones and are known as composite operatorsThe necessity of developing a formulation to treat compositeoperators as fundamental objects of the many-body problemin condensedmatter physics has been deeply understood andsystematically noticed since quite long time Recent yearshave seen remarkable achievements in the development ofa modern many-body theory in solid-state physics in theform of an assortment of techniques that may be termedcomposite particle methods The foundations of these typesof techniques may be traced back to the work of Bogoliubov[1] and later to that of Dancoff [2] The work of Zwanzig[3] Mori [4ndash9] and Umezawa [10] definitely deserves to bementioned too Closely related to this work is that ofHubbard[11ndash13] Rowe [14] Roth [15] and Tserkovnikov [16 17] Theslave boson method [18ndash20] the spectral density approach[21 22] the diagram technique for Hubbard operators [23]the cumulant expansion based diagram technique [24] thegeneralized tight-binding method [25ndash27] self-consistentprojection operator method [28 29] operator projectionmethod [30ndash32] and the composite operatormethod (COM)[33ndash35] are along the same lines This large class of theoriesis very promising as it is based on the firm convictionthat strong interactions call for an analysis in terms of newelementary fields embedding the greatest possible part ofthe correlations so permitting to overcome the problem offinding an appropriate expansion parameter However oneprice must be paid In general composite fields are neitherFermi nor Bose operators since they do not satisfy canonical(anti)commutation relations and their properties must bedetermined self-consistently They can only be recognized asfermionic or bosonic operators according to the number andtype of the constituting original particles Accordingly newtechniques have to be developed in order to deal with suchcomposite fields and to design diagrammatic schemes wherethe building blocks are the propagators of such compositefields standard diagrammatic expansions and the Wickrsquostheorem are no more valid The formulation of the Greenrsquosfunctionmethod itself must be revisited and new frameworksof calculations have to be devised

Following these ideas we have been developing a sys-tematic approach the composite operator method (COM)[33ndash35] to study highly correlated systems The formalism isbased on two main ideas (i) use of propagators of relevantcomposite operators as building blocks for any subsequentapproximate calculations and (ii) use of algebra constraints tofix the representation of the relevant propagators in order toproperly preserve algebraic and symmetry properties theseconstraints will also determine the unknown parametersappearing in the formulation due to the noncanonical algebra

satisfied by the composite operators In the last fifteen yearsCOM has been applied to several models and materialsHubbard [36ndash47] 119901-119889 [48ndash51] 119905-119869 [52] 119905-1199051015840-119880 [53ndash55]extended Hubbard (119905-119880-119881) [56 57] Kondo [58] Anderson[59 60] two-orbital Hubbard [61ndash63] Ising [64 65] 119869

1minus

1198692[66ndash71] Hubbard-Kondo [72] Cuprates [73ndash78] and so

forth

12 UnderdopedCuprates Cuprate superconductors [79] dis-play a full range of anomalous features mainly appearingin the underdoped region in almost all experimentallymeasurable physical properties [80ndash84] According to thistheir microscopic description is still an open problem non-Fermi-liquid response quantum criticality pseudogap for-mation ill-defined Fermi surface kinks in the electronicdispersion and so forth still remain unexplained (or at leastcontroversially debated) anomalous features [84ndash87] In thelast years the attention of the community has been focusingon threemain experimental facts [84] the dramatic change inshape and nature of the Fermi surface between underdopedand overdoped regimes the appearance of a psedudogapin the underdoped regime and the striking differentiationbetween the physics at the nodes and at the antinodes in thepseudogap regime The topological transition of the Fermisurface has been first detected by means of ARPES [82]and reflects the noteworthy differences between the quite-ordinary large Fermi surface measured in the overdopedregime [88ndash90] and quite well described by LDA calculations[91] and quantum oscillations measurements [92] and theill-defined Fermi arcs appearing in the underdoped regime[85 93ndash98] The enormous relevance of these experimentalfindings not only for the microscopic comprehension ofthe high-119879

119888superconductivity phenomenology but also for

the drafting of a general microscopic theory for stronglycorrelated materials called for many more measurements inorder to explore all possible aspects of such extremely anoma-lous and peculiar behavior plenty of quantum oscillationsmeasurements in the underdoped regime [99ndash118] Hall effectmeasurements [100] Seebeck effect measurements [116] andheat capacity measurements [115] The presence of a quitestrong depletion in the electronic density of states knownas pseudogap [119] is well established thanks to ARPES [82]NMR [120] optical conductivity [121] and quantum oscilla-tions [122] measurements The microscopic origin of such aloss of single-particle electronic states is still unclear and thenumber of possible theoretical as well as phenomenologicalexplanations has grown quite large in the last few years Asa matter of fact this phenomenon affects any measurableproperties and accordingly was the first to be detected in theunderdoped regime granting to this latter the first evidencesof its exceptionality with respect to the other regimes inthe phase diagram The plethora of theoretical scenariospresent in the literature [85 123ndash125] tentatively explainingfew some or many of the anomalous features reported bythe experiments on underdoped cuprates can be coarselydivided into those not relying on any translational symmetrybreaking [126ndash131] and those instead proposing that it shouldbe some kind of charge andor spin arrangements to be

Advances in Condensed Matter Physics 3

held responsible for the whole range of anomalous featuresAmong these latter theories there are those focusing on thephysics at the antinodal region and those focusing on thenodal region We can account for proposals of (as regards theantinode) a collinear spin AF order [132] an AF quantumcritical point [133 134] and a 1D charge stripe order [132 135]with the addition of a smectic phase [136] Instead at thenode we have a 119889-density wave [137] amore-or-less ordinaryAF spin order [75ndash78 138ndash142] and a nodal pocket frombilayer low-119876 charge order slowly fluctuacting [135 143ndash145] This latter proposal which is among the newest on thetable relies onmany experimentalmeasurements STM [146ndash150] neutron scattering [122] X-ray diffraction [151] NMR[152] RXS [153] and phonon softening [154ndash156] A proposalregarding the emergence of a hidden Fermi liquid [157] is alsoworth mentioning

13 2D Hubbard Model Approximation Methods Since thevery beginning [158] the two-dimensional Hubbard model[11] has been universally recognized as the minimal modelcapable of describing the CundashO

2planes of cuprates super-

conductors It certainly contains many of the key ingredientsby construction strong electronic correlations competi-tion between localization and itineracy Mott physics andlow-energy spin excitations Unfortunately though beingfundamental for benchmarking and fine tuning analyticaltheories numerical approaches [159] cannot be of help tosolve the puzzle of underdoped cuprates owing to theirlimited resolution in frequency and momentum On theother hand there are not so many analytical approachescapable of dealing with the quite complex aspects of under-doped cuprates phenomenology [9] Among others the two-particle self-consistent (TPSC) approach [86 87 160] hasbeen the first completely microscopic approach to obtainresults comparable with the experimental findings Almostall other promising approaches available in the literaturecan be essentially divided into two classes One class makesuse of phenomenological expressions for the electronic self-energy and the electronic spin susceptibility [5ndash9 161 162]The electronic self-energy is usually computed as the con-volution of the electronic propagator and of the electronicspin susceptibility Then the electronic spin susceptibilityis modeled phenomenologically parameterizing correlationlength and damping as functions of doping and temper-ature according to the common belief that the electronicspin susceptibility should present a well-developed mode at119872 = (120587 120587) with a damping of Landau type The DMFT +

Σ approach [163ndash167] and the DGammaA approach [168169] should also be mentioned All cluster-dynamical-mean-field-like theories (cluster-DMFT theories) [86 87 170 171](the cellular dynamical mean-field theory (C-DMFT) [172173] the dynamical cluster approximation (DCA) [174] andthe cluster perturbation theory (CPT) [86 87 175 176])belong to the second class The dynamical mean-field theory(DMFT) [177ndash181] cannot tackle the underdoped cupratespuzzle because its self-energy has the same identical valueat each point on the Fermi surface without any possibledifferentiation between nodal and antinodal physics or visible

and phantom portion of the Fermi surface The cluster-DMFT theories instead can in principle deal with bothcoherent quasi-particles and marginal ones within the sameFermi surface These theories usually self-consistently mapthe generic Hubbard problem to a few-site lattice Andersonproblem and solve this latter by means of mainly numericaltechniques What really distinguishes one formulation fromanother within this second class is the procedure usedto map the small cluster on the infinite lattice Anywayit is worth noticing that these approaches often rely onnumerical methods in order to close their self-consistencycycles (with the abovementioned limitations in frequencyand momentum resolutions and with the obvious difficultiesin the physical interpretation of their results) and alwaysface the emergence of a quite serious periodization problemsince a cluster embedded in the lattice violates its periodicityThe Composite Operator Method (COM) [33ndash35] does notbelong to any of these two classes of theoretical formulationsand has the advantage to be completely microscopic exclu-sively analytical and fully self-consistent COM recipe usesthree main ingredients [33ndash35] composite operators algebraconstraints and residual self-energy treatment Compositeoperators are products of electronic operators and describethe new elementary excitations appearing in the systemowing to strong correlations According to the system underanalysis [33ndash35] one has to choose a set of composite opera-tors as operatorial basis and rewrite the electronic operatorsand the electronic Greenrsquos function in terms of this basisOne should think of composite operators just as a more con-venient starting point with respect to electronic operatorsfor any mean-field-like approximationperturbation schemeAlgebra constraints are relations among correlation functionsdictated by the noncanonical operatorial algebra closed bythe chosen operatorial basis [33ndash35] Other ways to obtainalgebra constraints rely on the symmetries enjoined by theHamiltonian under study the Ward-Takahashi identities thehydrodynamics and so forth [33ndash35] One should think ofalgebra constraints as a way to restrict the Fock space onwhich the chosen operatorial basis acts to the Fock space ofphysical electrons Algebra constraints are used to computeunknown correlation functions appearing in the calculationsInteractions among the elements of the chosen operatorialbasis are described by the residual self-energy that is thepropagator of the residual term of the current after this latterhas been projected on the chosen operatorial basis [33ndash35]According to the physical properties under analysis and therange of temperatures dopings and interactions you want toexplore one has to choose an approximation to compute theresidual self-energy In the last years we have been using the119899-pole Approximation [36ndash43 46 48ndash57 61ndash65 72ndash74] theAsymptotic Field Approach [58ndash60] the NCA [44 45 75ndash78 182] and the Two-Site Resolvent Approach [183 184] Youshould think of the residual self-energy as a measure in thefrequency and momentum space of how much well definedare as quasi-particles your composite operators It is reallyworth noticing that although the description of some of theanomalous features of underdoped cuprates given by COMqualitatively coincides with those obtained by TPSC [86 87]and by the two classes of formulations mentioned above


the results obtained by means of COM greatly differ fromthose obtained within the other methods as regards theevolution with doping of the dispersion and of the Fermisurface and at the moment no experimental result can tellwhich is the unique and distinctive choice nature made

14 Outline To study the underdoped cuprates modeled bythe 2DHubbardmodel (see Section 21) we start from a basisof two composite operators (the two Hubbard operators) andformulate the Dyson equation (see Section 22) in terms ofthe 2-pole approximated Greenrsquos function (see Section 23)According to this the self-energy is the propagator ofnonlocal composite operators describing the electronic fielddressed by charge spin and pair fluctuations on the nearest-neighbor sites Then within the noncrossing approximation(NCA) [185] we obtain a microscopic self-energy written interms of the convolution of the electronic propagator and ofthe charge spin and pair susceptibilities (see Section 24)Finally we close fully analytically the self-consistency cyclefor the electronic propagator by computing microscopic sus-ceptibilities within a 2-pole approximation (see Section 25)Our results (see Section 24) show that within COM thetwo-dimensional Hubbard model can describe some of theanomalous features experimentally observed in underdopedcuprates phenomenology In particular we show how Fermiarcs can develop out of a large Fermi surface (see Section 32)how pseudogap can show itself in the dispersion (seeSection 31) and in the density of states (see Section 33)how non-Fermi-liquid features can become apparent in themomentum distribution function (see Section 34) and in thefrequency and temperature dependences of the self-energy(see Section 35) how much kinked the dispersion can get onvarying doping (see Section 31) andwhy or at least how spindynamics can be held responsible for all this (see Section 36)Finally (see Section 4) we summarize the current status of thescenario emerging by these theoretical findings andwhich arethe perspectives

2 Framework

21 Hamiltonian The Hamiltonian of the two-dimensionalHubbard model reads as

119867 = sum

ij(minus120583120575ij minus 4119905120572ij) 119888

dagger

(119894) 119888 (119895) + 119880sum

i119899uarr(119894) 119899

darr(119894) (1)

where

119888 (119894) = (

119888uarr(119894)

119888darr(119894)

) (2)

is the electron field operator in spinorial notation and Hei-senberg picture (119894 = (i 119905

119894)) i = Ri is a vector of the Bravais

lattice 119899120590(119894) = 119888

dagger

120590(119894)119888

120590(119894) is the particle density operator for

spin 120590 119899(119894) = sum120590119899120590(119894) is the total particle density opera-

tor 120583 is the chemical potential 119905 is the hopping integral and

the energy unit 119880 is the Coulomb on-site repulsion and 120572ijis the projector on the nearest-neighbor sites

120572ij =1

119873

sum

keiksdot(RiminusRj)

120572 (k)

120572 (k) = 1

2

[cos (119896119909119886) + cos (119896

119910119886)]

(3)

where k runs over the first Brillouin zone119873 is the number ofsites and 119886 is the lattice constant

22 Greenrsquos Functions and Dyson Equation Following COMprescriptions [33ndash35] we chose a basic field in particular weselect the composite doublet field operator

120595 (119894) = (

120585 (119894)

120578 (119894)

) (4)

where 120578(119894) = 119899(119894)119888(119894) and 120585(119894) = 119888(119894) minus 120578(119894) are the Hubbardoperators describing the main subbands This choice isguided by the hierarchy of the equations of motion and bythe fact that 120585(119894) and 120578(119894) are eigenoperators of the interactingterm in the Hamiltonian (1) The field 120595(119894) satisfies theHeisenberg equation

i 120597120597119905

120595 (119894) = 119869 (119894) = (

minus120583120585 (119894) minus 4119905119888

120572(119894) minus 4119905120587 (119894)

(119880 minus 120583) 120578 (119894) + 4119905120587 (119894)

) (5)

where the higher-order composite field 120587(119894) is defined by

120587 (119894) =

1

2

120590

120583119899120583(119894) 119888

120572

(119894) + 119888 (119894) 119888

dagger120572

(119894) 119888 (119894) (6)

with the following notation 119899120583(119894) = 119888

dagger(119894)120590

120583119888(119894) is the particle

(120583 = 0) and spin (120583 = 1 2 3) density operator and120590120583= (1 ) 120590120583 = (minus1 ) and 120590

119896(119896 = 1 2 3) are the Pauli

matrices Hereafter for any operatorΦ(119894) we use the notationΦ

120572(i 119905) = sumj 120572ijΦ(j 119905)It is always possible to decompose the source 119869(119894) under

the form

119869 (119894) = 120576 (minusinabla)120595 (119894) + 120575119869 (119894) (7)

where the linear term represents the projection of the sourceon the basis 120595(119894) and is calculated by means of the equation

⟨120575119869 (i 119905) 120595dagger(j 119905)⟩ = 0 (8)

where ⟨sdot sdot sdot ⟩ stands for the thermal average taken in the grand-canonical ensemble

This constraint assures that the residual current 120575119869(119894)

contains all and only the physics orthogonal to the chosenbasis120595(119894)The action of the derivative operator 120576(minusinabla) on120595(119894)is defined in momentum space

120576 (minusinabla)120595 (119894) = 120576 (minusinabla) 1

radic119873

sum

keiksdotRi

120595 (k 119905)

=

1

radic119873

sum

keiksdotRi

120576 (k) 120595 (k 119905) (9)

where 120576(k) is namedenergy matrix


The constraint (8) gives

119898(k) = 120576 (k) 119868 (k) (10)

after defining the normalization matrix

119868 (i j) = ⟨120595 (i 119905) 120595dagger(j 119905)⟩

=

1

119873

sum

keiksdot(RiminusRj)

119868 (k) (11)

and the119898-matrix

119898(i j) = ⟨119869 (i 119905) 120595dagger(j 119905)⟩

=

1

119873

sum

keiksdot(RiminusRj)

119898(k) (12)

Since the components of 120595(119894) contain composite opera-tors the normalization matrix 119868(k) is not the identity matrixand defines the spectral content of the excitations In fact thecomposite operator method has the advantage of describingcrossover phenomena as the phenomena in which the weightof some operator is shifted to another one

By considering the two-time thermodynamic Greenrsquosfunctions [186ndash188] let us define the retarded function

119866 (119894 119895) = ⟨119877 [120595 (119894) 120595

dagger(119895)]⟩

= 120579 (119905119894minus 119905

119895) ⟨120595 (119894) 120595

dagger(119895)⟩

(13)

By means of the Heisenberg equation (5) and using thedecomposition (7) the Greenrsquos function 119866(119894 119895) satisfies theequation

Λ (120597119894) 119866 (119894 119895) Λ

dagger(120597119895) = Λ (120597

119894) 119866

0(119894 119895) Λ

dagger(120597119895)

+ ⟨119877 [120575119869 (119894) 120575119869

dagger(119895)]⟩

(14)

where the derivative operator Λ(120597119894) is defined as

Λ (120597119894) = i 120597

120597119905119894

minus 120576 (minusinabla119894) (15)

and the propagator 1198660(119894 119895) is defined by the equation

Λ (120597119894) 119866

0(119894 119895) = i120575 (119905

119894minus 119905

119895) 119868 (119894 119895) (16)

By introducing the Fourier transform

119866 (119894 119895) =

1

119873

sum

k

i2120587

int119889120596eiksdot(RiminusRj)minusi120596(119905119894minus119905119895)119866 (k 120596) (17)

equation (14) in momentum space can be written as

119866 (k 120596) = 119866

0

(k 120596) + 119866

0

(k 120596) 119868minus1 (k) Σ (k 120596) 119866 (k 120596) (18)

and can be formally solved as

119866 (k 120596) = 1

120596 minus 120576 (k) minus Σ (k 120596)119868 (k) (19)

where the self-energy Σ(k 120596) has the expression

Σ (k 120596) = 119861irr (k 120596) 119868minus1

(k) (20)

with

119861 (k 120596) = F ⟨119877 [120575119869 (119894) 120575119869

dagger(119895)]⟩ (21)

The notation F denotes the Fourier transform andthe subscript irr indicates that the irreducible part of thepropagator 119861(k 120596) is taken Equation (18) is nothing elsethan the Dyson equation for composite fields and representsthe starting point for a perturbative calculation in terms ofthe propagator 1198660

(k 120596) This quantity will be calculated inthe next section Then the attention will be given to thecalculation of the self-energy Σ(k 120596) It should be notedthat the computation of the two quantities 119866

0(k 120596) and

Σ(k 120596) is intimately related The total weight of the self-energy corrections is bounded by the weight of the residualsource operator 120575119869(119894) According to this it can be madesmaller and smaller by increasing the components of the basis120595(119894) (eg by including higher-order composite operatorsappearing in 120575119869(119894)) The result of such a procedure will bethe inclusion in the energy matrix of part of the self-energyas an expansion in terms of coupling constants multiplied bythe weights of the newly included basis operators In generalthe enlargement of the basis leads to a new self-energy witha smaller total weight However it is necessary pointing outthat this process can be quite cumbersome and the inclusionof fully momentum and frequency dependent self-energycorrections can be necessary to effectively take into accountlow-energy and virtual processes According to this one canchoose a reasonable number of components for the basic setand then use another approximation method to evaluate theresidual dynamical corrections

23 Two-Pole Approximation According to (16) the freepropagator 1198660

(k 120596) is determined by the following expres-sion

119866

0

(k 120596) = 1

120596 minus 120576 (k)119868 (k) (22)

For a paramagnetic state straightforward calculationsgive the following expressions for the normalization 119868(k) andenergy 120576(k)matrices

119868 (k) = (

1 minus

119899

2

0

0

119899

2

) = (

11986811

0

0 11986822

)

12057611(k) = minus120583 minus 4119905119868

minus1

11[Δ + (1 minus 119899 + 119901) 120572 (k)]

12057612(k) = 4119905119868

minus1

22[Δ + (119901 minus 119868

22) 120572 (k)]

12057621(k) = 4119905119868

minus1

11[Δ + (119901 minus 119868

22) 120572 (k)]

12057622(k) = 119880 minus 120583 minus 4119905119868

minus1

22[Δ + 119901120572 (k)]

(23)


where 119899 = ⟨119899 (119894)⟩ is the filling and

Δ = ⟨120585

120572

(119894) 120585

dagger

(119894)⟩ minus ⟨120578

120572

(119894) 120578

dagger

(119894)⟩

119901 =

1

4

⟨119899

120572

120583(119894) 119899

120583(119894)⟩ minus ⟨[119888

uarr(119894) 119888

darr(119894)]

120572

119888

dagger

darr(119894) 119888

dagger

uarr(119894)⟩

(24)

Then (22) can be written in spectral form as

119866

0

(k 120596) =2

sum

119899=1

120590

(119899)(k)

120596 minus 119864119899(k) + i120575

(25)

The energy spectra 119864119899(k) and the spectral functions

120590

(119899)(k) are given by

1198641(k) = 119877 (k) + 119876 (k)

1198642(k) = 119877 (k) minus 119876 (k)

120590

(1)

11(k) = 119868

11

2

[1 +

119892 (k)2119876 (k)

]

120590

(2)

11(k) = 119868

11

2

[1 minus

119892 (k)2119876 (k)

]

120590

(1)

12(k) = 119898

12(k)

2119876 (k)

120590

(2)

12(k) = minus

11989812(k)

2119876 (k)

120590

(1)

22(k) = 119868

22

2

[1 minus

119892 (k)2119876 (k)

]

120590

(2)

22(k) = 119868

22

2

[1 +

119892 (k)2119876 (k)

]

(26)

where

119877 (k) = minus 120583 minus 4119905120572 (k) + 1

2

119880 minus

12057612(k)

211986811

119876 (k) = 1

2

radic119892

2(k) +

4120576

2

12(k) 119868

22

11986811

119892 (k) = minus 119880 +

1 minus 119899

11986811

12057612(k)

(27)

The energy matrix 120576(k) contains three parameters 120583the chemical potential Δ the difference between upper andlower intrasubband contributions to kinetic energy and 119901a combination of the nearest-neighbor charge-charge spin-spin and pair-pair correlation functions These parameterswill be determined in a self-consistent way by means ofalgebra constraints in terms of the external parameters 119899 119880and 119879

24 Noncrossing Approximation The calculation of the self-energy Σ(k 120596) requires the calculation of the higher-orderpropagator119861(k 120596) [cf (21)]Wewill compute this quantity by

using the noncrossing approximation (NCA) By neglectingthe pair term 119888(119894)119888

dagger120572(119894)119888(119894) the source 119869(119894) can be written as

119869 (i 119905) = sum

j119886 (i j 119905) 120595 (j 119905) (28)

where

11988611(i j 119905) = minus120583120575ij minus 4119905120572ij minus 2119905120590

120583119899120583(119894) 120572ij

11988612(i j 119905) = minus4119905120572ij minus 2119905120590

120583119899120583(119894) 120572ij

11988621(i j 119905) = 2119905120590

120583119899120583(119894) 120572ij

11988622(i j 119905) = (119880 minus 120583) 120575ij + 2119905120590

120583119899120583(119894) 120572ij

(29)

Then for the calculation of 119861irr(119894 119895) = ⟨119877[120575119869(119894)120575119869

dagger(119895)]⟩irr

we approximate

120575119869 (i 119905) asymp sum

j[119886 (i j 119905) minus ⟨119886 (i j 119905)⟩] 120595 (j 119905) (30)

Therefore

119861irr (119894 119895) = 4119905

2119865 (119894 119895) (1 minus 120590

1) (31)

where we defined

119865 (119894 119895) = ⟨119877 [120590

120583120575119899

120583(119894) 119888

120572

(119894) 119888

dagger120572(119895) 120575119899

120582(119895) 120590

120582]⟩ (32)

with 120575119899120583(119894) = 119899

120583(119894) minus ⟨119899

120583(119894)⟩ The self-energy (20) is written

as

Σ (k 120596) = 4119905

2119865 (k 120596)(

119868

minus2

11minus119868

minus1

11119868

minus1

22

minus119868

minus1

11119868

minus1

22119868

minus2

22

) (33)

In order to calculate the retarded function 119865(119894 119895) first weuse the spectral theorem to express

119865 (119894 119895) =

119894

2120587

int

+infin

minusinfin

119889120596eminusi120596(119905119894minus119905119895)

times

1

2120587

int

+infin

minusinfin

119889120596

1015840 1 + eminus1205731205961015840

120596 minus 120596

1015840+ i120576

119862 (i minus j 1205961015840)

(34)

where 119862(i minus j 1205961015840) is the correlation function

119862 (119894 119895) = ⟨120590

120583120575119899

120583(119894) 119888

120572

(119894) 119888

dagger120572(119895) 120575119899

120582(119895) 120590

120582⟩

=

1

2120587

int119889120596eminusi120596(119905119894minus119905119895)119862 (i minus j 120596) (35)

Next we use the noncrossing approximation (NCA) andapproximate

⟨120590

120583120575119899

120583(119894) 119888

120572

(119894) 119888

dagger120572(119895) 120575119899

120582(119895) 120590

120582⟩

asymp ⟨120575119899120583(119894) 120575119899

120583(119895)⟩ ⟨119888

120572

(119894) 119888

dagger120572(119895)⟩

(36)


By means of this decoupling and using again the spectraltheorem we finally have

119865 (k 120596) = 1

120587

int

+infin

minusinfin

119889120596

1015840 1

120596 minus 120596

1015840+ i120575

119886

2

(2120587)

3int119889

2119901119889Ω120572

2(119901)

times [tanh120573Ω

2

+ coth120573 (120596

1015840minus Ω)

2

]

timesI [119866119888119888(p Ω)]I [120594 (k minus p 1205961015840

minus Ω)]

(37)

where119866119888119888(k 120596) is the retarded electronicGreenrsquos function [cf

(13)]

119866119888119888(k 120596) =

2

sum

119886119887=1

119866119886119887(k 120596) (38)

120594 (k 120596) = sum

120583

F ⟨119877 [120575119899120583(119894) 120575119899

120583(119895)]⟩ (39)

is the total charge and spin dynamical susceptibility Theresult (37) shows that the calculation of the self-energyrequires the knowledge of the bosonic propagator (39) Thisproblem will be considered in the following section

It is worth noting that the NCA can also be applied tothe casual propagators giving the same result In general theknowledge of the self-energy requires the calculation of thehigher-order propagator 119861119876

(119894 119895) = ⟨Q[120575119869(119894)120575119869dagger(119895)]⟩ where119876 can be 119877 (retarded propagator) o 119879 (causal propagator)Then typically we have to calculate propagator of the form

119867

119877(119894 119895) = ⟨R [119861 (119894) 119865 (119894) 119865

dagger(119895) 119861

dagger(119895)]⟩ (40)

where119865(119894) and119861(119894) are fermionic and bosonic field operatorsrespectively By means of the spectral representation we canwrite

119867

119877

(k 120596)

= minus

1

120587

int

+infin

minusinfin

119889120596

1015840 1

120596 minus 120596

1015840+ i120575

coth120573120596

1015840

2

I [119867

119862(k 1205961015840

)]

(41)

where 119867119862(119894 119895) = ⟨T[119861(119894)119865(119894)119865

dagger(119895)119861

dagger(119895)]⟩ is the causal pro-

pagator In the NCA we approximate

119867

119862(119894 119895) asymp 119891

119862(119894 119895) 119887

119862(119894 119895)

119891

119862(119894 119895) = ⟨T [119865 (119894) 119865

dagger(119895)]⟩

119887

119862(119894 119895) = ⟨T [119861 (119894) 119861

dagger(119895)]⟩

(42)

Then we can use the spectral representation to obtain

119891

119862

(k 120596) = minus

1

120587

int

+infin

minusinfin

119889120596

1015840[

1 minus 119891F (1205961015840)

120596 minus 120596

1015840+ i120575

+

119891F (1205961015840)

120596 minus 120596

1015840minus i120575

]

timesI [119891

119877(k 1205961015840

)]

119887

119862

(k 120596) = minus

1

120587

int

+infin

minusinfin

119889120596

1015840[

1 + 119891119861(120596

1015840)

120596 minus 120596

1015840+ i120575

minus

119891119861(120596

1015840)

120596 minus 120596

1015840minus i120575

]

timesI [119887

119877(k 1205961015840

)]

(43)

which leads to

119867

119877

(k 120596) = 1

120587

int

+infin

minusinfin

119889120596

1015840 1

120596 minus 120596

1015840+ i120575

119886

119889

(2120587)

119889+1

times int

Ω119861

119889

119889119901119889ΩI [119891

119877(119901 Ω)]

timesI [119887

119877(119896 minus 119901 120596

1015840minus Ω)]

times [tanh120573Ω

2

+ coth120573 (120596

1015840minus Ω)

2

]

(44)

It is worth noting that up to this point the system ofequations for the Greenrsquos function and the anomalous self-energy is similar to the one derived in the two-particle self-consistent approach (TPSC) [86 87 160] the DMFT + Σ

approach [163ndash167] and a Mori-like approach by Plakidaand coworkers [6 7 9] It would be the way to compute thedynamical spin and charge susceptibilities to be completelydifferent as instead of relying on a phenomenological modeland neglecting the charge susceptibility as these approchesdo we will use a self-consistent two-pole approximationObviously a proper description of the spin and chargedynamics would definitely require the inclusion of a properself-energy term in the charge and spin propagators too inorder to go beyond both any phenomenological approch andthe two-pole approximation (in preparation) On the otherhand the description of the electronic anomalous featurescould (actuallywill see in the following) not need this furtherand definitely not trivial complication

25 Dynamical Susceptibility In this section we will presenta calculation of the charge-charge and spin-spin propagators(39) within the two-pole approximation This approximationhas shown to be capable of catching correctly some of thephysical features of Hubbard model dynamics (for all detailssee [40])

Let us define the composite bosonic field

119873

(120583)

(119894) = (

119899120583(119894)

120588120583(119894)

)

119899120583(119894) = 119888

dagger

(119894) 120590120583119888 (119894)

120588120583(119894) = 119888

dagger

(119894) 120590120583119888

120572

(119894) minus 119888

dagger120572

(119894) 120590120583119888 (119894)

(45)


This field satisfies the Heisenberg equation

i 120597120597119905

119873

(120583)

(119894) = 119869

(120583)

(119894) = (

119869

(120583)

1(119894)

119869

(120583)

2(119894)

)

119869

(120583)

1(119894) = minus4119905120588

120583(119894)

119869

(120583)

2(119894) = 119880120581

120583(119894) minus 4119905119897

120583(119894)

(46)

where the higher-order composite fields 120581120583(119894) and 119897

120583(119894) are

defined as

120581120583(119894) = 119888

dagger

(119894) 120590120583120578

120572

(119894) minus 120578

dagger

(119894) 120590120583119888

120572

(119894)

+ 120578

dagger120572

(119894) 120590120583119888 (119894) minus 119888

dagger120572

(119894) 120590120583120578 (119894)

119897120583(119894) = 119888

dagger

(119894) 120590120583119888

1205722

(119894) + 119888

dagger1205722

(119894) 120590120583119888 (119894)

minus 2119888

dagger120572

(119894) 120590120583119888

120572

(119894)

(47)

and we are using the notation

119888

1205722

(i 119905) = sum

j120572

2

ij119888 (j 119905) = sum

jl120572il120572lj119888 (j 119905) (48)

We linearize the equation of motion (46) for the compos-ite field 119873

(120583)(119894) by using the same criterion as in Section 22

(ie the neglected residual current 120575119869(120583)(119894) is orthogonal tothe chosen basis (45))

i 120597120597119905

119873

(120583)

(i 119905) = sum

j120576

(120583)(i j)119873(120583)

(j 119905) (49)

where the energy matrix is given by

119898

(120583)(i j) = sum

l120576

(120583)

(i l) 119868(120583) (l j) (50)

and the normalization matrix 119868

(120583) and the 119898(120583) matrix havethe following definitions

119868

(120583)(i j) = ⟨[119873

(120583)

(i 119905) 119873(120583)dagger(j 119905)]⟩

119898

(120583)(i j) = ⟨[119869

(120583)

(i 119905) 119873(120583)dagger(j 119905)]⟩

(51)

As it can be easily verified in the paramagnetic phase thenormalization matrix 119868

(120583) does not depend on the index 120583charge and spin operators have the same weight The twomatrices 119868(120583) and119898(120583) have the following form inmomentumspace

119868

(120583)

(k) = (

0 119868

(120583)

12(k)

119868

(120583)

12(k) 0

)

119898

(120583)

(k) = (

119898

(120583)

11(k) 0

0 119898

(120583)

22(k)

)

(52)

where

119868

(120583)

12(k) = 4 [1 minus 120572 (k)] 119862120572

119888119888

119898

(120583)

11(k) = minus 4119905119868

(120583)

12(k)

119898

(120583)

22(k) = minus 4119905119868

119897120583120588120583(k) + 119880119868

120581120583120588120583(k)

(53)

The parameter 119862120572 is the electronic correlation function119862

120572= ⟨119888

120572(119894)119888

dagger(119894)⟩ The quantities 119868

119897120583120588120583(k) and 119868

120581120583120588120583(k) are

defined as

119868119897120583120588120583

(k) = F ⟨[119897120583(i 119905) 120588dagger

120583(j 119905)]⟩

119868120581120583120588120583

(k) = F ⟨[120581120583(i 119905) 120588dagger

120583(j 119905)]⟩

(54)

Let us define the causal Greenrsquos function (for bosonic-likefields we have to compute the casual Greenrsquos function anddeduce from this latter the retarded one according to theprescriptions in [33ndash35])

119866

(120583)(119894 119895) = ⟨119879 [119873

(120583)

(119894)119873

(120583)dagger(119895)]⟩

=

i1198862

(2120587)

3int119889

2119896119889120596eiksdot(RiminusRj)minusi120596(119905119894minus119905119895)

119866

(120583)

(k 120596)

(55)

By means of the equation of motion (49) the Fourier trans-form of 119866(120583)

(119894 119895) satisfies the following equation

[120596 minus 120576

(120583)

(k)] 119866(120583)

(k 120596) = 119868

(120583)

(k) (56)

where the energy matrix has the explicit form

120576

(120583)

(k) = (

0 120576

(120583)

12(k)

120576

(120583)

21(k) 0

)

120576

(120583)

12(k) = minus4119905

120576

(120583)

21(k) =

119898

(120583)

22(k)

119868

(120583)

12(k)

(57)

The solution of (56) is

119866

(120583)

(k 120596)

= Γ

(120583)

(k) [ 1

120596 + i120575minus

1

120596 minus i120575]

+

2

sum

119899=1

120590

(119899120583)

(k) [1 + 119891B (120596)

120596 minus 120596

(120583)

119899 (k) + i120575minus

119891B (120596)

120596 minus 120596

(120583)

119899 (k) minus i120575]

(58)


where Γ(120583)(k) is the zero frequency function (2 times 2 matrix)[33ndash35] and119891B(120596) = [e120573120596minus1]minus1 is the Bose distribution func-tion Correspondingly the correlation function 119862

(120583)(k 120596) =

⟨119873

(120583)(119894)119873

(120583)dagger(119895)⟩ has the expression

119862

(120583)

(k 120596) = 2120587Γ

(120583)

(k) 120575 (120596)

+ 2120587

2

sum

119899=1

120575 [120596 minus 120596

(120583)

119899(k)] [1 + 119891B (120596)] 120590

(119899120583)

(k)

(59)

The energy spectra 120596(120583)

119899(k) are given by

120596

(120583)

119899(k) = (minus)

119899120596

(120583)

(k)

120596

(120583)

(k) = radic120576

(120583)

12(k) 120576(120583)

21(k)

(60)

and the spectral functions120590(119899120583)(k) have the following expres-sion

120590

(119899120583)

(k) =119868

(120583)

12(k)2

(

120576

(120583)

12(k)

120596

(120583)

119899 (k)1

1

120576

(120583)

21(k)

120596

(120583)

119899 (k)

) (61)

Straightforward but lengthy calculations (see [40]) give forthe 2D system the following expressions for the commutatorsin (54)

119868119897120583120588120583

(k) = 3

4

[1 minus 120572 (k)] (12119862120572+ 119862

120582+ 6119862

120583)

minus 3 [1 minus 120573 (k)] (119862120572+ 119862

120583)

minus

3

4

[1 minus 120578 (k)] (119862120572+ 119862

120582+ 2119862

120583)

+

1

4

[1 minus 120582 (k)] 119862120582+

3

2

[1 minus 120583 (k)] 119862120583

119868120581120583120588120583

(k) = minus2 [1 minus 120572 (k)] 119863 + [1 minus 2120572 (k)] (2119864120573+ 119864

120578)

+ 2120573 (k) 119864120573+ 120578 (k) 119864120578

+ [1 minus 2120572 (k)] 119886120583

+

1

4

[119887120583+ 2120573 (k) 119888

120583+ 120578 (k) 119889

120583]

(62)

where 120572(k) 120573(k) 120578(k) 120583(k) and 120582(k) are the Fouriertransforms of the projectors on the first second third fourth

and fifth nearest-neighbor sitesThe parameters appearing in(62) are defined by

119864 = ⟨119888 (119894) 120578

dagger

(119894)⟩ 119862

120572= ⟨119888

120572

(119894) 119888

dagger

(119894)⟩

119864

120573= ⟨119888

120573

(119894) 120578

dagger

(119894)⟩ 119862

120582= ⟨119888

120582

(119894) 119888

dagger

(119894)⟩

119864

120578= ⟨119888

120578

(119894) 120578

dagger

(119894)⟩ 119862

120583= ⟨119888

120583

(119894) 119888

dagger

(119894)⟩

(63)

119886120583= 2 ⟨119888

dagger

(119894) 120590120583119888

120572

(119894) 119888

dagger

(119894) 120590120583119888

120572

(119894)⟩

minus ⟨119888

dagger120572

(119894) 120590120583120590

120582120590120583119888

120572

(119894) 119899120582(119894)⟩

119887120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger

(119894) 120590120583[119888 (119894) 119888 (119894)]

120572⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894) 119899

120572

120582(119894)⟩

119888120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger(119894

120578) 120590

120583119888 (119894

120572) 119888 (119894

120572)⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894

120578) 119899

120582(119894

120572)⟩

119889120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger(119894

120573) 120590

120583119888 (119894

120572) 119888 (119894

120572)⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894

120573) 119899

120582(119894

120572)⟩

(64)

where we used the notation

119894 = (119894119909 119894119910 119905) 119894

120573= (119894

119909+ 119886 119894

119910+ 119886 119905)

119894

120572= (119894

119909+ 119886 119894

119910 119905) 119894

120578= (119894

119909+ 2119886 119894

119910 119905)

(65)

We see that the bosonic Greenrsquos function 119866

(120583)(119894 119895) =

⟨119879[119873

(120583)(119894)119873

(120583)dagger(119895)]⟩ depends on the following set of param-

eters Fermionic correlators 119862120572 119862120582 119862120583 119864120573 119864120578 and 119863bosonic correlators 119886

120583 119887

120583 119888

120583 and 119889

120583 zero frequency matrix

Γ

(120583)(k) The fermionic parameters are calculated through

the Fermionic correlation function 119862(119894 119895) = ⟨120595(119894)120595

dagger(119895)⟩

The bosonic parameters are determined through symmetryrequirements In particular the requirement that the conti-nuity equation should be satisfied and that the susceptibilityshould be a single-value function at k = 0 leads to thefollowing equations

119887120583= 119886

120583+ 3119863 + 2119864

120573+ 119864

120578minus 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

119888120583= 119886

120583minus 119863 minus 2119864

120573+ 119864

120578+ 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

119889120583= 119886

120583minus 119863 + 2119864

120573minus 3119864

120578minus 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

(66)

The remaining parameters 119886120583and Γ

(120583)

11(k) are fixed by

means of the Pauli principle

⟨119899120583(119894) 119899

120583(119894)⟩ =

119899 + 2119863 for 120583 = 0

119899 minus 2119863 for 120583 = 1 2 3

(67)


B0(ac0 as0)

G0(1205830 Δ0 p0)

n = 0

n = n + 1

Yes No

Σn[Bn Gn]

Gn(120583n Δn pn)[Σnminus1] Bn(acn asn)

Self-consistency 120583nminus1 Δnminus1 pnminus1

120583n Δn pn ==

Figure 1 (Color online) Self-consistency scheme to compute the propagator119866 in terms of the charge-charge and spin-spin propagator 119861 andthe residual self-energy Σ

where 119863 = ⟨119899uarr(119894)119899

darr(119894)⟩ is the double occupancy and by the

ergodic value

Γ

(120583)

11(k) = 120575

1205830

(2120587)

2

119886

2120575

(2)

(k) ⟨119899⟩2 (68)

By putting (68) into (58) and (59) we obtain

⟨120575119899120583(119894) 120575119899

120583(119895)⟩

=

119886

2

2 (2120587)

2

2

sum

119899=1

int119889

2119896eiksdot(RiminusRj)minusi120596

(120583)

119899(k)(119905119894minus119905119895)

times [1 + coth120596

(120583)

119899(k)

2119896B119879]120590

(119899120583)

11(k)

(69)

⟨119877 [120575119899120583(119894) 120575119899

120583(119895)]⟩

=

i1198862

(2120587)

3

2

sum

119899=1

int119889


(70)

times

120590

(119899120583)

11(k)

120596 minus 120596

(120583)

119899 (k) + i120575 (71)

In conclusion the dynamical susceptibility 120594120583(k 120596)

which is independent of Γ(120583) by construction reads as

120594120583(k 120596) = minusF [⟨119877 [120575119899

120583(119894) 120575119899

120583(119895)]⟩]

=

16119905 [1 minus 120572 (k)] 119862120572

120596

2minus (120596

(120583)(k))2

(72)

where no summation is implied on the 120583 index It is reallyworth noticing the very good agreement between the resultswe obtained within this framework for the charge and spindynamics of the Hubbard model and the related numericalones present in the literature (see [40])

26 Self-Consistency In this section we will give a sketch ofthe procedure used to calculate the Greenrsquos function 119866(k 120596)The starting point is (19) where the two matrices 119868(k) and120576(k) are computed by means of the expressions (23) Theenergy matrix 120576(k) depends on three parameters 120583 Δ and119901 To determine these parameters we use the following set ofalgebra constraints

119899 = 2 (1 minus 11986211minus 119862

22)

Δ = 119862

120572

11minus 119862

120572

22

11986212

= ⟨120585 (119894) 120578

dagger

(119894)⟩ = 0

(73)

where 119862119899119898

and 119862

120572

119899119898are the time-independent correlation

functions 119862119899119898

= ⟨120595119899(119894)120595

dagger

119898(119894)⟩ and 119862

120572

119899119898= ⟨120595

120572

119899(119894)120595

dagger

119898(119894)⟩

To calculate Σ(k 120596) we use the NCA the results given inSection 24 show that within this approximation Σ(k 120596) isexpressed in terms of the fermionic 119866

119888119888(k 120596) [cf (38)] and

of the bosonic 120594120583(k 120596) [cf (39)] propagators The bosonic

propagator is calculated within the two-pole approximationusing the expression (70) As shown in Section 25 120594

120583(k 120596)

depends on both electronic correlation functions [see (63)]which can be straightforwardly computed from 119866(k 120596) andbosonic correlation functions one per each channel (chargeand spin) 119886

0and 119886

3The latter are determined bymeans of the

local algebra constraints (67) where 119899 is the filling and 119863 isthe double occupancy determined in terms of the electroniccorrelation function as119863 = 1198992 minus 119862

22

According to this the electronic Greenrsquos function119866(k 120596)is computed through the self-consistency scheme depicted inFigure 1 we first compute 1198660

(k 120596) and 120594120583(k 120596) in the two-

pole approximation then Σ(k 120596) and consequently 119866(k 120596)Finally we check how much the fermionic parameters (120583 Δand 119901) changed and decide whether to stop or to continueby computing new 120594

120583(k 120596) andΣ(k 120596) after119866(k 120596) Usually

to get 6 digits precision for fermionic parameters we need 8full cycles to reach self-consistency on a 3D grid of 128 times 128


points in momentum space and 4096Matsubara frequenciesActually many more cycles (almost twice) are needed at lowdoping and low temperatures

Summarizing within the NCA and the two-pole approx-imation for the computation of 120594

120583(k 120596) we have constructed

an analytical completely self-consistent scheme of calcula-tion of the electronic propagator119866

119888119888(k 120596) = F⟨119879[119888(119894)119888

dagger(119895)]⟩

where dynamical contributions of the self-energy Σ(k 120596) areincluded All the internal parameters are self-consistentlycalculated by means of algebra constraints [cf (67)-(68)and (73)] No adjustable parameters or phenomenologicalexpressions are introduced

3 Results

In the following we analyze some electronic properties bycomputing the spectral function

119860 (k 120596) = minus

1

120587

I [119866119888119888(k 120596)] (74)

the momentum distribution function per spin

119899 (k) = int119889120596119891F (120596)119860 (k 120596) (75)

and the density of states per spin

119873(120596) =

1

(2120587)

2int119889

2119896119860 (k 120596) (76)

where 119866119888119888(k 120596) = 119866

11(k 120596) + 119866

12(k 120596) + 119866

21(k 120596) +

11986622(k 120596) is the electronic propagator and 119891F(120596) is the Fermi

function We also study the electronic self-energy Σ119888119888(k 120596)

which is defined through the equation

119866119888119888(k 120596) = 1

120596 minus 1205980(k) minus Σ

119888119888(k 120596)

(77)

where 1205980(k) = minus120583 minus 4119905120572(k) is the noninteracting dispersion

Moreover we define the quantity 119903(k) = 1205980(k) + Σ

1015840

119888119888(k 120596 = 0)

that determines the Fermi surface locus inmomentum space119903(k) = 0 in a Fermi liquid that is when lim

120596rarr0Σ

10158401015840

119888119888(k 120596 119879 =

0) prop 120596

2 and lim119879rarr0

Σ

10158401015840

119888119888(k 120596 = 0 119879) prop 119879

2 The actualFermi surface (or its relic in a non-Fermi-liquid) is givenby the relative maxima of 119860(k 120596 = 0) which takes intoaccount at the same time and on equal footing both thereal and the imaginary parts of the self-energy and is directlyrelated within the sudden approximation and forgetting anyselection rules to what ARPES effectively measures

Finally the spin-spin correlation function ⟨119899119911119899

120572

119911⟩ the

pole120596(3)(k = Q = (120587 120587)) of the spin-spin propagator and the

antiferromagnetic correlation length 120585 are discussed Usuallythe latter is defined by supposing the following asymptoticexpression for the static susceptibility

limkrarrQ

120594

(3)

(k 0) =120594

(3)(Q 0)

1 + 120585

2|k minusQ|

2 (78)

where 120594(3)(k 0) = minus119866

(3)(k 0) It is worth noting that in our

case (78) is not assumed but it exactly holds [189]

31 Spectral Function andDispersion According to its overallrelevance in thewhole analysis performed hereinafter we firstdiscuss the electronic dispersion of the model under analysisor better its relic in a strongly correlated system In generalthe dispersion and its more or less anomalous features canbe inferred by looking at the maxima of the spectral function119860(k 120596) In Figures 2 and 3 the spectral function is shown inscale of grays (increasing fromwhite to black red is for above-scale values) along the principal directions (Γ = (0 0) rarr

119872 = (120587 120587) 119872 rarr 119883 = (120587 0) 119883 rarr 119884 = (0 120587) and 119884 rarr Γ)for 119880 = 8 119879 = 001 and (a) 119899 = 070 (b) 119899 = 078 (c)119899 = 085 and (d) 119899 = 092 (119879 = 002) In Figure 2 thewhole range of frequencies with finite values of 119860(k 120596) isreported while in Figure 3 a zoom in the proximity of thechemical potential is shownThe light gray lines and uniformareas are labeled with the values of the imaginary part of theself-energy Σ

10158401015840(k 120596) and give one of the most relevant keys

to interpret the characteristics of the dispersion The darkgreen lines in Figure 3 are just guides to the eye and indicatethe direction of the dispersion just before the visible kinkseparating the black and the red areas of the dispersion

The red areas as they mark the relative maxima of119860(k 120596) can be considered as the best possible estimates forthe dispersion In a non- (or weakly-) interacting system thedispersion would be a single continuos and (quite-) sharpline representing some function 120576 (k) being the simple poleof 119866

119888119888(k 120596) In this case (for a strongly correlated system)

instead we can clearly see that the dispersion is well-defined(red areas) only in some of the regions it crosses in the (k 120596)plane the regions where Σ10158401015840

(k 120596) is zero or almost negligibleIn the crossed regionswhereΣ10158401015840

(k 120596) is instead finite119860(k 120596)obviously assumes very low values whichwould be extremelydifficult (actually almost impossible) to detect by ARPESAccordingly ARPES would report only the red areas in thepicture This fact is fundamental to understand and describethe experimental findings regarding the Fermi surface inthe underdoped regime as they will be discussed in thenext section and to reconcile ARPES findings with those ofquantum oscillations measurements

The two Hubbard subbands separated by a gap of theorder 119880 and with a reduced bandwidth of order 4119905 areclearly visible the lower one (LHB) is partly occupied asit is crossed by the chemical potential and the upper one(UHB) is empty and very far from the chemical potentialThe lower subband systematically (for each value of doping)loses significance close to Γ and to 119872 although this effectis more and more pronounced on reducing doping In bothcases (close to Γ and to 119872) 119860(k 120596) loses weight as Σ10158401015840

(k 120596)increases its own this can be easily understood if we recallthat both 120594

0(k 120596) and 120594

3(k 120596) have a vanishing pole at

Γ (due to hydrodynamics) and that 1205943(k 120596) is strongly

peaked at119872 due to the strong antiferromagnetic correlationspresent in the system (see Section 36) The upper subbandaccording to the complementary effect induced by the evidentshadow-bands appearance (signaled by the relative maximaof Σ10158401015840

(k 120596) marked by 100 and the dark gray area inside thegap) due again to the strong antiferromagnetic correlationspresent in the system displays awell-defined dispersion at119872at least for high enough doping and practically no dispersion


9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 07

T = 001

Σ998400998400(k 120596)

(a)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 078

T = 001

Σ998400998400(k 120596)

(b)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596A(k120596)

00

06

12

18

24

30

U = 8n = 085

T = 001

Σ998400998400(k 120596)

(c)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 092

T = 002

Σ998400998400(k 120596)

(d)

Figure 2 (Color online) Spectral function119860(k 120596) along the principal directions (Γ = (0 0) rarr 119872 = (120587 120587)119872 rarr 119883 = (120587 0)119883 rarr 119884 = (0 120587)

and 119884 rarr Γ) for 119880 = 8 119879 = 001 and (a) 119899 = 070 (b) 119899 = 078 (c) 119899 = 085 and (d) 119899 = 092 (119879 = 002)

at all close to Γ For low doping the great majority of theupper subband weight is simply transferred to the lowersubband The growth on reducing doping of the undefined-dispersion regions close to Γ in the lower subband cuts downits already reduced bandwidth of order 4119905 to values of theorder 119869 = 4119905

2119880 = 05119905 as one would expect for the disper-

sion of few holes in a strong antiferromagnetic backgroundThe shape of the dispersion is too compatible with thisscenario the sequence of minima and maxima is compatibleactually driven by the doubling of the Brillouin zone inducedby the strong antiferromagnetic correlations as well as thedynamical generation of a 1199051015840 diagonal hopping (absent in theHamiltonian currently under study) clearly signaled by themore andmore on reducing doping pronounced warping ofthe dispersion along the 119883 rarr 119884 direction (more evident inFigure 3) whichwould be perfectly flat otherwise (for 1199051015840 = 0)

Moving to the zooms (Figure 3) they show much moreclearly the systematic reduction of the bandwidth on reducingdoping the doubling of the zone the systematic increase ofthe warping along 119883 rarr 119884 on reducing doping and theextreme flatness of the dispersion at 119883 coming from bothΓ and 119872 This latter feature is in very good agreement withquantum Monte Carlo calculations (see [159] and referencestherein) as well as with ARPES experiments [190] whichreport a similar behavior in the overdoped region Moreoverthey show that in contrast with the scenario for a non-

(or weakly-) interacting system where the doubling of thezone happens with the 119883 rarr 119884 direction as pivot herethe doubling is confined to the region close to 119872 Thelower subband is completely filled at half-filling while for anordinary Slater antiferromagnet the gap opens at half-fillingjust on top of the van-Hove singularity along 119883 rarr 119884 Inaddition the effective finite value of 1199051015840 makes the dispersionmaximum close to 119878 higher than the one present along the119872 rarr 119883 direction opening the possibility for the appearanceof hole pockets close to 119878 Finally it is now evident that thewarping of the dispersion along 119883 rarr 119884 will also inducethe presence of two maxima in the density of states one dueto the van-Hove singularities at 119883 and 119884 and one due tothe maximum in the dispersion close to 119878 How deep is thedip between these two maxima just depends on the numberof available well-defined (red) states in momentum presentbetween these two values of frequency This will determinethe appearance of a more or less pronounced pseudogap inthe density of states but let us come back to this after havinganalyzed the region close to119872

As a matter of fact it is just the absence of spectralweight in the region close to 119872 and in particular and moresurprisingly at the chemical potential (ie on the Fermisurface in contradiction with the Fermi-liquid picture) themain and more relevant result of this analysis it will deter-mine almost all interesting and anomalousunconventional


050

025

000

minus025

minus050

minus075

minus100

minus125

minus150Γ S M X S Y Γ

k

00

10

20

30

40

50

U = 8n = 07

T = 001

A(k120596)

Σ998400998400(k 120596)

120596

(a)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 078

T = 001

120596

(b)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 085

T = 001

120596

(c)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 092

T = 002

120596

(d)

Figure 3 (Color online) Spectral function 119860(k 120596) close to the chemical potential (120596 = 0) along the principal directions (Γ = (0 0) rarr 119872 =

(120587 120587) 119872 rarr 119883 = (120587 0) 119883 rarr 119884 = (0 120587) and 119884 rarr Γ) for 119880 = 8 119879 = 001 and (a) 119899 = 070 (b) 119899 = 078 (c) 119899 = 085 and (d) 119899 = 092

(119879 = 002)

features of the single-particle properties of the model Thescenario emerging from this analysis can be relevant notonly for the understanding of the physics of the Hubbardmodel and for the microscopical description of the cupratehigh-119879

119888superconductors but also for the drafting of a

general microscopic theory of strongly correlated systemsThe strong antiferromagnetic correlations (through Σ10158401015840

(k 120596)which mainly follows 120594

3(k 120596)) cause a significative and

anomalousunconventional loss of spectral weight around119872which induces in turn the deconstruction of the Fermi surface(Section 32) the emergence of momentum selective non-Fermi-liquid features (Sections 34 and 35) and the openingof a well-developed (deep) pseudogap in the density of states(Section 33)

Last but not least it is also remarkable the presence ofkinks in the dispersion in both the nodal (Γ rarr 119872) and theantinodal (119883 rarr Γ) directions as highlighted by the darkgreen guidelines in qualitative agreement with some ARPESexperiments [82] Such a phenomenon clearly signals thecoupling of the electrons to a bosonic mode In this scenariothe mode is clearly magnetic in nature The frequency ofthe kink with respect to the chemical potential reducessystematically and quite drastically on reducing doping fol-lowing the behavior of the pole of 120594

3(k 120596) (see Section 36)

Combining the presence of kinks and the strong reduction

of spectral weight below them we see the appearance ofwaterfalls in particular along the antinodal (119883 rarr Γ)direction as found in some ARPES experiments [82] Finallythe extension of the flat region in the dispersion aroundthe antinodal points (119883 and 119884) that is at the van-Hovepoints increases systematically on decreasing doping Thisclearly signals the transfer of spectral weight from the Fermisurface which is depleted by the strong antiferromagneticfluctuations which are also responsible for the remarkableflatness of the band edge

Beforemoving to the next section it is worth noticing thatsimilar results for the single-particle excitation spectrum (flatbands close to 119883 weight transfer from the LHB to the UHBat119872 and splitting of the band close to 119883 ) were obtainedwithin the self-consistent projection operator method [2829] the operator projection method [30ndash32] and within aMori-like approach by Plakida and coworkers [7 9]

32 Spectral Function and Fermi Surface Focusing on thevalue of the spectral function at the chemical potential119860(k 120596 = 0) we can discuss the closest concept to Fermisurface available in a strongly correlated system In Figure 4119860(k 120596 = 0) is plotted as a function of the momentum k ina quarter of the Brillouin zone for 119880 = 8 and four differentcouples of values of temperature and filling (a) 119899 = 07 and


00 20 40 60 80 100 120 120

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

5120587831205874

120587

120587

120587

120587

U = 8

T = 001

n = 070

A(k120596

=0)

31205878

31205878

31205878

51205878

51205878

51205878

31205874 31205874

31205874

31205874

71205878

71205878

71205878

71205878

000

0

12058781205878

4

120587444444444444444444444

1205874

1205874

1205872

1205871205871205871205871205871205871205871205871205871205872222222222222222222222222222222222222

1205872

1205872

55555555555555555555555555555555555555555555555555555555555555555552222212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444

120587

1205878

31205878

333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

51205878

51205878

51205878

31205874 31205874

3120587

33333333333333333333333331205871205871205871205871205871205871205871205871205874

71205878

71205878

7

74 1205878

kx

ky

(a)

00 20 40 60 80 100

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

3120587471205878

120587

120587

120587

120587

U = 8

T = 001

n = 078

kx

ky

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

3120587851205878

31205878

51205878

31205874

71205878

0

0

12058781205878

4

1205874 1205874

1205872

1205872

1205872

1205872

33333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444444444444444444447777777777777777777777777777777777777777777777777777777712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333333333333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

512058783120587

7

5120587831205874 71205878

3120587852212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888

1205878

51205878

31205874

71205878

(b)

00 10 20 30 40 50 60 70 80 90

00

0

0

123456789

1011

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

71205878

120587

120587

120587

120587

U = 8

T = 001n = 085

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

312058785120587831205874

00

000000000000000000000000000000000000000000

12058781205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

4

1205874 1205874

1205872

1205872

1205872

1205872

777777777777777777777777777777777777777777777777777777777771205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333331205878

512058783120587

7

3120587851205878

31205874 71205878

8

51205878

31205874

71205878

3120587852212058783333333333312058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058744444444444444444444444444444444444444444444444

kx

ky

(c)

00 10 20 30 40 50 60 70

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

120587

120587

120587

120587

U = 8

T = 002n = 092

A(k120596

=0)

31205878

5120587831205874

71205878

71205878

312058785120587831205874

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

0000

0

12058781205878

4

1205874

1205874

1205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874

1205872

1205872

1205872

1205872

120587

333331205878

512058783120587

7

77777777777777777777777120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587888888888888888888888888888888888888888

31205878522120587831205874

3120587851205878

31205874 71205878

1205878

51205878

31205874

71205878

kx

ky

(d)

Figure 4 (Color online) Spectral function at the chemical potential 119860(k 120596 = 0) as a function of momentum k for 119880 = 8 119879 = 001 and (a)119899 = 070 (b) 119899 = 078 (c) 119899 = 085 and (d) 119899 = 092 (119879 = 002)


119879 = 001 (b) 119899 = 078 and 119879 = 001 (c) 119899 = 085 and119879 = 001 and (d) 119899 = 092 and 119879 = 002 The Fermi surfacein agreement with the interpretation of the ARPES mea-surements within the sudden approximation can be definedas the locus in momentum space of the relative maxima of119860(k 120596 = 0) Such a definition opens up the possibility of notonly explainingARPESmeasurements but also going beyondthem and their finite instrumental resolution and sensitivitywith the aim at filling the gap with other kinds of measure-ments in particular quantum oscillations ones which seemsto report results in disagreement up to dichotomy in somecases with the scenario depicted by ARPES

For each value of the filling reported we can easilydistinguish two wallsarcs for 119899 = 092 they somewhat joinFirst let us focus on the arc with the larger (by far) intensitiesgiven the current sensitivities this is the only one among thetwo possibly visible to ARPES At 119899 = 07 (see Figure 4(a))the 3Dperspective allows to better appreciate the difference inthe intensities of the relative maxima of 119860(k 120596 = 0) betweenthe region close to the main diagonal (119872 rarr 119883) where thesignal is weaker and the regions close to the main axes (Γ rarr

119883 and Γ rarr 119884) where the signal is stronger this behaviorhas been also reported by ARPES experiments [82 190] aswell as the electron-like nature of the Fermi surface [190]On decreasing doping this trend reverses passing through119899 = 078 where the intensities almost match and up to119899 = 092 where the region in proximity of 119878 is the onlyone with an appreciable signal The less-intense (by far) arcreported in [7] too is the relic of a shadow band as canbe clearly seen in Figure 3 and consequently never changesits curvature in contrast to what happens to the other arcwhich is subject to the crossing of the van Hove singularity(119899 cong 082) instead Although the ratio between themaximumvalues of the intensities at the two arcs never goes belowtwo (see Figure 6(b)) there is an evident decrease of themaximum value of the intensity at the larger-intensity arc ondecreasing doping which clearly signals an overall increase ofthe intensity andor of the effectiveness (in terms of capabilityof affecting the relevant quasi-particles which are those at theFermi surface) of the correlations

Moving to a 2D perspective (see Figure 5) we can addthree ingredients to our discussion that can help us betterunderstand the evolution with doping of the Fermi surface(i) the 119899(k) = 05 locus (solid line) that is the Fermi surfaceif the system would be noninteracting (ii) the 119903(k) = 0 locus(dashed line) that is the Fermi surface if the system wouldbe a Fermi liquid or somewhat close to it conceptually (iii)the values (grey lines and labels) of the imaginary part of theself-energy at the chemical potentialΣ10158401015840

119888119888(k 120596 = 0) (notice that

119879 = 0) Combining these three ingredients with the positionsand intensities of the the relative maxima of 119860(k 120596 = 0) wecan try to better understand what these latter signify and toclassify the behavior of the system on changing doping At119899 = 07 (see Figure 5(a)) the positions of the two arcs areexactly matching 119903(k) = 0 lines this will stay valid at eachvalue of the filling reported with a fine but very relevantdistinction at 119899 = 092 This occurrence makes our definitionof Fermi surface robust but also versatile as it permits togo beyond Fermi liquid picture without contradicting this

latter The almost perfect coincidence for the higher valueof doping reported 119899 = 07 of the 119899(k) = 05 line with thelarger-intensity arc clearly asserts that we are dealing with avery-weakly-interacting Fermi metal Σ10158401015840

119888119888(k 120596 = 0) is quite

large close to 119872 (see Figure 3(a)) and eats up the weight ofthe second arc which becomes a ghost band more than ashadow oneThe antiferromagnetic correlations are definitelyfinite (see Section 36) and consequently lead to the doublingof the zone but not strong enough to affect the behaviorof the ordinary quasiparticles safely living at the ordinaryFermi surface Decreasing the doping we can witness a firsttopological transition from a Fermi surface closed aroundΓ (electron-like hole-like in cuprates language) to a Fermisurface closed around119872 (hole-like electron-like in cuprateslanguage) at 119899 cong 082 where the chemical potentialcrosses the van Hove singularity (see Figure 3) The chemicalpotential presents an inflection point at this doping (notshown) which allowed us to determine its value with greataccuracy In proximity of the antinodal points (119883 and 119884) anet discrepancy between the position of the relative maximaof 119860(k 120596 = 0) and the locus 119899(k) = 05 becomes moreandmore evident on decreasing doping (see Figures 5(b) and5(c)) This occurrence does not only allow the topologicaltransition which is absent for the 119899(k) = 05 locus thatreaches the antidiagonal (119883 rarr 119884) at half-filling in agreementwith the Luttinger theorem but also accounts for the apparentbroadening of the relative maxima of 119860(k 120596 = 0) closeto the antinodal points (119883 and 119884) The broadening is dueto the small but finite value of Σ10158401015840

119888119888(k 120596 = 0) in those

momentum regions (see Figures 3(b) and 3(c)) signaling thenet increase of the correlation strength and accordingly theimpossibility of describing the system in this regime as aconventional non- (or weakly-) interacting system within aFermi-liquid scenario or its ordinary extensions for orderedphases What is really interesting and goes beyond the actualproblem under analysis (cuprates 2D Hubbard model) is theemergence of such features only in well-defined regions inmomentum spaceThis selectiveness in momentum is quite anew feature in condensed matter physics and its understand-ing and description require quite new theoretical approachesIn this system almost independently from their effectivestrength the correlations play a so fundamental role to cometo shape and determine qualitatively the response of thesystemAccordingly any attempt to treat correlationswithouttaking into account the level of entanglement between alldegrees of freedom present in the system is bound to fail orat least to miss the most relevant features

Let us come to themost interesting result At 119899 = 092 therelative maxima of 119860(k 120596 = 0) detach also from the 119903(k) = 0

locus at least partially opening a completely new scenario119903(k) = 0 defines a pocket (close but absolutely not identicalread below to that of an antiferromagnet) while the relativemaxima of119860(k 120596 = 0) feature the very same pocket togetherwith quite broad but still well-definedwings closingwith onehalf of the pocket (the most intense one) what can be safelyconsidered the relic of a large Fermi surfaceThis is the secondand most surprising topological transition occurring to theactual Fermi surface the two arcs clearly visible for all othervalues of the filling join and instead of closing just a pocket as


00

25

50

75

100

125

r(k) = 0

n(k) = 05 U = 8

n = 070

T = 001

minus010 minus020

minus030

minus040minus050

Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

ky

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(a)

minus010

minus020minus030minus040

minus050

0

1205878

1205874

31205878

1205872

51205878

31205874

71205878

120587

0 1205878 1205874 31205878 1205872 51205878 31205874 71205878 120587

r(k) = 0

n(k) = 05

kx

k y

00

21

42

63

84

105

A(k120596

=0)

U = 8

n = 078

T = 001Σ998400998400cc(k 120596 = 0)

(b)

00

18

36

54

72

90

r(k) = 0

n(k) = 05 U = 8

n = 085

T = 001Σ998400998400cc(k 120596 = 0)

A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(c)

r(k) = 0

n(k) = 05 U = 8

n = 092

T = 002Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y00

14

28

42

56

70

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(d)

Figure 5 (Color online) Spectral function at the chemical potential 119860(k 120596 = 0) as a function of momentum k for 119880 = 8 119879 = 001 and(a) 119899 = 070 (b) 119899 = 078 (c) 119899 = 085 and (d) 119899 = 092 (119879 = 002) The solid line marks the locus 119899(k) = 05 the dashed line marks thelocus 119903(k) = 0 the gray lines are labeled with the values of Σ10158401015840

119888119888(k 120596 = 0) and the dotted line is a guide to the eye and marks the reduced

(antiferromagnetic) Brillouin zone

one would expect on the basis of the conventional theory foran antiferromagnetmdashhere mimed by 119903(k) = 0 locus develop(or keep) a completely independent branch The actualFermi surface is neither a pocket nor a large Fermi surfacefor a more expressive representation see Figure 4(d) Thisvery unexpected result can be connected to the dichotomybetween those experiments (eg ARPES) pointing to a smalland those ones (eg quantum oscillations) pointing to alarge Fermi surfaceThis result can be understood by lookingonce more at the dispersion for this value of the filling (see

Figure 3(d)) the difference in height induced by the effectivefinite value of 1199051015840 which increases with decreasing dopingbetween the twohighestmaxima in the dispersionmdashone closeto 119878 and the other along the 119872 rarr 119883 directionmdashmakes thelatter to cross the chemical potential for larger values of thedoping than the first but given the significative broadeningof the dispersion even when the center mass of the secondleaves the Fermi surface (disappearing from 119903(k) = 0

locus) its shoulders are still active and well-identifiable at thechemical potentialmdashthat is on the actual Fermi surface


00

02

04

06

08

10

12

n = 070

n = 078n = 085

n = 092

minus03 minus02 minus01 00 01 02 03

120596

N(120596)

U = 8 T = 001 U = 8 T = 002

(a)

1

2

3

4

5

6

7

8

0minus02 minus01 00 01 02

U = 8T = 002n = 092

A(k

=k120596)

120596

k = S

k = S

k = X

(b)

Figure 6 (Color online) (a) Density of states119873(120596) as a function of frequency 120596 for119880 = 8 (black squares) 119899 = 07 and 119879 = 001 (red circles)119899 = 078 and 119879 = 001 (blue up triangles) 119899 = 085 and 119879 = 001 and (green down triangles) 119899 = 092 and 119879 = 002 (b) Spectral functionin the proximity of the chemical potential 119860(k 120596 sim 0) at (black squares) k = 119878 = (1205872 1205872) (red circles) 119878 (in the text) and (blue triangles)119883 = (120587 0) for 119880 = 8 119899 = 092 and 119879 = 002

The pocket is too far from being conventional It is clearlyevident in Figure 5(d) that there are two distinct halvesof the pocket one with very high intensity pinned at 119878

(again the only possibly visible to ARPES) and another withvery low intensity (visible only to theoreticians and somequantum oscillations experiments)This is our interpretationfor the Fermi arcs reported by many ARPES experiments[82 93] and unaccountable for any ordinary theory relayingon the Fermi liquid picture though being modified by thepresence of an incipient spin or charge ordering Obviouslylooking only at the Fermi arc (as ARPES is forced to do)the Fermi surface looks ill defined as it does not enclose adefinite region of momentum space but having access alsoto the other half of the pocket such a problem is greatlyalleviatedThepoint is that the antiferromagnetic fluctuationsare so strong to destroy the coherence of the quasiparticlesin that region of momentum space as similarly reportedwithin the DMFT + Σ approach [163ndash167] and a Mori-like approach by Plakida and coworkers [7 9] Moreoverwe will see that (see Section 35) the phantom half is notsimply lower in intensity because it belongs to a shadowband depleted by a finite imaginary part of the self-energy(see Figure 5(d)) but that it lives in a region of momentumwhere the imaginary part of the self-energy shows clearsigns of non-Fermi-liquid behavior The lack of next-nearesthopping terms (ie 1199051015840 and 11990510158401015840) in the chosen Hamiltonian (1)although they are evidently generated dynamically does not

allow us to perform a quantitative comparison between ourresults and the experimental ones which refers to a specificmaterial characterized by a specific set of hoppings Thisalso explains why our Fermi arc is pinned at 119878 and occupiesthe outer reduced Brillouin zone while many experimentalresults that report a Fermi arc occupying the inner reducedBrillouin zone Actually the pinning (with respect to dopingwithin the underdoped region) of the center of mass of theARPES-visible Fermi arc has been reported also fromARPESexperiments [191]

33 Density of States and Pseudogap The other main issuein the underdoped regime of cuprates superconductors isthe presence of a quite strong depletion in the electronicdensity of states known as pseudogap In Figure 6(a) wereport the density of states119873(120596) for 119880 = 8 and four couplesof values of filling and temperature 119899 = 07 and 119879 =

001 119899 = 078 and 119879 = 001 119899 = 085 and 119879 = 001and 119899 = 092 and 119879 = 002 in the frequency region inproximity of the chemical potential As a reference we alsoreport in Figure 6(b) the spectral function in proximity ofthe chemical potential 119860(k 120596 sim 0) at k = 119878 = (1205872 1205872)k = 119878which lies where the phantom half of the pocket touchesthe main diagonal Γ rarr 119872 (ie where the dispersion cutsthe main diagonal Γ rarr 119872 closer to 119872) and k = 119883 =

(120587 0) for 119880 = 8 119899 = 092 and 119879 = 002 As it can beclearly seen in Figure 6(a) the density of states presents two


YX SS M

k

n(k)

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

n = 070

n = 075

n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

00

01

02

03

04

05

06

07

08

09

10

(a)

S M

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

kΓ

n = 070n = 075n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

(b)

Figure 7 (Color online)Momentum distribution function 119899(k) for119880 = 8 and 119899 = 07 075 08 119899 = 085 (119879 = 001) and 119899 = 092 (119879 = 002)(a) along the principal directions (Γ = (0 0) rarr 119872 = (120587 120587)119872 rarr 119883 = (120587 0) 119883 rarr 119884 = (0 120587) and 119884 rarr Γ) (b) along the principal diagonal(Γ rarr 119872)

maxima separated by a dip which plays the role of pseudogapin this scenario Its presence is due to the warping in thedispersion along the 119883 rarr 119884 direction (see Figure 3) whichinduces the presence of the two maxima (one due to thevan-Hove singularity at 119883 and one due to the maximum inthe dispersion close to 119878mdashsee Figure 6(b)) and to the lossof states within this window in frequency in the region inmomentum close to119872 as discussed in detail in the previoussections As a measure of how much weight is lost because ofthe finite value of the imaginary part of the self-energy in theregion in momentum close to119872 one can look at the strikingdifference between the value of 119860(119878 120596 = 0) in comparison tothat of 119860(119878 120596 = 0) see Figure 6(b) On reducing the dopingthere is an evident transfer of spectral weight between thetwo maxima in particular the weight is transferred from thetop of the dispersion close to 119878 to the antinodal point 119883where the van Hove singularity resides At the lowest doping(119899 = 092) a well-developed pseudogap is visible belowthe chemical potential and will clearly affect all measurableproperties of the system For this doping we do not observeany divergence ofΣ1015840

119888119888(k 120596 = 0) in contrast to what is reported

in [192] where this feature is presented as the ultimate reasonfor the opening of the pseudogap In our scenario thepseudogap is just the result of the transfer of weight fromthe single-particle density of states to the two-particle onerelated to the (antiferro) magnetic excitations developing inthe system on decreasing doping at low temperatures (seeSection 36)

It is worthmentioning that an analogous doping behaviorof the pseudogap has been found by the DMFT+Σ approach[163ndash167] a Mori-like approach by Plakida and coworkers [79] and the cluster perturbation theory [86 87 175 176]

34 Momentum Distribution Function To analyze a possiblecrossover from a Fermi liquid to a non-Fermi-liquid behaviorin certain regions of momentum space at small dopingsand low temperatures and to better characterize the pocketforming on the Fermi surface at the lowest reported dopingwe study the electronic momentum distribution function119899(k)per spin along the principal directions (Γ rarr 119872119872 rarr 119883119883 rarr 119884 and 119884 rarr Γ) and report it in Figure 7 for 119880 = 8

and 119899 = 07 075 08 and 085 (119879 = 001) and for 119899 =

092 (119879 = 002) Figure 7(b) reports a zoom along the maindiagonal (Γ rarr 119872) At the highest studied doping (119899 =

07) 119899(k) shows the usual features of a quasinoninteractingsystem except for one single but very important featurethe dip along the main diagonal (Γ rarr 119872) signaling thepresence of a shadow band due to the weak but anyway finiteantiferromagnetic correlations (see Section 36) as alreadydiscussed many times hereinbefore The quite small heightof the secondary jumps along 119878 rarr 119872 and 119872 rarr 119883

directions (with respect to the height of the main jumpsalong Γ rarr 119878 and 119884 rarr Γ directions) gives a measureof the relevance of this feature in the overall picture notreally much relevant except at 119899 = 092 where it changesqualitatively On increasing filling (reducing doping) thefeatures related to the shadow band do not change muchtheir positions and intensity while the features related to theordinary band change their positions as expected in orderto accommodate (ie to activate states in momentum for)the increasing number of particles Finally at 119899 = 092 thetwo sets of features close a pocket At this final stage what isvery relevant as it is very unconventional is the evident andremarkable difference in behavior between the main jumpalong the Γ rarr 119878 direction and the secondary jump along


09

0807 06

05

0403

0202

00

01

02

03

04

05

06

07

08

09

10

n(k)

U = 8n = 092T = 001

Y

X

M

Γ

Figure 8 (Color online) The momentum distribution function 119899(k) for 119899 = 092 119879 = 001 and 119880 = 8

the 119878 rarr 119872 direction (see Figure 7(b)) the former stays quitesharp (just a bit skewed by the slightly higher temperature)as the Fermi liquid theory requires the latter instead losescompletely its sharpness much more than what would bereasonable because of the finite value of the temperature asit can be deduced by the comparison with the behavior ofthe main jump Such a strong qualitative modification is theevidence of a non-Fermi-liquid-like kind of behavior butonly combining this occurrence with a detailed study of thefrequency and temperature dependence of the imaginary partof the self-energy in the very same region of momentumspace (see Section 35) we will be able to make a definitivestatement about this

In Figure 8 we try to summarize the scenario in theextreme case (119899 = 092 119879 = 001 and 119880 = 8) where all theanomalous features are present and well formed by reportingthe full 2D scan of the momentum distribution function 119899(k)in a quarter of the Brillouin zone We can clearly see now thepocket with its center along the main diagonal and the lowerborder touching the border of the magnetic zone at exactly119878 = (1205872 1205872) Actually the 2D prospective makes moreevident that there is a second underlying Fermi surface thatcorresponds to the ordinary paramagnetic one for this filling119899 = 092 (large and hole-like) and touching the border of thezone between 119872 = (120587 120587) and 119883 = (120587 0) (119884 = (0 120587)) Thiscorresponds to the very small jump in Figure 9(a) along thesame direction It is worthmentioning that a similar behaviorof the momentum distribution function has been found bymeans of a Mori-like approach by Plakida and coworkers[7 9]

In Figure 9 we study the dependence of the momentumdistribution function 119899(k) on the temperature 119879 (a) and theon-site Coulomb repulsion 119880 (b) by keeping the filling 119899

fixed at themost interesting value 092 At high temperatures

(in particular down to 119879 = 04) the behavior of 119899(k) isthat of a weakly correlated paramagnet (no pocket along the119878 rarr 119872 direction no warping along the 119883 rarr 119884 direction)For lower temperatures the pocket develops along the 119878 rarr

119872 direction and the signal along the 119883 rarr 119884 direction isno more constant signaling the dynamical generation of adiagonal hopping term 119905

1015840 connecting same-spin sites in anewly developed antiferromagnetic background unwilling tobe disturbed In fact such a behavior is what one expectswhen quite strong magnetic fluctuations develop in thesystem and corresponds to a well-defined tendency towardsan antiferromagnetic phase (see Section 36) The 119872 pointbecomes another minimum in the dispersion in competitionwith Γ and the dispersion should feature amaximumbetweenthem in correspondence to the center of the pocket Thewhole bending of the dispersion confines the van Hovesingularity below the Fermi surface in an open pocket (itcloses out of the actually chosen Brillouin zone) visible in themomentum distribution as a new quite broadmaximum at119883and119884The dependence on119880 shows that for 119899 = 092 and119879 =

001 our solution presents quite strong antiferromagneticfluctuations for every finite value of the Coulomb repulsionalthough the two kinds of pockets discussed just above arenot well formed for values of 119880 less than 119880 = 3 divide 4

35 Self-Energy To obtain clear-cut pieces of informationabout the lifetime of the quasiparticles generated by the verystrong interactions within this scenario but even more tounderstand if it is still reasonable or not to discuss in termsof quasiparticles at all (ie if this Fermi-liquid-like conceptstill holds for each region of the momentum and frequencyspace) we analyze the imaginary part of the self-energyΣ

10158401015840(k 120596) as a function of both frequency and temperature

In the top panels of Figure 10 we plot the imaginary part of


YXS M

U= 8[[

n = 092

T = 100

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrT= 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

(a)

YXS M

U = 01 [[

n = 092

T = 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrU = 7

(b)

Figure 9 (Color online) The momentum distribution function 119899(k) along the main directions (Γ = (0 0) rarr 119872 = (120587 120587) rarr 119883 = (120587 0) rarr

119884 = (0 120587) rarr Γ = (0 0)) for different values of temperature 119879 (a) and on-site Coulomb repulsion 119880 (b) at 119899 = 092

the self-energy Σ10158401015840(k 120596) together with its real part Σ1015840

(k 120596)the spectral function 119860(k 120596) and the noninteracting disper-sion 120576

0(k 120596) as functions of the frequency at the nodal point

k = 119878 (a) and at its companion position on the phantom halfof the pocket k = 119878 (b) along the main diagonal Γ rarr 119872In both cases although k = 119878 is not visible in the picturebut is very clear for k = 119878 the position of the relativelocalmaximum of 119860(k 120596) coincides with the chemical potential(ie on the Fermi surface) and it is determined by the sum of1205760(k 120596 = 0) and Σ

1015840(k 120596 = 0) as expected (ie both points

belong to the 119903(k) = 0 locus) What is very interesting andsomewhat unexpected and peculiar is that at the nodal point aparabolic-like (ie a Fermi-liquid-like) behavior of Σ10158401015840

(k 120596)is clearly apparent whereas at k = 119878 the dependence ofΣ

10158401015840(k 120596) on frequency shows a predominance of a linear

term giving a definite proof that this region in momentumspace is interested to a kind of physics very different fromwhat can be considered even by far Fermi-liquid-like

In order to remove any possible doubt regarding thenon-Fermi-liquid-like nature of the physics going on at thephantom half of the pocket in the bottom panels of Figure 10the imaginary part of the self-energy at the Fermi surfaceΣ

10158401015840(k 120596 = 0) is reported as a function of the temperature at

the nodal point k = 119878 and at its companion k = 119878 The bluestraight line in the right panel is just a guide to the eye Weclearly see that in this case too although it requires to movefrom a119879 to a1198792 representation (from (a) to (b)) the behaviorof Σ

10158401015840(k 120596 = 0) shows rather different behaviors at the

selected points In particular the temperature dependence ofΣ

10158401015840(k 120596 = 0) is exactly parabolic (ie exactly Fermi-liquid)

at the nodal point while it exhibits a predominance of linearand logarithmic contributions at 119878 This is one of the mostrelevant results of this analysis and characterizes this scenariowith respect to the others present in the literature

36 Spin Dynamics Finally to analyze the way the systemapproaches the antiferromagnetic phase on decreasing dop-ing and temperature and increasing correlation strength wereport the behavior of the nearest neighbor spin-spin corre-lation function ⟨119899

119911119899

120572

119911⟩ of the antiferromagnetic correlation

length 120585 and of the pole 120596(3)(Q) (as defined in Section 3) as

functions of filling 119899 temperature 119879 and on-site Coulombrepulsion 119880 In Figures 11 12 and 13 respectively suchquantities are presented for values in the ranges 07 lt 119899 lt

092 001 lt 119879 lt 1 and 01 lt 119880 lt 8 The choice for theextremal values of low doping 119899 = 092 low temperature 119879 =

001 and strong on-site Coulomb repulsion 119880 = 8 has beenmade as for these values we find that all investigated single-particle properties (spectral density function Fermi surfacedispersion relation density of states momentum distributionfunction self-energy ) present anomalous behaviors Thenearest-neighbor spin-spin correlation function ⟨119899

119911119899

120572

119911⟩ is

always antiferromagnetic in character (ie negative) andincreases its absolute value on decreasing doping and temper-ature 119879 and on increasing119880 as expectedThe signature of theexchange scale of energy 119869 asymp 4119905

2119880 asymp 05 in the temperature

dependence as a significative enhancement in the slope israther evident The analysis of the filling dependence unveilsquite strong correlations at the higher value of doping too120585 is always larger than one for all values of fillings showingthat at 119879 = 001 and 119880 = 8 we should expect antiferro-magnetic fluctuations in the overdoped regime too as alsoclaimed by recent experiments [193ndash195] In the overdopedregion the antiferromagnetic fluctuations are quite less welldefined in terms of magnonparamagnon width than in theunderdoped region [196ndash198] and than what is found in thecurrent two-pole approximation In fact a proper descriptionof the paramagnons dynamics would definitely require theinclusion of a proper self-energy term in the charge and


minus10 minus05 00 05 10

120596

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

minus08

minus06

minus04

minus02

00

02

04

06

08

(a)


120596

minus08

minus06

minus04

minus02

00

02

04

06

08

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

(b)

000 001 002 003 004 005minus012

minus010

minus008

minus006

minus004

minus002

000

T

Σ998400998400(k120596

=0)

n = 092

U = 8k = Sk = S

(c)

minus012

minus010

minus008

minus006

minus004

minus002

000

Σ998400998400(k120596

=0)

0000 0001 0002

T2

n = 092

U = 8k = Sk = S

(d)

Figure 10 (Color online) Spectral density function 119860 (k 120596) real (1015840) and imaginary (

10158401015840) parts of the self-energy Σ (k 120596) noninteracting

dispersion 1205760(k 120596) as functions of frequency at (a) k = 119878 and (b) k = 119878 for 119899 = 092 119879 = 002 and 119880 = 8 ((c) and (d)) Imaginary part of

the self-energy Σ10158401015840(k 120596 = 0) as a function of temperature at (squares) k = 119878 and (circles) k = 119878 for 119899 = 092 and 119880 = 8 The blue line is just a

guide to the eye

spin propagators too (in preparation) Coming back to results(Figure 12) 120585 overcomes one lattice constant at temperaturesbelow 119869 and tends to diverge for low enough temperaturesOn the other hand 120585 seems to saturate for low enough values

of doping 120585 equals one between 119880 = 3 and 119880 = 4 andagain rapidly increases for large enough values of119880 The pole120596

(3)(Q) decreases on decreasing doping 119899 and temperature

119879 and on increasing 119880 In particular it is very sensitive to


T

n

00 01 02 03 04 05 06 07 08 09 10minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009090 087 084 081 078 075 072

n = 092

U = 8

T = 001

⟨nzn120572 z⟩

8 7 6 5 4 3 2 1 0

U

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8T = 001

⟨nzn120572 z⟩

Figure 11 (Color online)The spin-spin correlation function ⟨119899119911119899

120572

119911⟩ as a function of filling 119899 temperature 119879 and on-site Coulomb repulsion

119880 in the ranges 07 lt 119899 lt 092 001 lt 119879 lt 1 and 01 lt 119880 lt 8

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

n

090 087 084 081 078 075 072

120585

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

78 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

120585

n = 092

U = 8

T = 001

Figure 12 (Color online)The antiferromagnetic correlation length 120585 as a function of filling 119899 temperature 119879 and on-site Coulomb repulsion119880 in the ranges 07 lt 119899 lt 092 001 lt 119879 lt 1 and 01 lt 119880 lt 8

00

02

04

06

08

10

12

14

16

00

02

04

06

08

10

12

14

16

n

090 087 084 081 078 075 072

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

00

04

08

12

16

20

24

28

32

00

04

08

12

16

20

24

28

32

8 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

120596(Q

)

120596(Q

)

Figure 13 (Color online)The pole120596(3)(Q) of the spin-spin propagator as a function of filling 119899 temperature119879 and on-site Coulomb repulsion



the variations in temperature 119879 and in on-site Coulombrepulsion 119880 which make the mode softer and softer clearlyshowing the definite tendency towards an antiferromagneticinstability

4 Conclusions and Perspectives

We have reviewed and systematized the theory and theresults for the single-particle and magnetic-response proper-ties microscopically derived for the 2D Hubbard model asminimal model for high-119879

119888cuprate superconductors within

the Composite Operator Method with the residual self-energy computed in the noncrossing approximation (NCA)

Among the several scenarios proposed for the pseudogaporigin [119] COM definitely falls into the AF scenario (thepseudogap is a precursor of the AF long-range order) [199ndash201] as well as the two-particle self-consistent approach(TPSC) [86 87 160] the DMFT + Σ approach [163ndash167] anda Mori-like approach by Plakida and coworkers [6 7 9]

In the limit of strong on-site Coulomb repulsion and lowdoping such results show the emergence of a pseudogapscenario the deconstruction of the Fermi surface in ill-defined open arcs and the clear signatures of non-Fermi-liquid features similarly to what has been found by ARPESexperiments [82] and not only [84] In particular we haveshown that a very low-intensity signal develops around 119872

point andmoves towards 119878 nodal point on decreasing dopingcoming to close together with the ordinary Fermi surfaceboundary a pocket in the underdoped region Whenever thepocket develops it is just the remarkable difference in theintensity of the signal between the two halves of the pocket tomake a Fermi arc apparent As the doping decreases furtherthe arc shrinks into a point at 119878 exactly at half-fillingmaking itpossible to reconcile the large-small Fermi surface dichotomyonce the Fermi surface is defined as the relativemaxima of thespectral function and the relic of the ordinary paramagneticFermi surface are also taken into account The pseudogapdevelops since a region inmomentum (and frequency) with avery low-intensity signal (it corresponds to the phantom halfof the pocket at the chemical potential and to the shadowband out of it) is present between the van Hove singularityand the quite flat band edge (quite flat after the doubling ofthe Brillouin zone due to the very strong antiferromagneticfluctuations) On changing doping a spectral weight transfertakes place in the density of states between the two maximacorresponding to the van Hove singularity and the bandedge respectively A crossover between a Fermi liquid and anon-Fermi-liquid can be clearly observed in the momentumdistribution function (definitely not featuring a sharp jumpon the phantom half of the pocket) and in the imaginarypart of the self-energy (featuring linear and logarithmic termsin the frequency and temperature dependence instead ofthe ordinary parabolic term) on decreasing doping at lowtemperatures and large interaction strength This crossoverexactly corresponds to the process of deconstruction of theFermi surface We also report kinks in the dispersion alongnodal and antinodal directions In order to properly interpretthe behavior of the spectral density function and of themomentum distribution function we have also analyzed the

characteristic features in the spin-spin correlation functionthe antiferromagnetic correlation length and the pole of thespin-spin propagator As expected on reducing doping ortemperature and on increasing 119880 the correlations becomestronger and stronger The exchange scale of energy 119869 isclearly visible in the temperature dependence of the spin-spin correlation function and drives the overall behavior ofthe magnetic response These results also demonstrate thata properly microscopically derived susceptibility can giveresults practically identical or at least very similar to thoseattainable by means of phenomenological susceptibilitiesespecially tailored to describe experimentsThis is even moreremarkable since COMbrings the benefice of amicroscopicaldetermination of the temperature and filling dependencies ofthe correlation length

Many other issues should be addressed in the next futureto improve the Hamiltonian description by adding and fine-tuning longer-range hopping terms in order to quantitativelyand not only qualitatively describe specific materials toverify the stability and the modifications of this scenariowith respect to the inclusion of a residual self-energy inthe calculation of the charge and spin propagators closing afully self-consistent cycle to investigate the charge and spinresponses in the full range of momentum and frequency rele-vant to these systems searching for hourglasses thresholdsand all other peculiar features experimentally observed toinvestigate the superconducting phase and establish that it isnature and relationship with the anomalous features of thenormal phases to analyze in detail the transitioncrossoverbetween the quasiordinary antiferromagnetic phase at half-filling and the underdoped regime

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author gratefully acknowledges many stimulating andenlightening discussions with A Chubukov F Mancini NM Plakida P Prelovsek J Tranquada and R Zayer He alsowishes to thank E Piegari for her careful reading of the paper

References

[1] N N Bogoliubov ldquoOn the theory of superfluidityrdquo Journal ofPhysics vol 11 no 1 pp 23ndash32 1947 (English)

[2] S M Dancoff ldquoNon-adiabatic meson theory of nuclear forcesrdquoPhysical Review vol 78 no 4 pp 382ndash385 1950

[3] R W Zwanzig ldquoStatistical mechanics of irreversibilityrdquo inLectures in Theoretical Physics W Britton B Downs and JDowns Eds vol 3 pp 106ndash141 Wiley-Interscience New YorkNY USA 1961

[4] H Mori ldquoTransport collective motion and Brownian motionrdquoProgress of Theoretical Physics vol 33 no 3 pp 423ndash455 1965

[5] N M Plakida R Hayn and J-L Richard ldquoTwo-band singlet-hole model for the copper oxide planerdquo Physical Review B vol51 no 23 pp 16599ndash16607 1995


[6] NM Plakida L Anton S Adam andG Adam ldquoExchange andspin-fluctuation mechanisms of superconductivity in cupratesrdquoJournal of Experimental and Theoretical Physics vol 97 no 2pp 331ndash342 2003

[7] N M Plakida and V S Oudovenko ldquoElectron spectrum inhigh-temperature cuprate superconductorsrdquo Journal of Experi-mental andTheoretical Physics vol 104 no 2 pp 230ndash244 2007

[8] G Adam and S Adam ldquoRigorous derivation of the mean-fieldGreen functions of the two-band Hubbard model of supercon-ductivityrdquo Journal of Physics A Mathematical and Theoreticalvol 40 p 11205 2007

[9] N Plakida High-Temperature Cuprate Superconductors Exper-iment Theory and Applications vol 166 of Springer Series inSolid-State Sciences Springer Berlin Germany 2010

[10] HUmezawaAdvanced FieldTheoryMicroMacro andThermalPhysics American Institute of Physics New York NY USA1993

[11] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society A vol 276 no 1365 pp 238ndash257 1963

[12] J Hubbard ldquoPergamon presselectron correlations in narrowenergy bands II The degenerate band caserdquo Proceedings of theRoyal Society A vol 277 no 1369 pp 237ndash259 1964

[13] J Hubbard ldquoElectron correlations in narrow energy bands IIIAn improved solutionrdquo Proceedings of the Royal Society A vol281 no 1386 pp 401ndash419 1964

[14] D J Rowe ldquoEquations-of-motion method and the extendedshell modelrdquo Reviews of Modern Physics vol 40 pp 153ndash1661968

[15] LM Roth ldquoElectron correlation in narrow energy bands IThetwo-pole approximation in a narrow S bandrdquo Physical ReviewLetters Physical Review and Reviews ofModern Physics vol 184no 2 pp 451ndash459 1969

[16] Y A Tserkovnikov ldquoA method for solving infinite systemsof equations for two-time temperature Green functionsrdquo Teo-reticheskaya i Matematicheskaya Fizika vol 49 no 2 pp 219ndash233 1981

[17] Y A Tserkovnikov ldquoTwo-time temperature Greenrsquos functionsfor commuting dynamical variablesrdquo Akademiya Nauk SSSRTeoreticheskaya i Matematicheskaya Fizika vol 50 no 2 pp261ndash271 1982

[18] S E Barnes ldquoNew method for the Anderson modelrdquo Journal ofPhysics F Metal Physics vol 6 no 7 pp 1375ndash1383 1976

[19] P Coleman ldquoNew approach to the mixed-valence problemrdquoPhysical Review B vol 29 no 6 pp 3035ndash3044 1984

[20] G Kotliar and A E Ruckenstein ldquoNew functional integralapproach to strongly correlated Fermi systems the Gutzwillerapproximation as a saddle pointrdquo Physical Review Letters vol57 no 11 pp 1362ndash1365 1986

[21] O K Kalashnikov and E S Fradkin ldquoThe method of spectraldensities in quantum statistical mechanicsrdquo Soviet PhysicsmdashJETP vol 28 p 317 1969

[22] W Nolting ldquoMethod of spectral moments for the hubbard-model of a narrow S-bandrdquo Zeitschrift fur Physik vol 255 no 1pp 25ndash39 1972

[23] Y A Izyumov and Y N Skryabin Statistical Mechanics ofMagnetically Ordered Systems Consultants Bureau New YorkNY USA 1988

[24] V Moskalenko and M Vladimir ldquoTopological part of theinteraction of phonons with dislocations and disclinationsrdquofA]aAgK9AcQ1uK M1gAW1gK9AcQ1uCKoKQ1 vol 83 no 3 pp428ndash446 1990

[25] S G Ovchinnikov and V V Valrsquokov Hubbard Operators in theTheory of Strongly Correlated Electrons Imperial College PressLondon UK 2004

[26] M M Korshunov V A Gavrichkov S G Ovchinnikov I ANekrasov Z V Pchelkina and V I Anisimov ldquoHybrid LDAand generalized tight-binding method for electronic structurecalculations of strongly correlated electron systemsrdquo PhysicalReview B Condensed Matter and Materials Physics vol 72 no16 Article ID 165104 2005

[27] S Ovchinnikov V GavrichkovM Korshunov and E ShneyderldquoElectronic structure and electron-phonon interaction in thestrongly correlated electron system of cupratesrdquo Fizika NizkikhTemperatur vol 32 p 634 2006

[28] Y Kakehashi and P Fulde ldquoSelf-consistent projection operatorapproach to nonlocal excitation spectrardquo Physical Review B vol70 no 19 Article ID 195102 12 pages 2004

[29] Y Kakehashi and P Fulde ldquoMarginal Fermi liquid theory inthe Hubbard modelrdquo Physical Review Letters vol 94 Article ID156401 2005

[30] S Onoda and M Imada ldquoOperator projection method appliedto the single-particle greenrsquos function in the hubbard modelrdquoJournal of the Physical Society of Japan vol 70 no 3 pp 632ndash635 2001

[31] S Onoda and M Imada ldquoFilling-control metal-insulator tran-sition in the hubbard model studied by the operator projectionmethodrdquo Journal of the Physical Society of Japan vol 70 no 11pp 3398ndash3418 2001

[32] S Onoda and M Imada ldquoMott transitions in the two-dimensional half-filled Hubbard model correlator projectionmethod with projective dynamical mean-field approximationrdquoPhysical Review B vol 67 no 16 Article ID 161102(R) 4 pages2003

[33] FMancini andA Avella ldquoTheHubbardmodel within the equa-tions of motion approachrdquo Advances in Physics vol 53 no 5-6pp 537ndash768 2004

[34] F Mancini and A Avella ldquoEquation of motion method forcomposite field operatorsrdquo European Physical Journal B vol 36no 1 pp 37ndash56 2003

[35] A Avella and F Mancini ldquoThe Composite Operator Method(COM)rdquo in Strongly Correlated SystemsTheoretical Methods AAvella and F Mancini Eds vol 171 of Springer Series in Solid-State Sciences pp 103ndash141 Springer Berlin Germany 2012

[36] A Avella F Mancini D Villani and H Matsumoto ldquoThesuperconducting gap in the two-dimensional Hubbard modelrdquoPhysica C Superconductivity and Its Applications vol 282ndash287part 3 pp 1757ndash1758 1997

[37] A Avella F Mancini D Villani L Siurakshina and V YYushankhai ldquoThe Hubbard model in the two-pole approxima-tionrdquo International Journal of Modern Physics B vol 12 p 811998

[38] A Avella F Mancini and R Munzner ldquoAntiferromagneticphase in the Hubbard model by means of the composite opera-tor methodrdquo Physical Review B vol 63 Article ID 245117 2001

[39] A Avella F Mancini andM Sanchez-Lopez ldquoThe 1DHubbardmodel within the composite operator methodrdquo The EuropeanPhysical Journal B vol 29 no 3 pp 399ndash417 2002

[40] A Avella F Mancini and V Turkowski ldquoBosonic sector ofthe two-dimensional Hubbardmodel studied within a two-poleapproximationrdquo Physical Review B vol 67 Article ID 1151232003


[41] A Avella F Mancini and T Saikawa ldquoThe 2-site Hubbard andt-J modelsrdquo European Physical Journal B vol 36 no 4 pp 445ndash473 2003

[42] A Avella F Mancini and E Plekhanov ldquoCOM framework ford-wave superconductivity in the 2DHubbardmodelrdquoPhysicaCSuperconductivity and its Applications vol 470 no 1 pp S930ndashS931 2010

[43] A Avella F Mancini F Paolo Mancini and E PlekhanovldquoFilling and temperature dependence of the spin susceptibilityof the two-dimensional Hubbardmodel in the superconductingd-wave phaserdquo Journal of Physics andChemistry of Solids vol 72no 5 pp 362ndash365 2011

[44] A Avella S Krivenko F Mancini and N M Plakida ldquoSelf-energy corrections to the electronic spectrum of the Hubbardmodelrdquo Journal of Magnetism andMagnetic Materials vol 272ndash276 part 1 pp 456ndash457 2004

[45] S Krivenko A Avella F Mancini and N Plakida ldquoSCBAwithin composite operator method for the Hubbard modelrdquoPhysica B Condensed Matter vol 359ndash361 pp 666ndash668 2005

[46] S Odashima A Avella and F Mancini ldquoHigh-order correla-tion effects in the two-dimensional Hubbard modelrdquo PhysicalReview B vol 72 Article ID 205121 2005

[47] A Avella ldquoThe Hubbard model beyond the two-pole approx-imation a composite operator method studyrdquo The EuropeanPhysical Journal B vol 87 article 45 2014

[48] A Avella F Mancini F Paolo Mancini and E PlekhanovldquoSingle-particle dispersion of the 2D p-d modelrdquo Journal ofPhysics and Chemistry of Solids vol 72 no 5 pp 384ndash387 2011

[49] E Plekhanov A Avella F Mancini and F P MancinildquoCorrelation-induced band suppression in the two-orbital Hub-bard modelrdquo Journal of Physics Conference Series vol 273 no1 Article ID 012147 2011

[50] AAvella FMancini F PMancini andE Plekhanov ldquoCompos-ite operator candidates for a study of the p-d modelrdquo Journal ofPhysics Conference Series vol 391 no 1 Article ID 012121 2012

[51] A Avella F Mancini F P Mancini and E Plekhanov ldquoEmeryvs Hubbard model for cuprate superconductors a compositeoperator method studyrdquo The European Physical Journal BCondensed Matter and Complex Systems vol 86 article 2652013

[52] A Avella S-S Feng and F Mancini ldquoThe 2D t-J model aproposal for an analytical studyrdquo Physica B Condensed Mattervol 312-313 pp 537ndash538 2002

[53] A Avella F Mancini and D Villani ldquoIncommensurate spinfluctuations in the two-dimensional t-t-U modelrdquo PhysicsLetters A vol 240 no 4-5 pp 235ndash240 1998

[54] A Avella F Mancini D Villani and H Matsumoto ldquoThetwo-dimensional t-trsquo-U model as a minimal model for cupratematerialsrdquo The European Physical Journal B vol 20 no 3 pp303ndash311 2001

[55] A Avella and F Mancini ldquoThe charge and spin sectors ofthe t-t1015840 Hubbard modelrdquo Physica C Superconductivity and ItsApplications vol 408ndash410 no 1ndash4 pp 284ndash286 2004

[56] A Avella and F Mancini ldquoThe hubbard model with intersiteinteraction within the composite operator methodrdquo EuropeanPhysical Journal B vol 41 no 2 pp 149ndash162 2004

[57] A Avella and F Mancini ldquoPhase diagrams of half-filled 1D and2D extended Hubbard model within COMrdquo Journal of Physicsand Chemistry of Solids vol 67 no 1ndash3 pp 142ndash145 2006

[58] D Villani E Lange A Avella and G Kotliar ldquoTwo-scaleanalysis of the SU(N) Kondo modelrdquo Physical Review Lettersvol 85 no 4 pp 804ndash807 2000

[59] A Avella F Mancini and R Hayn ldquoThe energy-scale-depend-ent composite operator method for the single-impurity Ander-son modelrdquo European Physical Journal B vol 37 no 4 pp 465ndash471 2004

[60] A Avella F Mancini and R Hayn ldquoThe composite operatormethod for impurity modelsrdquo Acta Physica Polonica B vol 34no 22 p 1345 2003

[61] A Avella F Mancini S Odashima and G Scelza ldquoThe two-orbitalHubbardmodel and theOSMTrdquoPhysicaC Superconduc-tivity and its Applications vol 460ndash462 pp 1068ndash1069 2007

[62] A Avella F Mancini G Scelza and S Chaturvedi ldquoEntangle-ment properties and phase diagram of the two-orbital atomichubbardmodelrdquoActa Physica Polonica A vol 113 no 1 pp 417ndash420 2008

[63] AAvella FMancini F PMancini andE Plekhanov ldquoRelation-ship between band populations and band structure in the three-bandHubbardmodelrdquo Journal of Physics Conference Series vol273 Article ID 012091 2011

[64] A Avella and F Mancini ldquoExact solution of the one-dimen-sional spin-32 Ising model in magnetic fieldrdquo European Physi-cal Journal B vol 50 no 4 pp 527ndash539 2006

[65] A Avella and F Mancini ldquoCharge ordering in the extendedHubbardmodel in the ionic limitrdquoPhysica B CondensedMattervol 378ndash380 pp 311ndash312 2006

[66] M Bak A Avella and F Mancini ldquoNon-ergodicity of the 1DHeisenberg modelrdquo Physica Status Solidi (B) Basic Research vol236 no 2 pp 396ndash399 2003

[67] E Plekhanov A Avella and F Mancini ldquoNonergodic dynamicsof the extended anisotropic Heisenberg chainrdquo Physical ReviewB vol 74 Article ID 115120 2006

[68] E Plekhanov A Avella and F Mancini ldquoEntanglement inthe 1D extended anisotropic Heisenberg modelrdquo Physica BCondensed Matter vol 403 no 5ndash9 pp 1282ndash1283 2008

[69] E Plekhanov A Avella and F Mancini ldquo119879 = 0 phase diagramof 1D extended anisotropic spin-iexcl Heisenberg modelrdquo Journalof Physics Conference Series vol 145 no 1 Article ID 0120632009

[70] E Plekhanov A Avella and F Mancini ldquoThe phase diagramof the extended anisotropic ferromagnetic- antiferromagneticHeisenberg chainrdquo European Physical Journal B vol 77 no 3pp 381ndash392 2010

[71] A Avella F Mancini and E Plekhanov ldquo119883119883119885-like phase inthe F-AF anisotropic Heisenberg chainrdquoThe European PhysicalJournal B Condensed Matter and Complex Systems vol 66 no3 pp 295ndash299 2008

[72] A Avella and F Mancini ldquoStudy of the spin- frac-32 Hubbard-Kondo lattice model by means of the Composite OperatorMethodrdquo Physica B Condensed Matter vol 378ndash380 pp 700ndash701 2006

[73] A Avella F Mancini and D Villani ldquoThe overdoped regime inLa

2minus119909Sr

119909CuO

4rdquo Solid State Communications vol 108 no 10 pp

723ndash725 1998[74] A Avella and FMancini ldquoA theoretical analysis of themagnetic

properties of La2CuO

4rdquo The European Physical Journal B vol

32 no 1 pp 27ndash33 2003[75] A Avella and F Mancini ldquoUnderdoped cuprate phenomenol-

ogy in the two-dimensional Hubbard model within the com-posite operator methodrdquo Physical Review B vol 75 Article ID134518 2007

[76] A Avella and F Mancini ldquoThe 2D Hubbard model and thepseudogap aCOM(SCBA) studyrdquo Journal of Physics CondensedMatter vol 19 no 25 Article ID 255209 2007


[77] AAvella and FMancini ldquoAnomalous self-energy features in the2D hubbard modelrdquo Acta Physica Polonica A vol 113 no 1 pp395ndash398 2008

[78] A Avella and F Mancini ldquoStrong antiferromagnetic correlationeffects on the momentum distribution function of the Hubbardmodelrdquo Journal of Physics Condensed Matter vol 21 no 25Article ID 254209 2009

[79] J G Bednorz and K A Muller ldquoPossible high Tc supercon-ductivity in the Ba-La-Cu-O systemrdquo Zeitschrift fur Physik BCondensed Matter vol 64 no 2 pp 189ndash193 1986

[80] T Timusk and B Statt ldquoThe pseudogap in high-temperaturesuperconductors an experimental surveyrdquo Reports on Progressin Physics vol 62 no 1 pp 61ndash122 1999

[81] J Orenstein and A J Millis ldquoAdvances in the physics of high-temperature superconductivityrdquo Science vol 288 no 5465 pp468ndash474 2000

[82] A Damascelli Z Hussain and Z-X Shen ldquoAngle-resolvedphotoemission studies of the cuprate superconductorsrdquoReviewsof Modern Physics vol 75 no 2 pp 473ndash541 2003

[83] M Eschrig ldquoThe effect of collective spin-1 excitations onelectronic spectra in high-119879

119862superconductorsrdquo Advances in

Physics vol 55 no 1-2 pp 47ndash183 2006[84] S E Sebastian N Harrison and G G Lonzarich ldquoTowards

resolution of the Fermi surface in underdoped high-T119888super-

conductorsrdquo Reports on Progress in Physics vol 75 no 10Article ID 102501 2012

[85] P A Lee N Nagaosa andX-GWen ldquoDoping aMott insulatorphysics of high-temperature superconductivityrdquo Reviews ofModern Physics vol 78 no 1 p 17 2006

[86] A-M S Tremblay B Kyung and D Senechal ldquoPseudogap andhigh-temperature superconductivity from weak to strong cou-pling Towards quantitative theoryrdquo Fizika Nizkikh Temperaturvol 32 no 4-5 pp 561ndash595 2006

[87] A-M S Tremblay B Kyung and D Senechal ldquoPseudogapand high-temperature superconductivity from weak to strongcoupling Towards a quantitative theoryrdquo Low TemperaturePhysics vol 32 p 424 2006

[88] N E Hussey M Abdel-Jawad A Carrington A P Mackenzieand L Balicas ldquoA coherent three-dimensional Fermi surface in ahigh-transition-temperature superconductorrdquo Nature vol 425no 6960 pp 814ndash817 2003

[89] M Plate J D F Mottershead I S Elfimov et al ldquoFermi surfaceand quasiparticle excitations of overdoped Tl2Ba2CuO6+120575rdquoPhysical Review Letters vol 95 Article ID 077001 2005

[90] D C Peets J D F Mottershead B Wu et al ldquoTl2Ba

2CuO

6+120575

brings spectroscopic probes deep into the overdoped regime ofthe high-119879

119888cupratesrdquo New Journal of Physics vol 9 article 28

2007[91] O K Andersen A I Liechtenstein O Jepsen and F Paulsen

ldquoLDA energy bands low-energy hamiltonians t1015840 t10158401015840 tperp (k) andJperprdquo Journal of Physics and Chemistry of Solids vol 56 no 12 pp1573ndash1591 1995

[92] B Vignolle A Carrington R A Cooper et al ldquoQuantumoscillations in an overdoped high-119879

119888superconductorrdquo Nature

vol 455 no 7215 pp 952ndash955 2008[93] K M Shen F Ronning D H Lu et al ldquoNodal quasiparticles

and antinodal charge ordering in Ca2minus119909

Na119909CuO

2Cl

2rdquo Science

vol 307 no 5711 pp 901ndash904 2005[94] T Valla A V Fedorov J Lee J C Davis and G D Gu ldquoThe

ground state of the pseudogap in cuprate supercondustorsrdquoScience vol 314 no 5807 pp 1914ndash1916 2006

[95] A Kanigel M R Norman M Randeria et al ldquoEvolution of thepseudogap from Fermi arcs to the nodal liquidrdquoNature Physicsvol 2 no 7 pp 447ndash451 2006

[96] M A Hossain J D F Mottershead D Fournier et al ldquoIn situdoping control of the surface of high-temperature superconduc-torsrdquo Nature Physics vol 4 pp 527ndash531 2008

[97] J Q Meng G Liu W Zhang et al ldquoCoexistence of Fermi arcsand Fermi pockets in a high-T

119888copper oxide superconductorrdquo

Nature vol 462 pp 335ndash338 2009[98] P D C King J A Rosen W Meevasana et al ldquoStructural

origin of apparent fermi surface pockets in angle-resolvedphotoemission of Bi

2Sr

2minus119909LaxCuO

6+120575rdquo Physical Review Letters

vol 106 no 12 Article ID 127005 4 pages 2011[99] N Doiron-Leyraud C Proust D LeBoeuf et al ldquoQuantum

oscillations and the Fermi surface in an underdoped high-Tcsuperconductorrdquo Nature vol 447 no 7144 pp 565ndash568 2007

[100] D LeBoeuf N Doiron-Leyraud J Levallois et al ldquoElectronpockets in the Fermi surface of hole-doped high-Tc supercon-ductorsrdquo Nature vol 450 no 7169 pp 533ndash536 2007

[101] E A Yelland J Singleton C H Mielke et al ldquoQuantumoscillations in the underdoped cuprate YBa

2Cu

4O

8rdquo Physical

Review Letters vol 100 no 4 Article ID 047003 2008[102] A F Bangura J D Fletcher A Carrington et al ldquoSmall

fermi surface pockets in underdoped high temperature super-conductors observation of Shubnikov-de Haas oscillationsin YBa2Cu4O8rdquo Physical Review Letters vol 100 Article ID047004 2008

[103] S E Sebastian N Harrison E Palm et al ldquoAmulti-componentFermi surface in the vortex state of an underdoped high-119879

119888

superconductorrdquo Nature vol 454 no 7201 pp 200ndash203 2008[104] A Audouard C Jaudet D Vignolles et al ldquoMultiple quan-

tum oscillations in the de haas-van alphen spectra of theunderdoped high-temperature superconductor YBa

2Cu

3O

65rdquo

Physical Review Letters vol 103 no 15 Article ID 157003 2009[105] S E Sebastian N Harrison C H Altarawneh et al ldquoFermi-

liquid behavior in an underdoped high-119879119862superconductorrdquo

Physical Review B vol 81 Article ID 140505(R) 4 pages 2010[106] S E Sebastian N Harrison P A Goddard et al ldquoCompensated

electron and hole pockets in an underdoped high-119879119888super-

conductorrdquo Physical Review B Condensed Matter and MaterialsPhysics vol 81 no 21 Article ID 214524 2010

[107] S E Sebastian N Harrison M M Altarawneh et al ldquoMetal-insulator quantum critical point beneath the high Tc supercon-ducting domerdquo Proceedings of the National Academy of Sciencesof the United States of America vol 107 no 14 pp 6175ndash61792010

[108] J Singleton C de la Cruz R D McDonald et al ldquoMagneticquantum oscillations in YBa

2Cu

3O

661and YBa

2Cu

3O

669

in fields of up to 85 T patching the hole in the roof ofthe superconducting domerdquo Physical Review Letters vol 104Article ID 086403 2010

[109] I M Vishik W S Lee F Schmitt et al ldquoDoping-dependentnodal fermi velocity of the high-temperature superconduc-tor Bi

2Sr

2CaCu

2O

8+120575revealed using high-resolution angle-

resolved photoemission spectroscopyrdquo Physical Review Lettersvol 104 no 20 Article ID 207002 4 pages 2010

[110] H Anzai A Ino T Kamo et al ldquoEnergy-dependent enhance-ment of the electron-coupling spectrum of the underdopedBi

2Sr

2CaCu

2O

8+120575superconductorrdquo Physical Review Letters vol

105 no 22 Article ID 227002 4 pages 2010


[111] B J Ramshaw B Vignolle J Day et al ldquoAngle dependenceof quantum oscillations in YBa

2Cu

3O

659shows free-spin

behaviour of quasiparticlesrdquo Nature Physics vol 7 no 3 pp234ndash238 2011

[112] S E Sebastian N Harrison M M Altarawneh et al ldquoMetal-insulator quantum critical point beneath the high 119879

119888supercon-

ducting domerdquo Proceedings of the National Academy of Sciencesof the United States of America vol 107 no 14 pp 6175ndash61792010

[113] S E Sebastian N HarrisonMM Altarawneh et al ldquoChemicalpotential oscillations from nodal Fermi surface pocket inthe underdoped high-temperature superconductor YBa

2Cu

3

O6+119909

rdquo Nature Communications vol 2 no 1 article 471 2011[114] S E Sebastian N Harrison and G G Lonzarich ldquoQuantum

oscillations in the high-119879119888cupratesrdquo Philosophical Transactions

of the Royal Society A Mathematical Physical and EngineeringSciences vol 369 no 1941 pp 1687ndash1711 2011

[115] S C Riggs O Vafek J B Kemper et al ldquoHeat capacity throughthe magnetic-field-induced resistive transition in an under-doped high-temperature superconductorrdquo Nature Physics vol7 no 4 pp 332ndash335 2011

[116] F Laliberte J Chang N Doiron-Leyraud et al ldquoFermi-surfacereconstruction by stripe order in cuprate superconductorsrdquoNature Communications vol 2 article 432 2011

[117] B Vignolle D Vignolles D LeBoeuf et al ldquoQuantum oscil-lations and the Fermi surface of high-temperature cupratesuperconductorsrdquoComptes Rendus Physique vol 12 no 5-6 pp446ndash460 2011

[118] S E Sebastian N Harrison R Liang et al ldquoQuantumoscillations from nodal bilayer magnetic breakdown in theunderdoped high temperature superconductor YBa

2Cu

3O

6+119909rdquo

Physical Review Letters vol 108 no 19 Article ID 196403 2012[119] M R Norman D Pines and C Kallin ldquoThe pseudogap friend

or foe of high Tcrdquo Advances in Physics vol 54 no 8 pp 715ndash733 2005

[120] H Alloul T Ohno and P Mendels ldquoY89 NMR evidence for afermi-liquid behavior inYBa

2Cu

3O

6+119909rdquoPhysical Review Letters

vol 63 no 16 pp 1700ndash1703 1989[121] D N Basov S I Woods A S Katz et al ldquoSum rules and

interlayer conductivity of high-119879119888cupratesrdquo Science vol 283

no 5398 pp 49ndash52 1999[122] J M Tranquada D N Basov A D LaForge and A A

Schafgans ldquoInterpreting quantum oscillation experiments onunderdoped YBa2Cu3O6+119909rdquo Physical Review B vol 81 ArticleID 060506 2010

[123] S A Kivelson I P Bindloss E Fradkin et al ldquoHow to detectfluctuating stripes in the high-temperature superconductorsrdquoReviews of Modern Physics vol 75 no 4 pp 1201ndash1241 2003

[124] A Carrington and E Yelland ldquoBand-structure calculationsof Fermi-surface pockets in ortho-II YBa

2Cu

3O

65rdquo Physical

Review B vol 76 Article ID 140508 2007[125] O P Sushkov ldquoMagnetic properties of lightly doped antiferro-

magnetic YBa2Cu

3O

119910rdquo Physical Review B vol 84 no 9 Article

ID 09453 10 pages 2011[126] A S Alexandrov and A M Bratkovsky ldquode Haas-van Alphen

effect in canonical and grand canonicalmultiband fermi liquidrdquoPhysical Review Letters vol 76 no 8 pp 1308ndash1311 1996

[127] K-Y Yang T M Rice and F-C Zhang ldquoPhenomenologicaltheory of the pseudogap staterdquo Physical Review B vol 73 no17 Article ID 174501 10 pages 2006

[128] A Melikyan and O Vafek ldquoQuantum oscillations in the mixedstate of 119889-wave superconductorsrdquo Physical Review B vol 78Article ID 020502 2008

[129] C M Varma ldquoMagneto-oscillations in underdoped cupratesrdquoPhysical Review B vol 79 Article ID 085110 2009

[130] J A Wilson ldquoElucidation of the origins of transport behaviourand quantum oscillations in high temperature superconductingcupratesrdquo Journal of Physics Condensed Matter vol 21 no 24Article ID 245702 2009

[131] T Pereg-Barnea H Weber G Refael and M Franz ldquoQuantumoscillations from Fermi arcsrdquo Nature Physics vol 6 no 1 pp44ndash49 2010

[132] A J Millis and M R Norman ldquoAntiphase stripe order as theorigin of electron pockets observed in 18-hole-doped cupratesrdquoPhysical Review B vol 76 no 22 Article ID 220503 4 pages2007

[133] D Haug V Hinkov Y Sidis et al ldquoNeutron scattering studyof the magnetic phase diagram of underdoped YBa

2Cu

3O

6+119909rdquo

New Journal of Physics vol 12 Article ID 105006 2010[134] V Baledent D Haug Y Sidis V Hinkov C T Lin and

P Bourges ldquoEvidence for competing magnetic instabilitiesin underdoped YBa

2Cu

3O

6+119909rdquo Physical Review B Condensed

Matter and Materials Physics vol 83 no 10 Article ID 1045042011

[135] N Harrison and S E Sebastian ldquoProtected nodal electronpocket frommultiple-Q ordering in underdoped high tempera-ture Superconductorsrdquo Physical Review Letters vol 106 ArticleID 226402 2011

[136] H Yao D-H Lee and S Kivelson ldquoFermi-surface reconstruc-tion in a smectic phase of a high-temperature superconductorrdquoPhysical Review B vol 84 Article ID 012507 2011

[137] S Chakravarty and H-Y Kee ldquoFermi pockets and quantumoscillations of the Hall coefficient in high-temperature super-conductorsrdquo Proceedings of the National Academy of Sciences ofthe United States of America vol 105 no 26 pp 8835ndash88392008

[138] A V Chubukov and D K Morr ldquoElectronic structure ofunderdoped cupratesrdquo Physics Report vol 288 no 1ndash6 pp 355ndash387 1997

[139] A Avella F Mancini and E Plekhanov ldquoNon-Fermi liquidbehavior in the 2D Hubbard model within COM(SCBA)rdquoJournal of Magnetism andMagnetic Materials vol 310 no 2 pp999ndash1001 2007

[140] A Avella and F Mancini ldquoPseudogap opening in the 2D Hub-bardmodel within COM (SCBA)rdquo Physica C Superconductivityand Its Applications vol 460ndash462 part 2 pp 1096ndash1097 2007

[141] W-Q Chen K-Y Yang T M Rice and F C Zhang ldquoQuantumoscillations inmagnetic-fieldndashinduced antiferromagnetic phaseof underdoped cuprates application to ortho-II YBa

2Cu

3O

65rdquo

Europhysics Letters vol 82 no 1 Article ID 17004 2008[142] H Oh H J Choi S G Louie and M L Cohen ldquoFermi

surfaces and quantum oscillations in the underdoped high-119879119888superconductors YBa2Cu3O65 and YBa2Cu4O8rdquo Physical

Review B vol 84 Article ID 014518 2011[143] J-X Li C-QWu andD-H Lee ldquoCheckerboard charge density

wave and pseudogap of high-119879119888cupraterdquo Physical Review B vol

74 Article ID 184515 2006[144] N Harrison ldquoNear doping-independent pocket area from an

antinodal fermi surface instability in underdoped high temper-ature superconductorsrdquo Physical Review Letters vol 107 no 18Article ID 186408 4 pages 2011


[145] N Harrison and S E Sebastian ldquoFermi surface reconstructionfrom bilayer charge ordering in the underdoped high tempera-ture superconductor YBa

2Cu

3O

6+119909rdquoNew Journal of Physics vol

14 Article ID 095023 2012[146] H L Edwards A L Barr J T Martkert and A L de Lozanne

ldquoModulations in the CuO chain layer of YBa2Cu

3O

7minus120575charge

density wavesrdquo Physical Review Letters vol 73 p 1154 1994[147] J E Hoffman E W Hudson K M Lang et al ldquoA four unit

cell periodic pattern of quasi-particle states surrounding vortexcores in Bi

2Sr

2CaCu

2O

8+120575rdquo Science vol 295 no 5554 pp 466ndash

469 2002[148] T Hanaguri C Lupien Y Kohsaka et al ldquoA rsquocheckerboardrsquo elec-

tronic crystal state in lightly hole-doped Ca2minus119909

Na119909CuO

2Cl

2rdquo

Nature vol 430 no 7003 pp 1001ndash1005 2004[149] K McElroy D H Lee J E Hoffman et al ldquoCoincidence of

checkerboard charge order and antinodal state decoherence instrongly underdoped superconducting Bi

2Sr

2Ca Cu

2O

8+120575rdquo

Physical Review Letters vol 94 Article ID 197005 2005[150] M Maki T Nishizaki K Shibata and N Kobayashi

ldquoLayered charge-density waves with nanoscale coherencein YBa

2Cu

3O

7minus120575rdquo Physical Review B vol 72 no 2 Article ID

024536 6 pages 2005[151] X Liu Z Islam S K Sinha et al ldquoFermi-surface-induced

lattice modulation and charge-density wave in optimally dopedYBa

2Cu

3O

7minus119909rdquo Physical Review B CondensedMatter andMate-

rials Physics vol 78 no 13 Article ID 134526 2008[152] T Wu H Mayaffre S Kramer et al ldquoMagnetic-field-induced

charge-stripe order in the high-temperature superconductorYBa2Cu3O119910

rdquo Nature vol 477 pp 191ndash194 2011[153] G Ghiringhelli M Le Tacon M Minola et al ldquoLong-range

incommensurate charge fluctuations in (YNd)Ba2Cu

3O

6+119909rdquo

Science vol 337 no 6096 pp 821ndash825 2012[154] H A Mook P Dai K Salama et al ldquoIncommensurate one-

dimensional fluctuations in YBa2Cu

3O

693rdquo Physical Review

Letters vol 77 no 2 pp 370ndash373 1996[155] H A Mook and F Dogan ldquoCharge fluctuations in

YBa2Cu

3O

7minus119909high-temperature superconductorsrdquo Nature vol

401 no 6749 pp 145ndash148 1999[156] D Reznik L Pintschovius J M Tranquada et al ldquoTemperature

dependence of the bond-stretching phonon anomaly in YBa2

Cu3O

695rdquo Physical Review BmdashCondensedMatter andMaterials

Physics vol 78 no 9 Article ID 094507 2008[157] P A Casey and P W Anderson ldquoHidden Fermi liquid self-

consistent theory for the normal state of high-119879119888superconduc-

torsrdquo Physical Review Letters vol 106 no 9 Article ID 0970024 pages 2009

[158] PW Anderson ldquoThe resonating valence bond state in La2CuO

4

and Superconductivityrdquo Science vol 235 no 4793 pp 1196ndash1198 1987

[159] N Bulut ldquodx2-y2 superconductivity and the Hubbard modelrdquoAdvances in Physics vol 51 no 7 pp 1587ndash1667 2002

[160] Y M Vilk and A-M S Tremblay ldquoDestruction of the fermiliquid by spin fluctuations in two dimensionsrdquo Journal of Physicsand Chemistry of Solids vol 56 no 12 pp 1769ndash1771 1995

[161] A V Chubukov and M R Norman ldquoDispersion anomalies incuprate superconductorsrdquo Physical Review B vol 70 Article ID174505 2004

[162] P Prelovsek and A Ramsak ldquoSpin-fluctuation mechanismof superconductivity in cupratesrdquo Physical Review B vol 72Article ID 012510 2005

[163] M Sadovskii ldquoPseudogap in high-temperature superconduc-torsrdquo Physics-Uspekhi vol 44 no 5 pp 515ndash539 2001

[164] M V Sadovskii I A Nekrasov E Z Kuchinskii T Pruschkeand V I Anisimov ldquoPseudogaps in strongly correlated metalsa generalized dynamical mean-field theory approachrdquo PhysicalReview BmdashCondensed Matter and Materials Physics vol 72 no15 Article ID 155105 2005

[165] E Z Kuchinskii I A Nekrasov andM V Sadovskii ldquoDestruc-tion of the fermi surface due to pseudogap fluctuations instrongly correlated systemsrdquo JETP Letters vol 82 no 4 pp 198ndash203 2005

[166] E Z Kuchinskii I A Nekrasov and M V Sadovskii ldquoPseu-dogaps introducing the length scale into dynamical mean-fieldtheoryrdquo Fizika Nizkikh Temperatur vol 32 no 4-5 pp 528ndash5372006

[167] E Z Kuchinskii I A Nekrasov Z V Pchelkina and M VSadovskii ldquoPseudogap behavior in Bi

2Ca

2SrCu

2O

8 results of

the generalized dynamical mean-field approachrdquo Journal ofExperimental and Theoretical Physics vol 104 no 5 pp 792ndash804 2007

[168] A Toschi A A Katanin and K Held ldquoDynamical vertexapproximation a step beyond dynamical mean-field theoryrdquoPhysical Review B vol 75 no 4 Article ID 045118 8 pages 2007

[169] T Schafer F Geles D Rost et al ldquoFate of the false Mott-Hubbard transition in two dimensionsrdquo httparxivorgabs14057250

[170] T Maier M Jarrell T Pruschke and M H Hettler ldquoQuantumcluster theoriesrdquo Reviews of Modern Physics vol 77 no 3 pp1027ndash1080 2005

[171] T D Stanescu M Civelli K Haule and G Kotliar ldquoA cellulardynamical mean field theory approach to Mottnessrdquo Annals ofPhysics vol 321 no 7 pp 1682ndash1715 2006

[172] G Kotliar S Y Savrasov G Palsson and G Biroli ldquoCellulardynamical mean field approach to strongly correlated systemsrdquoPhysical Review Letters vol 87 Article ID 186401 2001

[173] KHaule andGKotliar ldquoStrongly correlated superconductivitya plaquette dynamicalmean-field theory studyrdquo Physical ReviewB vol 76 Article ID 104509 2007

[174] M H Hettler A N Tahvildar-Zadeh M Jarrell T Pruschkeand H R Krishnamurthy ldquoNonlocal dynamical correlationsof strongly interacting electron systemsrdquo Physical Review BCondensed Matter and Materials Physics vol 58 no 12 ppR7475ndashR7479 1998

[175] D Senechal D Perez and M Pioro-Ladriere ldquoSpectral weightof the Hubbard model through cluster perturbation theoryrdquoPhysical Review Letters vol 84 pp 522ndash525 2000

[176] D Senechal and A-M S Tremblay ldquoHot spots and pseudogapsfor hole- and electron-doped high-temperature superconduc-torsrdquo Physical Review Letters vol 92 no 12 Article ID 1264012004

[177] A Georges G Kotliar W Krauth and M J RozenbergldquoDynamical mean-field theory of strongly correlated fermionsystems and the limit of infinite dimensionsrdquoReviews ofModernPhysics vol 68 no 1 pp 13ndash125 1996

[178] G Kotliar and D Vollhardt ldquoStrongly correlated materialsinsights from dynamical mean-field theoryrdquo Physics Today vol57 no 3 pp 53ndash59 2004

[179] G Kotliar S Y Savrasov K Haule V S Oudovenko O Par-collet and C A Marianetti ldquoElectronic structure calculationswith dynamical mean-field theoryrdquo Reviews of Modern Physicsvol 78 no 3 pp 865ndash951 2006


[180] K Held ldquoElectronic structure calculations using dynamicalmean field theoryrdquo Advances in Physics vol 56 no 6 pp 829ndash926 2007

[181] Y Izyumov and V I Anisimov Electronic Structure of Com-poundswith Strong Correlations Regular andChaotic DynamicsNIC Moscow Russia 2008

[182] A Avella ldquoSelfminusenergy corrections within the composite oper-atormethodrdquoAIPConference Proceedings vol 695 p 258 2003

[183] H Matsumoto T Saikawa and F Mancini ldquoTemperaturedependence of electronic states in the t-J modelrdquo PhysicalReview B Condensed Matter and Materials Physics vol 54 no20 pp 14445ndash14454 1996

[184] HMatsumoto and FMancini ldquoTwo-site correlation in analysisof the Hubbard modelrdquo Physical Review B vol 55 no 4 pp2095ndash2106 1997

[185] J Bosse W Gotze and M Lucke ldquoMode-coupling theory ofsimple classical liquidsrdquo Physical Review A General Physics vol17 no 1 pp 434ndash446 1978

[186] N N Bogoliubov and S V Tyablikov Doklady Akademii NaukUSSR vol 126 p 53 1959

[187] D N Zubarev ldquoDouble-time Green functions in statisticalphysicsrdquo Soviet Physics Uspekhi vol 3 pp 320ndash345 1960

[188] D N Zubarev Non Equilibrium Statistical ThermodynamicsConsultants Bureau New York NY USA 1974

[189] A Avella and FMancini ldquoA theoretical analysis of themagneticproperties of La

2CuO


32 pp 27ndash33 2003[190] T Yoshida X J Zhou M Nakamura et al ldquoElectron-

like Fermi surface and remnant (1205870) feature in overdopedLa

178Sr

022CuO

4rdquo Physical Review BmdashCondensed Matter and

Materials Physics vol 63 no 22 Article ID 220501 2001[191] A Koitzsch S V Borisenko A A Kordyuk et al ldquoOrigin of the

shadow Fermi surface in Bi-based cupratesrdquo Physical Review BCondensed Matter and Materials Physics vol 69 no 22 ArticleID 220505 2004

[192] T D Stanescu and G Kotliar ldquoFermi arcs and hidden zeros ofthe Green function in the pseudogap staterdquo Physical Review Bvol 74 Article ID 125110 2006

[193] M Le Tacon M Minola D C Peets et al ldquoDispersive spinexcitations in highly overdoped cuprates revealed by resonantinelastic x-ray scatteringrdquo Physical Review B Condensed Matterand Materials Physics vol 88 no 2 Article ID 020501 2013

[194] M P M Dean G Dellea R S Springell et al ldquoPersistence ofmagnetic excitations in La

2minus119909Sr

119909CuO

4from the undoped insu-

lator to the heavily overdoped non-superconducting metalrdquoNature Materials vol 12 no 11 pp 1019ndash1023 2013

[195] C J Jia E A Nowadnick K Wohlfeld et al ldquoPersistent spinexcitations in doped antiferromagnets revealed by resonantinelastic light scatteringrdquoNature Communications vol 5 article4314 2014

[196] J Jaklic and P Prelovsek ldquoAnomalous spin dynamics in dopedquantum antiferromagnetsrdquo Physical Review Letters vol 74 pp3411ndash3414 1995

[197] J Jaklic and P Prelovsek ldquoUniversal charge and spin responsein optimally doped antiferromagnetsrdquo Physical Review Lettersvol 75 no 7 pp 1340ndash1343 1995

[198] A A Vladimirov D Ihle and N M Plakida ldquoDynamic spinsusceptibility in the t-J modelrdquo Physical Review B CondensedMatter and Materials Physics vol 80 no 10 Article ID 1044252009

[199] A Kampf and J R Schrieffer ldquoPseudogaps and the spin-bagapproach to high-119879

119888superconductivityrdquo Physical Review B vol

41 no 10 pp 6399ndash6408 1990[200] J Schmalian D Pines and B Stojkovic ldquoMicroscopic theory of

weak pseudogap behavior in the underdoped cuprate supercon-ductors general theory and quasiparticle propertiesrdquo PhysicalReview B vol 60 p 667 1999

[201] A Abanov A V Chubukov and J Schmalian ldquoQuantum-critical theory of the spin-fermion model and its application tocuprates normal state analysisrdquoAdvances in Physics vol 52 no3 pp 119ndash218 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


properties entirely determined by the dynamics and by theboundary conditions (ie the phase under study the externalfields ) These new objects appear as the final result of themodifications imposed by the interactions on the originalparticles and contain by the very beginning the effects ofcorrelations

On the basis of this evidence one is induced to move theattention from the original fields to the new fields generatedby the interactions The operators describing these excita-tions once they have been identified can be written in termsof the original ones and are known as composite operatorsThe necessity of developing a formulation to treat compositeoperators as fundamental objects of the many-body problemin condensedmatter physics has been deeply understood andsystematically noticed since quite long time Recent yearshave seen remarkable achievements in the development ofa modern many-body theory in solid-state physics in theform of an assortment of techniques that may be termedcomposite particle methods The foundations of these typesof techniques may be traced back to the work of Bogoliubov[1] and later to that of Dancoff [2] The work of Zwanzig[3] Mori [4ndash9] and Umezawa [10] definitely deserves to bementioned too Closely related to this work is that ofHubbard[11ndash13] Rowe [14] Roth [15] and Tserkovnikov [16 17] Theslave boson method [18ndash20] the spectral density approach[21 22] the diagram technique for Hubbard operators [23]the cumulant expansion based diagram technique [24] thegeneralized tight-binding method [25ndash27] self-consistentprojection operator method [28 29] operator projectionmethod [30ndash32] and the composite operatormethod (COM)[33ndash35] are along the same lines This large class of theoriesis very promising as it is based on the firm convictionthat strong interactions call for an analysis in terms of newelementary fields embedding the greatest possible part ofthe correlations so permitting to overcome the problem offinding an appropriate expansion parameter However oneprice must be paid In general composite fields are neitherFermi nor Bose operators since they do not satisfy canonical(anti)commutation relations and their properties must bedetermined self-consistently They can only be recognized asfermionic or bosonic operators according to the number andtype of the constituting original particles Accordingly newtechniques have to be developed in order to deal with suchcomposite fields and to design diagrammatic schemes wherethe building blocks are the propagators of such compositefields standard diagrammatic expansions and the Wickrsquostheorem are no more valid The formulation of the Greenrsquosfunctionmethod itself must be revisited and new frameworksof calculations have to be devised

Following these ideas we have been developing a sys-tematic approach the composite operator method (COM)[33ndash35] to study highly correlated systems The formalism isbased on two main ideas (i) use of propagators of relevantcomposite operators as building blocks for any subsequentapproximate calculations and (ii) use of algebra constraints tofix the representation of the relevant propagators in order toproperly preserve algebraic and symmetry properties theseconstraints will also determine the unknown parametersappearing in the formulation due to the noncanonical algebra

satisfied by the composite operators In the last fifteen yearsCOM has been applied to several models and materialsHubbard [36ndash47] 119901-119889 [48ndash51] 119905-119869 [52] 119905-1199051015840-119880 [53ndash55]extended Hubbard (119905-119880-119881) [56 57] Kondo [58] Anderson[59 60] two-orbital Hubbard [61ndash63] Ising [64 65] 119869

1minus

1198692[66ndash71] Hubbard-Kondo [72] Cuprates [73ndash78] and so

forth

12 UnderdopedCuprates Cuprate superconductors [79] dis-play a full range of anomalous features mainly appearingin the underdoped region in almost all experimentallymeasurable physical properties [80ndash84] According to thistheir microscopic description is still an open problem non-Fermi-liquid response quantum criticality pseudogap for-mation ill-defined Fermi surface kinks in the electronicdispersion and so forth still remain unexplained (or at leastcontroversially debated) anomalous features [84ndash87] In thelast years the attention of the community has been focusingon threemain experimental facts [84] the dramatic change inshape and nature of the Fermi surface between underdopedand overdoped regimes the appearance of a psedudogapin the underdoped regime and the striking differentiationbetween the physics at the nodes and at the antinodes in thepseudogap regime The topological transition of the Fermisurface has been first detected by means of ARPES [82]and reflects the noteworthy differences between the quite-ordinary large Fermi surface measured in the overdopedregime [88ndash90] and quite well described by LDA calculations[91] and quantum oscillations measurements [92] and theill-defined Fermi arcs appearing in the underdoped regime[85 93ndash98] The enormous relevance of these experimentalfindings not only for the microscopic comprehension ofthe high-119879

119888superconductivity phenomenology but also for

the drafting of a general microscopic theory for stronglycorrelated materials called for many more measurements inorder to explore all possible aspects of such extremely anoma-lous and peculiar behavior plenty of quantum oscillationsmeasurements in the underdoped regime [99ndash118] Hall effectmeasurements [100] Seebeck effect measurements [116] andheat capacity measurements [115] The presence of a quitestrong depletion in the electronic density of states knownas pseudogap [119] is well established thanks to ARPES [82]NMR [120] optical conductivity [121] and quantum oscilla-tions [122] measurements The microscopic origin of such aloss of single-particle electronic states is still unclear and thenumber of possible theoretical as well as phenomenologicalexplanations has grown quite large in the last few years Asa matter of fact this phenomenon affects any measurableproperties and accordingly was the first to be detected in theunderdoped regime granting to this latter the first evidencesof its exceptionality with respect to the other regimes inthe phase diagram The plethora of theoretical scenariospresent in the literature [85 123ndash125] tentatively explainingfew some or many of the anomalous features reported bythe experiments on underdoped cuprates can be coarselydivided into those not relying on any translational symmetrybreaking [126ndash131] and those instead proposing that it shouldbe some kind of charge andor spin arrangements to be











2 Framework


119867 = sum


dagger

(119894) 119888 (119895) + 119880sum

i119899uarr(119894) 119899

darr(119894) (1)

where

119888 (119894) = (

119888uarr(119894)

119888darr(119894)

) (2)



lattice 119899120590(119894) = 119888

dagger

120590(119894)119888





120572ij =1

119873

sum

keiksdot(RiminusRj)

120572 (k)

120572 (k) = 1

2

[cos (119896119909119886) + cos (119896

119910119886)]

(3)



120595 (119894) = (

120585 (119894)

120578 (119894)

) (4)


i 120597120597119905

120595 (119894) = 119869 (119894) = (

minus120583120585 (119894) minus 4119905119888

120572(119894) minus 4119905120587 (119894)

(119880 minus 120583) 120578 (119894) + 4119905120587 (119894)

) (5)


120587 (119894) =

1

2

120590

120583119899120583(119894) 119888

120572

(119894) + 119888 (119894) 119888

dagger120572

(119894) 119888 (119894) (6)


dagger(119894)120590

120583119888(119894) is the particle


119896(119896 = 1 2 3) are the Pauli



the form

119869 (119894) = 120576 (minusinabla)120595 (119894) + 120575119869 (119894) (7)


⟨120575119869 (i 119905) 120595dagger(j 119905)⟩ = 0 (8)





radic119873

sum

keiksdotRi

120595 (k 119905)

=

1

radic119873

sum

keiksdotRi

120576 (k) 120595 (k 119905) (9)




119898(k) = 120576 (k) 119868 (k) (10)


119868 (i j) = ⟨120595 (i 119905) 120595dagger(j 119905)⟩

=

1

119873

sum

keiksdot(RiminusRj)

119868 (k) (11)


119898(i j) = ⟨119869 (i 119905) 120595dagger(j 119905)⟩

=

1

119873

sum

keiksdot(RiminusRj)

119898(k) (12)



119866 (119894 119895) = ⟨119877 [120595 (119894) 120595

dagger(119895)]⟩

= 120579 (119905119894minus 119905

119895) ⟨120595 (119894) 120595

dagger(119895)⟩

(13)


Λ (120597119894) 119866 (119894 119895) Λ

dagger(120597119895) = Λ (120597

119894) 119866

0(119894 119895) Λ

dagger(120597119895)

+ ⟨119877 [120575119869 (119894) 120575119869

dagger(119895)]⟩

(14)


Λ (120597119894) = i 120597

120597119905119894



Λ (120597119894) 119866

0(119894 119895) = i120575 (119905

119894minus 119905

119895) 119868 (119894 119895) (16)


119866 (119894 119895) =

1

119873

sum

k

i2120587



119866 (k 120596) = 119866

0

(k 120596) + 119866

0

(k 120596) 119868minus1 (k) Σ (k 120596) 119866 (k 120596) (18)


119866 (k 120596) = 1

120596 minus 120576 (k) minus Σ (k 120596)119868 (k) (19)


Σ (k 120596) = 119861irr (k 120596) 119868minus1

(k) (20)

with

119861 (k 120596) = F ⟨119877 [120575119869 (119894) 120575119869

dagger(119895)]⟩ (21)



0(k 120596) and




119866

0

(k 120596) = 1

120596 minus 120576 (k)119868 (k) (22)


119868 (k) = (

1 minus

119899

2

0

0

119899

2

) = (

11986811

0

0 11986822

)

12057611(k) = minus120583 minus 4119905119868

minus1

11[Δ + (1 minus 119899 + 119901) 120572 (k)]

12057612(k) = 4119905119868

minus1

22[Δ + (119901 minus 119868

22) 120572 (k)]

12057621(k) = 4119905119868

minus1

11[Δ + (119901 minus 119868

22) 120572 (k)]

12057622(k) = 119880 minus 120583 minus 4119905119868

minus1

22[Δ + 119901120572 (k)]

(23)



Δ = ⟨120585

120572

(119894) 120585

dagger

(119894)⟩ minus ⟨120578

120572

(119894) 120578

dagger

(119894)⟩

119901 =

1

4

⟨119899

120572

120583(119894) 119899

120583(119894)⟩ minus ⟨[119888

uarr(119894) 119888

darr(119894)]

120572

119888

dagger

darr(119894) 119888

dagger

uarr(119894)⟩

(24)


119866

0

(k 120596) =2

sum

119899=1

120590

(119899)(k)

120596 minus 119864119899(k) + i120575

(25)


120590


1198641(k) = 119877 (k) + 119876 (k)

1198642(k) = 119877 (k) minus 119876 (k)

120590

(1)

11(k) = 119868

11

2

[1 +

119892 (k)2119876 (k)

]

120590

(2)

11(k) = 119868

11

2

[1 minus

119892 (k)2119876 (k)

]

120590

(1)

12(k) = 119898

12(k)

2119876 (k)

120590

(2)

12(k) = minus

11989812(k)

2119876 (k)

120590

(1)

22(k) = 119868

22

2

[1 minus

119892 (k)2119876 (k)

]

120590

(2)

22(k) = 119868

22

2

[1 +

119892 (k)2119876 (k)

]

(26)

where

119877 (k) = minus 120583 minus 4119905120572 (k) + 1

2

119880 minus

12057612(k)

211986811

119876 (k) = 1

2

radic119892

2(k) +

4120576

2

12(k) 119868

22

11986811

119892 (k) = minus 119880 +

1 minus 119899

11986811

12057612(k)

(27)





119869 (i 119905) = sum

j119886 (i j 119905) 120595 (j 119905) (28)

where


120583119899120583(119894) 120572ij

11988612(i j 119905) = minus4119905120572ij minus 2119905120590

120583119899120583(119894) 120572ij

11988621(i j 119905) = 2119905120590

120583119899120583(119894) 120572ij

11988622(i j 119905) = (119880 minus 120583) 120575ij + 2119905120590

120583119899120583(119894) 120572ij

(29)



we approximate

120575119869 (i 119905) asymp sum

j[119886 (i j 119905) minus ⟨119886 (i j 119905)⟩] 120595 (j 119905) (30)

Therefore

119861irr (119894 119895) = 4119905

2119865 (119894 119895) (1 minus 120590

1) (31)

where we defined

119865 (119894 119895) = ⟨119877 [120590

120583120575119899

120583(119894) 119888

120572

(119894) 119888

dagger120572(119895) 120575119899

120582(119895) 120590

120582]⟩ (32)

with 120575119899120583(119894) = 119899

120583(119894) minus ⟨119899


as

Σ (k 120596) = 4119905

2119865 (k 120596)(

119868

minus2

11minus119868

minus1

11119868

minus1

22

minus119868

minus1

11119868

minus1

22119868

minus2

22

) (33)


119865 (119894 119895) =

119894

2120587

int

+infin

minusinfin

119889120596eminusi120596(119905119894minus119905119895)

times

1

2120587

int

+infin

minusinfin

119889120596

1015840 1 + eminus1205731205961015840

120596 minus 120596

1015840+ i120576

119862 (i minus j 1205961015840)

(34)


119862 (119894 119895) = ⟨120590

120583120575119899

120583(119894) 119888

120572

(119894) 119888

dagger120572(119895) 120575119899

120582(119895) 120590

120582⟩

=

1

2120587



⟨120590

120583120575119899

120583(119894) 119888

120572

(119894) 119888

dagger120572(119895) 120575119899

120582(119895) 120590

120582⟩

asymp ⟨120575119899120583(119894) 120575119899

120583(119895)⟩ ⟨119888

120572

(119894) 119888

dagger120572(119895)⟩

(36)



119865 (k 120596) = 1

120587

int

+infin

minusinfin

119889120596

1015840 1

120596 minus 120596

1015840+ i120575

119886

2

(2120587)

3int119889

2119901119889Ω120572

2(119901)

times [tanh120573Ω

2

+ coth120573 (120596

1015840minus Ω)

2

]


minus Ω)]

(37)


(13)]

119866119888119888(k 120596) =

2

sum

119886119887=1

119866119886119887(k 120596) (38)

120594 (k 120596) = sum

120583

F ⟨119877 [120575119899120583(119894) 120575119899

120583(119895)]⟩ (39)




119867

119877(119894 119895) = ⟨R [119861 (119894) 119865 (119894) 119865

dagger(119895) 119861

dagger(119895)]⟩ (40)


119867

119877

(k 120596)

= minus

1

120587

int

+infin

minusinfin

119889120596

1015840 1

120596 minus 120596

1015840+ i120575

coth120573120596

1015840

2

I [119867

119862(k 1205961015840

)]

(41)

where 119867119862(119894 119895) = ⟨T[119861(119894)119865(119894)119865

dagger(119895)119861



119867

119862(119894 119895) asymp 119891

119862(119894 119895) 119887

119862(119894 119895)

119891

119862(119894 119895) = ⟨T [119865 (119894) 119865

dagger(119895)]⟩

119887

119862(119894 119895) = ⟨T [119861 (119894) 119861

dagger(119895)]⟩

(42)


119891

119862

(k 120596) = minus

1

120587

int

+infin

minusinfin

119889120596

1015840[

1 minus 119891F (1205961015840)

120596 minus 120596

1015840+ i120575

+

119891F (1205961015840)

120596 minus 120596

1015840minus i120575

]

timesI [119891

119877(k 1205961015840

)]

119887

119862

(k 120596) = minus

1

120587

int

+infin

minusinfin

119889120596

1015840[

1 + 119891119861(120596

1015840)

120596 minus 120596

1015840+ i120575

minus

119891119861(120596

1015840)

120596 minus 120596

1015840minus i120575

]

timesI [119887

119877(k 1205961015840

)]

(43)

which leads to

119867

119877

(k 120596) = 1

120587

int

+infin

minusinfin

119889120596

1015840 1

120596 minus 120596

1015840+ i120575

119886

119889

(2120587)

119889+1

times int

Ω119861

119889

119889119901119889ΩI [119891

119877(119901 Ω)]

timesI [119887

119877(119896 minus 119901 120596

1015840minus Ω)]

times [tanh120573Ω

2

+ coth120573 (120596

1015840minus Ω)

2

]

(44)





119873

(120583)

(119894) = (

119899120583(119894)

120588120583(119894)

)

119899120583(119894) = 119888

dagger

(119894) 120590120583119888 (119894)

120588120583(119894) = 119888

dagger

(119894) 120590120583119888

120572

(119894) minus 119888

dagger120572

(119894) 120590120583119888 (119894)

(45)



i 120597120597119905

119873

(120583)

(119894) = 119869

(120583)

(119894) = (

119869

(120583)

1(119894)

119869

(120583)

2(119894)

)

119869

(120583)

1(119894) = minus4119905120588

120583(119894)

119869

(120583)

2(119894) = 119880120581

120583(119894) minus 4119905119897

120583(119894)

(46)


120583(119894) are

defined as

120581120583(119894) = 119888

dagger

(119894) 120590120583120578

120572

(119894) minus 120578

dagger

(119894) 120590120583119888

120572

(119894)

+ 120578

dagger120572

(119894) 120590120583119888 (119894) minus 119888

dagger120572

(119894) 120590120583120578 (119894)

119897120583(119894) = 119888

dagger

(119894) 120590120583119888

1205722

(119894) + 119888

dagger1205722

(119894) 120590120583119888 (119894)

minus 2119888

dagger120572

(119894) 120590120583119888

120572

(119894)

(47)


119888

1205722

(i 119905) = sum

j120572

2

ij119888 (j 119905) = sum

jl120572il120572lj119888 (j 119905) (48)




i 120597120597119905

119873

(120583)

(i 119905) = sum

j120576

(120583)(i j)119873(120583)

(j 119905) (49)


119898

(120583)(i j) = sum

l120576

(120583)

(i l) 119868(120583) (l j) (50)



119868

(120583)(i j) = ⟨[119873

(120583)

(i 119905) 119873(120583)dagger(j 119905)]⟩

119898

(120583)(i j) = ⟨[119869

(120583)

(i 119905) 119873(120583)dagger(j 119905)]⟩

(51)



119868

(120583)

(k) = (

0 119868

(120583)

12(k)

119868

(120583)

12(k) 0

)

119898

(120583)

(k) = (

119898

(120583)

11(k) 0

0 119898

(120583)

22(k)

)

(52)

where

119868

(120583)

12(k) = 4 [1 minus 120572 (k)] 119862120572

119888119888

119898

(120583)

11(k) = minus 4119905119868

(120583)

12(k)

119898

(120583)

22(k) = minus 4119905119868

119897120583120588120583(k) + 119880119868

120581120583120588120583(k)

(53)


120572= ⟨119888

120572(119894)119888


119897120583120588120583(k) and 119868

120581120583120588120583(k) are

defined as

119868119897120583120588120583

(k) = F ⟨[119897120583(i 119905) 120588dagger

120583(j 119905)]⟩

119868120581120583120588120583

(k) = F ⟨[120581120583(i 119905) 120588dagger

120583(j 119905)]⟩

(54)


119866

(120583)(119894 119895) = ⟨119879 [119873

(120583)

(119894)119873

(120583)dagger(119895)]⟩

=

i1198862

(2120587)

3int119889


119866

(120583)

(k 120596)

(55)



[120596 minus 120576

(120583)

(k)] 119866(120583)

(k 120596) = 119868

(120583)

(k) (56)


120576

(120583)

(k) = (

0 120576

(120583)

12(k)

120576

(120583)

21(k) 0

)

120576

(120583)

12(k) = minus4119905

120576

(120583)

21(k) =

119898

(120583)

22(k)

119868

(120583)

12(k)

(57)


119866

(120583)

(k 120596)

= Γ

(120583)

(k) [ 1

120596 + i120575minus

1

120596 minus i120575]

+

2

sum

119899=1

120590

(119899120583)

(k) [1 + 119891B (120596)

120596 minus 120596

(120583)

119899 (k) + i120575minus

119891B (120596)

120596 minus 120596

(120583)

119899 (k) minus i120575]

(58)



(120583)(k 120596) =

⟨119873

(120583)(119894)119873


119862

(120583)

(k 120596) = 2120587Γ

(120583)

(k) 120575 (120596)

+ 2120587

2

sum

119899=1

120575 [120596 minus 120596

(120583)

119899(k)] [1 + 119891B (120596)] 120590

(119899120583)

(k)

(59)



120596

(120583)

119899(k) = (minus)

119899120596

(120583)

(k)

120596

(120583)

(k) = radic120576

(120583)

12(k) 120576(120583)

21(k)

(60)


120590

(119899120583)

(k) =119868

(120583)

12(k)2

(

120576

(120583)

12(k)

120596

(120583)

119899 (k)1

1

120576

(120583)

21(k)

120596

(120583)

119899 (k)

) (61)


119868119897120583120588120583

(k) = 3

4

[1 minus 120572 (k)] (12119862120572+ 119862

120582+ 6119862

120583)

minus 3 [1 minus 120573 (k)] (119862120572+ 119862

120583)

minus

3

4

[1 minus 120578 (k)] (119862120572+ 119862

120582+ 2119862

120583)

+

1

4

[1 minus 120582 (k)] 119862120582+

3

2

[1 minus 120583 (k)] 119862120583

119868120581120583120588120583


120578)

+ 2120573 (k) 119864120573+ 120578 (k) 119864120578

+ [1 minus 2120572 (k)] 119886120583

+

1

4

[119887120583+ 2120573 (k) 119888

120583+ 120578 (k) 119889

120583]

(62)



119864 = ⟨119888 (119894) 120578

dagger

(119894)⟩ 119862

120572= ⟨119888

120572

(119894) 119888

dagger

(119894)⟩

119864

120573= ⟨119888

120573

(119894) 120578

dagger

(119894)⟩ 119862

120582= ⟨119888

120582

(119894) 119888

dagger

(119894)⟩

119864

120578= ⟨119888

120578

(119894) 120578

dagger

(119894)⟩ 119862

120583= ⟨119888

120583

(119894) 119888

dagger

(119894)⟩

(63)

119886120583= 2 ⟨119888

dagger

(119894) 120590120583119888

120572

(119894) 119888

dagger

(119894) 120590120583119888

120572

(119894)⟩

minus ⟨119888

dagger120572

(119894) 120590120583120590

120582120590120583119888

120572

(119894) 119899120582(119894)⟩

119887120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger

(119894) 120590120583[119888 (119894) 119888 (119894)]

120572⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894) 119899

120572

120582(119894)⟩

119888120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger(119894

120578) 120590

120583119888 (119894

120572) 119888 (119894

120572)⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894

120578) 119899

120582(119894

120572)⟩

119889120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger(119894

120573) 120590

120583119888 (119894

120572) 119888 (119894

120572)⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894

120573) 119899

120582(119894

120572)⟩

(64)


119894 = (119894119909 119894119910 119905) 119894

120573= (119894

119909+ 119886 119894

119910+ 119886 119905)

119894

120572= (119894

119909+ 119886 119894

119910 119905) 119894

120578= (119894

119909+ 2119886 119894

119910 119905)

(65)


(120583)(119894 119895) =

⟨119879[119873

(120583)(119894)119873



120583 119887

120583 119888

120583 and 119889


Γ



dagger(119895)⟩


119887120583= 119886

120583+ 3119863 + 2119864

120573+ 119864

120578minus 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

119888120583= 119886

120583minus 119863 minus 2119864

120573+ 119864

120578+ 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

119889120583= 119886

120583minus 119863 + 2119864

120573minus 3119864

120578minus 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

(66)


(120583)

11(k) are fixed by


⟨119899120583(119894) 119899

120583(119894)⟩ =

119899 + 2119863 for 120583 = 0

119899 minus 2119863 for 120583 = 1 2 3

(67)


B0(ac0 as0)

G0(1205830 Δ0 p0)

n = 0

n = n + 1

Yes No

Σn[Bn Gn]



120583n Δn pn ==


where 119863 = ⟨119899uarr(119894)119899


ergodic value

Γ

(120583)

11(k) = 120575

1205830

(2120587)

2

119886

2120575

(2)

(k) ⟨119899⟩2 (68)


⟨120575119899120583(119894) 120575119899

120583(119895)⟩

=

119886

2

2 (2120587)

2

2

sum

119899=1

int119889


(120583)

119899(k)(119905119894minus119905119895)


(120583)

119899(k)

2119896B119879]120590

(119899120583)

11(k)

(69)

⟨119877 [120575119899120583(119894) 120575119899

120583(119895)]⟩

=

i1198862

(2120587)

3

2

sum

119899=1

int119889


(70)

times

120590

(119899120583)

11(k)

120596 minus 120596

(120583)

119899 (k) + i120575 (71)



120594120583(k 120596) = minusF [⟨119877 [120575119899

120583(119894) 120575119899

120583(119895)]⟩]

=

16119905 [1 minus 120572 (k)] 119862120572

120596

2minus (120596

(120583)(k))2

(72)



119899 = 2 (1 minus 11986211minus 119862

22)

Δ = 119862

120572

11minus 119862

120572

22

11986212

= ⟨120585 (119894) 120578

dagger

(119894)⟩ = 0

(73)

where 119862119899119898

and 119862

120572


functions 119862119899119898

= ⟨120595119899(119894)120595

dagger

119898(119894)⟩ and 119862

120572

119899119898= ⟨120595

120572

119899(119894)120595

dagger

119898(119894)⟩


119888119888(k 120596) [cf (38)] and



120583(k 120596)


0and 119886



22


(k 120596) and 120594120583(k 120596) in the two-









119888119888(k 120596) = F⟨119879[119888(119894)119888

dagger(119895)]⟩


3 Results


119860 (k 120596) = minus

1

120587

I [119866119888119888(k 120596)] (74)


119899 (k) = int119889120596119891F (120596)119860 (k 120596) (75)


119873(120596) =

1

(2120587)

2int119889

2119896119860 (k 120596) (76)

where 119866119888119888(k 120596) = 119866

11(k 120596) + 119866

12(k 120596) + 119866

21(k 120596) +




119866119888119888(k 120596) = 1

120596 minus 1205980(k) minus Σ

119888119888(k 120596)

(77)



1015840

119888119888(k 120596 = 0)


120596rarr0Σ

10158401015840

119888119888(k 120596 119879 =

0) prop 120596


Σ

10158401015840

119888119888(k 120596 = 0 119879) prop 119879



120572

119911⟩ the



limkrarrQ

120594

(3)

(k 0) =120594

(3)(Q 0)

1 + 120585

2|k minusQ|

2 (78)

where 120594(3)(k 0) = minus119866














0(k 120596) and 120594






9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 07

T = 001

Σ998400998400(k 120596)

(a)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 078

T = 001

Σ998400998400(k 120596)

(b)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596A(k120596)

00

06

12

18

24

30

U = 8n = 085

T = 001

Σ998400998400(k 120596)

(c)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 092

T = 002

Σ998400998400(k 120596)

(d)










050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 07

T = 001

A(k120596)

Σ998400998400(k 120596)

120596

(a)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 078

T = 001

120596

(b)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 085

T = 001

120596

(c)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 092

T = 002

120596

(d)



(119879 = 002)








3(k 120596) (see Section 36)






00 20 40 60 80 100 120 120

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

5120587831205874

120587

120587

120587

120587

U = 8

T = 001

n = 070

A(k120596

=0)

31205878

31205878

31205878

51205878

51205878

51205878

31205874 31205874

31205874

31205874

71205878

71205878

71205878

71205878

000

0

12058781205878

4

120587444444444444444444444

1205874

1205874

1205872

1205871205871205871205871205871205871205871205871205871205872222222222222222222222222222222222222

1205872

1205872

55555555555555555555555555555555555555555555555555555555555555555552222212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444

120587

1205878

31205878

333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

51205878

51205878

51205878

31205874 31205874

3120587

33333333333333333333333331205871205871205871205871205871205871205871205871205874

71205878

71205878

7

74 1205878

kx

ky

(a)

00 20 40 60 80 100

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

3120587471205878

120587

120587

120587

120587

U = 8

T = 001

n = 078

kx

ky

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

3120587851205878

31205878

51205878

31205874

71205878

0

0

12058781205878

4

1205874 1205874

1205872

1205872

1205872

1205872

33333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444444444444444444447777777777777777777777777777777777777777777777777777777712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333333333333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

512058783120587

7

5120587831205874 71205878

3120587852212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888

1205878

51205878

31205874

71205878

(b)

00 10 20 30 40 50 60 70 80 90

00

0

0

123456789

1011

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

71205878

120587

120587

120587

120587

U = 8

T = 001n = 085

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

312058785120587831205874

00

000000000000000000000000000000000000000000

12058781205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

4

1205874 1205874

1205872

1205872

1205872

1205872

777777777777777777777777777777777777777777777777777777777771205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333331205878

512058783120587

7

3120587851205878

31205874 71205878

8

51205878

31205874

71205878

3120587852212058783333333333312058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058744444444444444444444444444444444444444444444444

kx

ky

(c)

00 10 20 30 40 50 60 70

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

120587

120587

120587

120587

U = 8

T = 002n = 092

A(k120596

=0)

31205878

5120587831205874

71205878

71205878

312058785120587831205874

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

0000

0

12058781205878

4

1205874

1205874

1205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874

1205872

1205872

1205872

1205872

120587

333331205878

512058783120587

7

77777777777777777777777120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587888888888888888888888888888888888888888

31205878522120587831205874

3120587851205878

31205874 71205878

1205878

51205878

31205874

71205878

kx

ky

(d)







119888119888(k 120596 = 0) (notice that



119888119888(k 120596 = 0) is quite


119888119888(k 120596 = 0) in those





00

25

50

75

100

125

r(k) = 0

n(k) = 05 U = 8

n = 070

T = 001

minus010 minus020

minus030

minus040minus050

Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

ky

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(a)

minus010


minus050

0

1205878

1205874

31205878

1205872

51205878

31205874

71205878

120587

0 1205878 1205874 31205878 1205872 51205878 31205874 71205878 120587

r(k) = 0

n(k) = 05

kx

k y

00

21

42

63

84

105

A(k120596

=0)

U = 8

n = 078

T = 001Σ998400998400cc(k 120596 = 0)

(b)

00

18

36

54

72

90

r(k) = 0

n(k) = 05 U = 8

n = 085

T = 001Σ998400998400cc(k 120596 = 0)

A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(c)

r(k) = 0

n(k) = 05 U = 8

n = 092

T = 002Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y00

14

28

42

56

70

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(d)








00

02

04

06

08

10

12

n = 070

n = 078n = 085

n = 092


120596

N(120596)

U = 8 T = 001 U = 8 T = 002

(a)

1

2

3

4

5

6

7

8


U = 8T = 002n = 092

A(k

=k120596)

120596

k = S

k = S

k = X

(b)









YX SS M

k

n(k)

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

n = 070

n = 075

n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

00

01

02

03

04

05

06

07

08

09

10

(a)

S M

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

kΓ

n = 070n = 075n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

(b)







and 119899 = 07 075 08 and 085 (119879 = 001) and for 119899 =





09

0807 06

05

0403

0202

00

01

02

03

04

05

06

07

08

09

10

n(k)

U = 8n = 092T = 001

Y

X

M

Γ














YXS M

U= 8[[

n = 092

T = 100

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrT= 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

(a)

YXS M

U = 01 [[

n = 092

T = 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrU = 7

(b)




















119911119899

120572






119911119899

120572

119911⟩ is






120596

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

minus08

minus06

minus04

minus02

00

02

04

06

08

(a)


120596

minus08

minus06

minus04

minus02

00

02

04

06

08

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

(b)

000 001 002 003 004 005minus012

minus010

minus008

minus006

minus004

minus002

000

T

Σ998400998400(k120596

=0)

n = 092

U = 8k = Sk = S

(c)

minus012

minus010

minus008

minus006

minus004

minus002

000

Σ998400998400(k120596

=0)

0000 0001 0002

T2

n = 092

U = 8k = Sk = S

(d)





guide to the eye






T

n

00 01 02 03 04 05 06 07 08 09 10minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009090 087 084 081 078 075 072

n = 092

U = 8

T = 001

⟨nzn120572 z⟩

8 7 6 5 4 3 2 1 0

U

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8T = 001

⟨nzn120572 z⟩


120572



0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

n

090 087 084 081 078 075 072

120585

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

78 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

120585

n = 092

U = 8

T = 001


00

02

04

06

08

10

12

14

16

00

02

04

06

08

10

12

14

16

n

090 087 084 081 078 075 072

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

00

04

08

12

16

20

24

28

32

00

04

08

12

16

20

24

28

32

8 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

120596(Q

)

120596(Q

)
















Acknowledgments


References












































































2minus119909Sr

119909CuO


























2CuO

6+120575









Na119909CuO

2Cl

2rdquo Science









2Sr

2minus119909LaxCuO






2Cu

4O

8rdquo Physical




119888



2Cu

3O

65rdquo








2Cu

3O

661and YBa

2Cu

3O

669



2Sr

2CaCu

2O




2Sr

2CaCu

2O





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics













2 Framework


119867 = sum


dagger

(119894) 119888 (119895) + 119880sum

i119899uarr(119894) 119899

darr(119894) (1)

where

119888 (119894) = (

119888uarr(119894)

119888darr(119894)

) (2)



lattice 119899120590(119894) = 119888

dagger

120590(119894)119888





120572ij =1

119873

sum

keiksdot(RiminusRj)

120572 (k)

120572 (k) = 1

2

[cos (119896119909119886) + cos (119896

119910119886)]

(3)



120595 (119894) = (

120585 (119894)

120578 (119894)

) (4)


i 120597120597119905

120595 (119894) = 119869 (119894) = (

minus120583120585 (119894) minus 4119905119888

120572(119894) minus 4119905120587 (119894)

(119880 minus 120583) 120578 (119894) + 4119905120587 (119894)

) (5)


120587 (119894) =

1

2

120590

120583119899120583(119894) 119888

120572

(119894) + 119888 (119894) 119888

dagger120572

(119894) 119888 (119894) (6)


dagger(119894)120590

120583119888(119894) is the particle


119896(119896 = 1 2 3) are the Pauli



the form

119869 (119894) = 120576 (minusinabla)120595 (119894) + 120575119869 (119894) (7)


⟨120575119869 (i 119905) 120595dagger(j 119905)⟩ = 0 (8)





radic119873

sum

keiksdotRi

120595 (k 119905)

=

1

radic119873

sum

keiksdotRi

120576 (k) 120595 (k 119905) (9)




119898(k) = 120576 (k) 119868 (k) (10)


119868 (i j) = ⟨120595 (i 119905) 120595dagger(j 119905)⟩

=

1

119873

sum

keiksdot(RiminusRj)

119868 (k) (11)


119898(i j) = ⟨119869 (i 119905) 120595dagger(j 119905)⟩

=

1

119873

sum

keiksdot(RiminusRj)

119898(k) (12)



119866 (119894 119895) = ⟨119877 [120595 (119894) 120595

dagger(119895)]⟩

= 120579 (119905119894minus 119905

119895) ⟨120595 (119894) 120595

dagger(119895)⟩

(13)


Λ (120597119894) 119866 (119894 119895) Λ

dagger(120597119895) = Λ (120597

119894) 119866

0(119894 119895) Λ

dagger(120597119895)

+ ⟨119877 [120575119869 (119894) 120575119869

dagger(119895)]⟩

(14)


Λ (120597119894) = i 120597

120597119905119894



Λ (120597119894) 119866

0(119894 119895) = i120575 (119905

119894minus 119905

119895) 119868 (119894 119895) (16)


119866 (119894 119895) =

1

119873

sum

k

i2120587



119866 (k 120596) = 119866

0

(k 120596) + 119866

0

(k 120596) 119868minus1 (k) Σ (k 120596) 119866 (k 120596) (18)


119866 (k 120596) = 1

120596 minus 120576 (k) minus Σ (k 120596)119868 (k) (19)


Σ (k 120596) = 119861irr (k 120596) 119868minus1

(k) (20)

with

119861 (k 120596) = F ⟨119877 [120575119869 (119894) 120575119869

dagger(119895)]⟩ (21)



0(k 120596) and




119866

0

(k 120596) = 1

120596 minus 120576 (k)119868 (k) (22)


119868 (k) = (

1 minus

119899

2

0

0

119899

2

) = (

11986811

0

0 11986822

)

12057611(k) = minus120583 minus 4119905119868

minus1

11[Δ + (1 minus 119899 + 119901) 120572 (k)]

12057612(k) = 4119905119868

minus1

22[Δ + (119901 minus 119868

22) 120572 (k)]

12057621(k) = 4119905119868

minus1

11[Δ + (119901 minus 119868

22) 120572 (k)]

12057622(k) = 119880 minus 120583 minus 4119905119868

minus1

22[Δ + 119901120572 (k)]

(23)



Δ = ⟨120585

120572

(119894) 120585

dagger

(119894)⟩ minus ⟨120578

120572

(119894) 120578

dagger

(119894)⟩

119901 =

1

4

⟨119899

120572

120583(119894) 119899

120583(119894)⟩ minus ⟨[119888

uarr(119894) 119888

darr(119894)]

120572

119888

dagger

darr(119894) 119888

dagger

uarr(119894)⟩

(24)


119866

0

(k 120596) =2

sum

119899=1

120590

(119899)(k)

120596 minus 119864119899(k) + i120575

(25)


120590


1198641(k) = 119877 (k) + 119876 (k)

1198642(k) = 119877 (k) minus 119876 (k)

120590

(1)

11(k) = 119868

11

2

[1 +

119892 (k)2119876 (k)

]

120590

(2)

11(k) = 119868

11

2

[1 minus

119892 (k)2119876 (k)

]

120590

(1)

12(k) = 119898

12(k)

2119876 (k)

120590

(2)

12(k) = minus

11989812(k)

2119876 (k)

120590

(1)

22(k) = 119868

22

2

[1 minus

119892 (k)2119876 (k)

]

120590

(2)

22(k) = 119868

22

2

[1 +

119892 (k)2119876 (k)

]

(26)

where

119877 (k) = minus 120583 minus 4119905120572 (k) + 1

2

119880 minus

12057612(k)

211986811

119876 (k) = 1

2

radic119892

2(k) +

4120576

2

12(k) 119868

22

11986811

119892 (k) = minus 119880 +

1 minus 119899

11986811

12057612(k)

(27)





119869 (i 119905) = sum

j119886 (i j 119905) 120595 (j 119905) (28)

where


120583119899120583(119894) 120572ij

11988612(i j 119905) = minus4119905120572ij minus 2119905120590

120583119899120583(119894) 120572ij

11988621(i j 119905) = 2119905120590

120583119899120583(119894) 120572ij

11988622(i j 119905) = (119880 minus 120583) 120575ij + 2119905120590

120583119899120583(119894) 120572ij

(29)



we approximate

120575119869 (i 119905) asymp sum

j[119886 (i j 119905) minus ⟨119886 (i j 119905)⟩] 120595 (j 119905) (30)

Therefore

119861irr (119894 119895) = 4119905

2119865 (119894 119895) (1 minus 120590

1) (31)

where we defined

119865 (119894 119895) = ⟨119877 [120590

120583120575119899

120583(119894) 119888

120572

(119894) 119888

dagger120572(119895) 120575119899

120582(119895) 120590

120582]⟩ (32)

with 120575119899120583(119894) = 119899

120583(119894) minus ⟨119899


as

Σ (k 120596) = 4119905

2119865 (k 120596)(

119868

minus2

11minus119868

minus1

11119868

minus1

22

minus119868

minus1

11119868

minus1

22119868

minus2

22

) (33)


119865 (119894 119895) =

119894

2120587

int

+infin

minusinfin

119889120596eminusi120596(119905119894minus119905119895)

times

1

2120587

int

+infin

minusinfin

119889120596

1015840 1 + eminus1205731205961015840

120596 minus 120596

1015840+ i120576

119862 (i minus j 1205961015840)

(34)


119862 (119894 119895) = ⟨120590

120583120575119899

120583(119894) 119888

120572

(119894) 119888

dagger120572(119895) 120575119899

120582(119895) 120590

120582⟩

=

1

2120587



⟨120590

120583120575119899

120583(119894) 119888

120572

(119894) 119888

dagger120572(119895) 120575119899

120582(119895) 120590

120582⟩

asymp ⟨120575119899120583(119894) 120575119899

120583(119895)⟩ ⟨119888

120572

(119894) 119888

dagger120572(119895)⟩

(36)



119865 (k 120596) = 1

120587

int

+infin

minusinfin

119889120596

1015840 1

120596 minus 120596

1015840+ i120575

119886

2

(2120587)

3int119889

2119901119889Ω120572

2(119901)

times [tanh120573Ω

2

+ coth120573 (120596

1015840minus Ω)

2

]


minus Ω)]

(37)


(13)]

119866119888119888(k 120596) =

2

sum

119886119887=1

119866119886119887(k 120596) (38)

120594 (k 120596) = sum

120583

F ⟨119877 [120575119899120583(119894) 120575119899

120583(119895)]⟩ (39)




119867

119877(119894 119895) = ⟨R [119861 (119894) 119865 (119894) 119865

dagger(119895) 119861

dagger(119895)]⟩ (40)


119867

119877

(k 120596)

= minus

1

120587

int

+infin

minusinfin

119889120596

1015840 1

120596 minus 120596

1015840+ i120575

coth120573120596

1015840

2

I [119867

119862(k 1205961015840

)]

(41)

where 119867119862(119894 119895) = ⟨T[119861(119894)119865(119894)119865

dagger(119895)119861



119867

119862(119894 119895) asymp 119891

119862(119894 119895) 119887

119862(119894 119895)

119891

119862(119894 119895) = ⟨T [119865 (119894) 119865

dagger(119895)]⟩

119887

119862(119894 119895) = ⟨T [119861 (119894) 119861

dagger(119895)]⟩

(42)


119891

119862

(k 120596) = minus

1

120587

int

+infin

minusinfin

119889120596

1015840[

1 minus 119891F (1205961015840)

120596 minus 120596

1015840+ i120575

+

119891F (1205961015840)

120596 minus 120596

1015840minus i120575

]

timesI [119891

119877(k 1205961015840

)]

119887

119862

(k 120596) = minus

1

120587

int

+infin

minusinfin

119889120596

1015840[

1 + 119891119861(120596

1015840)

120596 minus 120596

1015840+ i120575

minus

119891119861(120596

1015840)

120596 minus 120596

1015840minus i120575

]

timesI [119887

119877(k 1205961015840

)]

(43)

which leads to

119867

119877

(k 120596) = 1

120587

int

+infin

minusinfin

119889120596

1015840 1

120596 minus 120596

1015840+ i120575

119886

119889

(2120587)

119889+1

times int

Ω119861

119889

119889119901119889ΩI [119891

119877(119901 Ω)]

timesI [119887

119877(119896 minus 119901 120596

1015840minus Ω)]

times [tanh120573Ω

2

+ coth120573 (120596

1015840minus Ω)

2

]

(44)





119873

(120583)

(119894) = (

119899120583(119894)

120588120583(119894)

)

119899120583(119894) = 119888

dagger

(119894) 120590120583119888 (119894)

120588120583(119894) = 119888

dagger

(119894) 120590120583119888

120572

(119894) minus 119888

dagger120572

(119894) 120590120583119888 (119894)

(45)



i 120597120597119905

119873

(120583)

(119894) = 119869

(120583)

(119894) = (

119869

(120583)

1(119894)

119869

(120583)

2(119894)

)

119869

(120583)

1(119894) = minus4119905120588

120583(119894)

119869

(120583)

2(119894) = 119880120581

120583(119894) minus 4119905119897

120583(119894)

(46)


120583(119894) are

defined as

120581120583(119894) = 119888

dagger

(119894) 120590120583120578

120572

(119894) minus 120578

dagger

(119894) 120590120583119888

120572

(119894)

+ 120578

dagger120572

(119894) 120590120583119888 (119894) minus 119888

dagger120572

(119894) 120590120583120578 (119894)

119897120583(119894) = 119888

dagger

(119894) 120590120583119888

1205722

(119894) + 119888

dagger1205722

(119894) 120590120583119888 (119894)

minus 2119888

dagger120572

(119894) 120590120583119888

120572

(119894)

(47)


119888

1205722

(i 119905) = sum

j120572

2

ij119888 (j 119905) = sum

jl120572il120572lj119888 (j 119905) (48)




i 120597120597119905

119873

(120583)

(i 119905) = sum

j120576

(120583)(i j)119873(120583)

(j 119905) (49)


119898

(120583)(i j) = sum

l120576

(120583)

(i l) 119868(120583) (l j) (50)



119868

(120583)(i j) = ⟨[119873

(120583)

(i 119905) 119873(120583)dagger(j 119905)]⟩

119898

(120583)(i j) = ⟨[119869

(120583)

(i 119905) 119873(120583)dagger(j 119905)]⟩

(51)



119868

(120583)

(k) = (

0 119868

(120583)

12(k)

119868

(120583)

12(k) 0

)

119898

(120583)

(k) = (

119898

(120583)

11(k) 0

0 119898

(120583)

22(k)

)

(52)

where

119868

(120583)

12(k) = 4 [1 minus 120572 (k)] 119862120572

119888119888

119898

(120583)

11(k) = minus 4119905119868

(120583)

12(k)

119898

(120583)

22(k) = minus 4119905119868

119897120583120588120583(k) + 119880119868

120581120583120588120583(k)

(53)


120572= ⟨119888

120572(119894)119888


119897120583120588120583(k) and 119868

120581120583120588120583(k) are

defined as

119868119897120583120588120583

(k) = F ⟨[119897120583(i 119905) 120588dagger

120583(j 119905)]⟩

119868120581120583120588120583

(k) = F ⟨[120581120583(i 119905) 120588dagger

120583(j 119905)]⟩

(54)


119866

(120583)(119894 119895) = ⟨119879 [119873

(120583)

(119894)119873

(120583)dagger(119895)]⟩

=

i1198862

(2120587)

3int119889


119866

(120583)

(k 120596)

(55)



[120596 minus 120576

(120583)

(k)] 119866(120583)

(k 120596) = 119868

(120583)

(k) (56)


120576

(120583)

(k) = (

0 120576

(120583)

12(k)

120576

(120583)

21(k) 0

)

120576

(120583)

12(k) = minus4119905

120576

(120583)

21(k) =

119898

(120583)

22(k)

119868

(120583)

12(k)

(57)


119866

(120583)

(k 120596)

= Γ

(120583)

(k) [ 1

120596 + i120575minus

1

120596 minus i120575]

+

2

sum

119899=1

120590

(119899120583)

(k) [1 + 119891B (120596)

120596 minus 120596

(120583)

119899 (k) + i120575minus

119891B (120596)

120596 minus 120596

(120583)

119899 (k) minus i120575]

(58)



(120583)(k 120596) =

⟨119873

(120583)(119894)119873


119862

(120583)

(k 120596) = 2120587Γ

(120583)

(k) 120575 (120596)

+ 2120587

2

sum

119899=1

120575 [120596 minus 120596

(120583)

119899(k)] [1 + 119891B (120596)] 120590

(119899120583)

(k)

(59)



120596

(120583)

119899(k) = (minus)

119899120596

(120583)

(k)

120596

(120583)

(k) = radic120576

(120583)

12(k) 120576(120583)

21(k)

(60)


120590

(119899120583)

(k) =119868

(120583)

12(k)2

(

120576

(120583)

12(k)

120596

(120583)

119899 (k)1

1

120576

(120583)

21(k)

120596

(120583)

119899 (k)

) (61)


119868119897120583120588120583

(k) = 3

4

[1 minus 120572 (k)] (12119862120572+ 119862

120582+ 6119862

120583)

minus 3 [1 minus 120573 (k)] (119862120572+ 119862

120583)

minus

3

4

[1 minus 120578 (k)] (119862120572+ 119862

120582+ 2119862

120583)

+

1

4

[1 minus 120582 (k)] 119862120582+

3

2

[1 minus 120583 (k)] 119862120583

119868120581120583120588120583


120578)

+ 2120573 (k) 119864120573+ 120578 (k) 119864120578

+ [1 minus 2120572 (k)] 119886120583

+

1

4

[119887120583+ 2120573 (k) 119888

120583+ 120578 (k) 119889

120583]

(62)



119864 = ⟨119888 (119894) 120578

dagger

(119894)⟩ 119862

120572= ⟨119888

120572

(119894) 119888

dagger

(119894)⟩

119864

120573= ⟨119888

120573

(119894) 120578

dagger

(119894)⟩ 119862

120582= ⟨119888

120582

(119894) 119888

dagger

(119894)⟩

119864

120578= ⟨119888

120578

(119894) 120578

dagger

(119894)⟩ 119862

120583= ⟨119888

120583

(119894) 119888

dagger

(119894)⟩

(63)

119886120583= 2 ⟨119888

dagger

(119894) 120590120583119888

120572

(119894) 119888

dagger

(119894) 120590120583119888

120572

(119894)⟩

minus ⟨119888

dagger120572

(119894) 120590120583120590

120582120590120583119888

120572

(119894) 119899120582(119894)⟩

119887120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger

(119894) 120590120583[119888 (119894) 119888 (119894)]

120572⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894) 119899

120572

120582(119894)⟩

119888120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger(119894

120578) 120590

120583119888 (119894

120572) 119888 (119894

120572)⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894

120578) 119899

120582(119894

120572)⟩

119889120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger(119894

120573) 120590

120583119888 (119894

120572) 119888 (119894

120572)⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894

120573) 119899

120582(119894

120572)⟩

(64)


119894 = (119894119909 119894119910 119905) 119894

120573= (119894

119909+ 119886 119894

119910+ 119886 119905)

119894

120572= (119894

119909+ 119886 119894

119910 119905) 119894

120578= (119894

119909+ 2119886 119894

119910 119905)

(65)


(120583)(119894 119895) =

⟨119879[119873

(120583)(119894)119873



120583 119887

120583 119888

120583 and 119889


Γ



dagger(119895)⟩


119887120583= 119886

120583+ 3119863 + 2119864

120573+ 119864

120578minus 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

119888120583= 119886

120583minus 119863 minus 2119864

120573+ 119864

120578+ 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

119889120583= 119886

120583minus 119863 + 2119864

120573minus 3119864

120578minus 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

(66)


(120583)

11(k) are fixed by


⟨119899120583(119894) 119899

120583(119894)⟩ =

119899 + 2119863 for 120583 = 0

119899 minus 2119863 for 120583 = 1 2 3

(67)


B0(ac0 as0)

G0(1205830 Δ0 p0)

n = 0

n = n + 1

Yes No

Σn[Bn Gn]



120583n Δn pn ==


where 119863 = ⟨119899uarr(119894)119899


ergodic value

Γ

(120583)

11(k) = 120575

1205830

(2120587)

2

119886

2120575

(2)

(k) ⟨119899⟩2 (68)


⟨120575119899120583(119894) 120575119899

120583(119895)⟩

=

119886

2

2 (2120587)

2

2

sum

119899=1

int119889


(120583)

119899(k)(119905119894minus119905119895)


(120583)

119899(k)

2119896B119879]120590

(119899120583)

11(k)

(69)

⟨119877 [120575119899120583(119894) 120575119899

120583(119895)]⟩

=

i1198862

(2120587)

3

2

sum

119899=1

int119889


(70)

times

120590

(119899120583)

11(k)

120596 minus 120596

(120583)

119899 (k) + i120575 (71)



120594120583(k 120596) = minusF [⟨119877 [120575119899

120583(119894) 120575119899

120583(119895)]⟩]

=

16119905 [1 minus 120572 (k)] 119862120572

120596

2minus (120596

(120583)(k))2

(72)



119899 = 2 (1 minus 11986211minus 119862

22)

Δ = 119862

120572

11minus 119862

120572

22

11986212

= ⟨120585 (119894) 120578

dagger

(119894)⟩ = 0

(73)

where 119862119899119898

and 119862

120572


functions 119862119899119898

= ⟨120595119899(119894)120595

dagger

119898(119894)⟩ and 119862

120572

119899119898= ⟨120595

120572

119899(119894)120595

dagger

119898(119894)⟩


119888119888(k 120596) [cf (38)] and



120583(k 120596)


0and 119886



22


(k 120596) and 120594120583(k 120596) in the two-









119888119888(k 120596) = F⟨119879[119888(119894)119888

dagger(119895)]⟩


3 Results


119860 (k 120596) = minus

1

120587

I [119866119888119888(k 120596)] (74)


119899 (k) = int119889120596119891F (120596)119860 (k 120596) (75)


119873(120596) =

1

(2120587)

2int119889

2119896119860 (k 120596) (76)

where 119866119888119888(k 120596) = 119866

11(k 120596) + 119866

12(k 120596) + 119866

21(k 120596) +




119866119888119888(k 120596) = 1

120596 minus 1205980(k) minus Σ

119888119888(k 120596)

(77)



1015840

119888119888(k 120596 = 0)


120596rarr0Σ

10158401015840

119888119888(k 120596 119879 =

0) prop 120596


Σ

10158401015840

119888119888(k 120596 = 0 119879) prop 119879



120572

119911⟩ the



limkrarrQ

120594

(3)

(k 0) =120594

(3)(Q 0)

1 + 120585

2|k minusQ|

2 (78)

where 120594(3)(k 0) = minus119866














0(k 120596) and 120594






9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 07

T = 001

Σ998400998400(k 120596)

(a)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 078

T = 001

Σ998400998400(k 120596)

(b)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596A(k120596)

00

06

12

18

24

30

U = 8n = 085

T = 001

Σ998400998400(k 120596)

(c)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 092

T = 002

Σ998400998400(k 120596)

(d)










050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 07

T = 001

A(k120596)

Σ998400998400(k 120596)

120596

(a)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 078

T = 001

120596

(b)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 085

T = 001

120596

(c)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 092

T = 002

120596

(d)



(119879 = 002)








3(k 120596) (see Section 36)






00 20 40 60 80 100 120 120

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

5120587831205874

120587

120587

120587

120587

U = 8

T = 001

n = 070

A(k120596

=0)

31205878

31205878

31205878

51205878

51205878

51205878

31205874 31205874

31205874

31205874

71205878

71205878

71205878

71205878

000

0

12058781205878

4

120587444444444444444444444

1205874

1205874

1205872

1205871205871205871205871205871205871205871205871205871205872222222222222222222222222222222222222

1205872

1205872

55555555555555555555555555555555555555555555555555555555555555555552222212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444

120587

1205878

31205878

333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

51205878

51205878

51205878

31205874 31205874

3120587

33333333333333333333333331205871205871205871205871205871205871205871205871205874

71205878

71205878

7

74 1205878

kx

ky

(a)

00 20 40 60 80 100

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

3120587471205878

120587

120587

120587

120587

U = 8

T = 001

n = 078

kx

ky

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

3120587851205878

31205878

51205878

31205874

71205878

0

0

12058781205878

4

1205874 1205874

1205872

1205872

1205872

1205872

33333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444444444444444444447777777777777777777777777777777777777777777777777777777712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333333333333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

512058783120587

7

5120587831205874 71205878

3120587852212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888

1205878

51205878

31205874

71205878

(b)

00 10 20 30 40 50 60 70 80 90

00

0

0

123456789

1011

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

71205878

120587

120587

120587

120587

U = 8

T = 001n = 085

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

312058785120587831205874

00

000000000000000000000000000000000000000000

12058781205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

4

1205874 1205874

1205872

1205872

1205872

1205872

777777777777777777777777777777777777777777777777777777777771205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333331205878

512058783120587

7

3120587851205878

31205874 71205878

8

51205878

31205874

71205878

3120587852212058783333333333312058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058744444444444444444444444444444444444444444444444

kx

ky

(c)

00 10 20 30 40 50 60 70

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

120587

120587

120587

120587

U = 8

T = 002n = 092

A(k120596

=0)

31205878

5120587831205874

71205878

71205878

312058785120587831205874

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

0000

0

12058781205878

4

1205874

1205874

1205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874

1205872

1205872

1205872

1205872

120587

333331205878

512058783120587

7

77777777777777777777777120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587888888888888888888888888888888888888888

31205878522120587831205874

3120587851205878

31205874 71205878

1205878

51205878

31205874

71205878

kx

ky

(d)







119888119888(k 120596 = 0) (notice that



119888119888(k 120596 = 0) is quite


119888119888(k 120596 = 0) in those





00

25

50

75

100

125

r(k) = 0

n(k) = 05 U = 8

n = 070

T = 001

minus010 minus020

minus030

minus040minus050

Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

ky

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(a)

minus010


minus050

0

1205878

1205874

31205878

1205872

51205878

31205874

71205878

120587

0 1205878 1205874 31205878 1205872 51205878 31205874 71205878 120587

r(k) = 0

n(k) = 05

kx

k y

00

21

42

63

84

105

A(k120596

=0)

U = 8

n = 078

T = 001Σ998400998400cc(k 120596 = 0)

(b)

00

18

36

54

72

90

r(k) = 0

n(k) = 05 U = 8

n = 085

T = 001Σ998400998400cc(k 120596 = 0)

A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(c)

r(k) = 0

n(k) = 05 U = 8

n = 092

T = 002Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y00

14

28

42

56

70

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(d)








00

02

04

06

08

10

12

n = 070

n = 078n = 085

n = 092


120596

N(120596)

U = 8 T = 001 U = 8 T = 002

(a)

1

2

3

4

5

6

7

8


U = 8T = 002n = 092

A(k

=k120596)

120596

k = S

k = S

k = X

(b)









YX SS M

k

n(k)

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

n = 070

n = 075

n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

00

01

02

03

04

05

06

07

08

09

10

(a)

S M

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

kΓ

n = 070n = 075n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

(b)







and 119899 = 07 075 08 and 085 (119879 = 001) and for 119899 =





09

0807 06

05

0403

0202

00

01

02

03

04

05

06

07

08

09

10

n(k)

U = 8n = 092T = 001

Y

X

M

Γ














YXS M

U= 8[[

n = 092

T = 100

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrT= 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

(a)

YXS M

U = 01 [[

n = 092

T = 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrU = 7

(b)




















119911119899

120572






119911119899

120572

119911⟩ is






120596

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

minus08

minus06

minus04

minus02

00

02

04

06

08

(a)


120596

minus08

minus06

minus04

minus02

00

02

04

06

08

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

(b)

000 001 002 003 004 005minus012

minus010

minus008

minus006

minus004

minus002

000

T

Σ998400998400(k120596

=0)

n = 092

U = 8k = Sk = S

(c)

minus012

minus010

minus008

minus006

minus004

minus002

000

Σ998400998400(k120596

=0)

0000 0001 0002

T2

n = 092

U = 8k = Sk = S

(d)





guide to the eye






T

n

00 01 02 03 04 05 06 07 08 09 10minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009090 087 084 081 078 075 072

n = 092

U = 8

T = 001

⟨nzn120572 z⟩

8 7 6 5 4 3 2 1 0

U

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8T = 001

⟨nzn120572 z⟩


120572



0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

n

090 087 084 081 078 075 072

120585

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

78 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

120585

n = 092

U = 8

T = 001


00

02

04

06

08

10

12

14

16

00

02

04

06

08

10

12

14

16

n

090 087 084 081 078 075 072

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

00

04

08

12

16

20

24

28

32

00

04

08

12

16

20

24

28

32

8 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

120596(Q

)

120596(Q

)
















Acknowledgments


References












































































2minus119909Sr

119909CuO


























2CuO

6+120575









Na119909CuO

2Cl

2rdquo Science









2Sr

2minus119909LaxCuO






2Cu

4O

8rdquo Physical




119888



2Cu

3O

65rdquo








2Cu

3O

661and YBa

2Cu

3O

669



2Sr

2CaCu

2O




2Sr

2CaCu

2O





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






2 Framework


119867 = sum


dagger

(119894) 119888 (119895) + 119880sum

i119899uarr(119894) 119899

darr(119894) (1)

where

119888 (119894) = (

119888uarr(119894)

119888darr(119894)

) (2)



lattice 119899120590(119894) = 119888

dagger

120590(119894)119888





120572ij =1

119873

sum

keiksdot(RiminusRj)

120572 (k)

120572 (k) = 1

2

[cos (119896119909119886) + cos (119896

119910119886)]

(3)



120595 (119894) = (

120585 (119894)

120578 (119894)

) (4)


i 120597120597119905

120595 (119894) = 119869 (119894) = (

minus120583120585 (119894) minus 4119905119888

120572(119894) minus 4119905120587 (119894)

(119880 minus 120583) 120578 (119894) + 4119905120587 (119894)

) (5)


120587 (119894) =

1

2

120590

120583119899120583(119894) 119888

120572

(119894) + 119888 (119894) 119888

dagger120572

(119894) 119888 (119894) (6)


dagger(119894)120590

120583119888(119894) is the particle


119896(119896 = 1 2 3) are the Pauli



the form

119869 (119894) = 120576 (minusinabla)120595 (119894) + 120575119869 (119894) (7)


⟨120575119869 (i 119905) 120595dagger(j 119905)⟩ = 0 (8)





radic119873

sum

keiksdotRi

120595 (k 119905)

=

1

radic119873

sum

keiksdotRi

120576 (k) 120595 (k 119905) (9)




119898(k) = 120576 (k) 119868 (k) (10)


119868 (i j) = ⟨120595 (i 119905) 120595dagger(j 119905)⟩

=

1

119873

sum

keiksdot(RiminusRj)

119868 (k) (11)


119898(i j) = ⟨119869 (i 119905) 120595dagger(j 119905)⟩

=

1

119873

sum

keiksdot(RiminusRj)

119898(k) (12)



119866 (119894 119895) = ⟨119877 [120595 (119894) 120595

dagger(119895)]⟩

= 120579 (119905119894minus 119905

119895) ⟨120595 (119894) 120595

dagger(119895)⟩

(13)


Λ (120597119894) 119866 (119894 119895) Λ

dagger(120597119895) = Λ (120597

119894) 119866

0(119894 119895) Λ

dagger(120597119895)

+ ⟨119877 [120575119869 (119894) 120575119869

dagger(119895)]⟩

(14)


Λ (120597119894) = i 120597

120597119905119894



Λ (120597119894) 119866

0(119894 119895) = i120575 (119905

119894minus 119905

119895) 119868 (119894 119895) (16)


119866 (119894 119895) =

1

119873

sum

k

i2120587



119866 (k 120596) = 119866

0

(k 120596) + 119866

0

(k 120596) 119868minus1 (k) Σ (k 120596) 119866 (k 120596) (18)


119866 (k 120596) = 1

120596 minus 120576 (k) minus Σ (k 120596)119868 (k) (19)


Σ (k 120596) = 119861irr (k 120596) 119868minus1

(k) (20)

with

119861 (k 120596) = F ⟨119877 [120575119869 (119894) 120575119869

dagger(119895)]⟩ (21)



0(k 120596) and




119866

0

(k 120596) = 1

120596 minus 120576 (k)119868 (k) (22)


119868 (k) = (

1 minus

119899

2

0

0

119899

2

) = (

11986811

0

0 11986822

)

12057611(k) = minus120583 minus 4119905119868

minus1

11[Δ + (1 minus 119899 + 119901) 120572 (k)]

12057612(k) = 4119905119868

minus1

22[Δ + (119901 minus 119868

22) 120572 (k)]

12057621(k) = 4119905119868

minus1

11[Δ + (119901 minus 119868

22) 120572 (k)]

12057622(k) = 119880 minus 120583 minus 4119905119868

minus1

22[Δ + 119901120572 (k)]

(23)



Δ = ⟨120585

120572

(119894) 120585

dagger

(119894)⟩ minus ⟨120578

120572

(119894) 120578

dagger

(119894)⟩

119901 =

1

4

⟨119899

120572

120583(119894) 119899

120583(119894)⟩ minus ⟨[119888

uarr(119894) 119888

darr(119894)]

120572

119888

dagger

darr(119894) 119888

dagger

uarr(119894)⟩

(24)


119866

0

(k 120596) =2

sum

119899=1

120590

(119899)(k)

120596 minus 119864119899(k) + i120575

(25)


120590


1198641(k) = 119877 (k) + 119876 (k)

1198642(k) = 119877 (k) minus 119876 (k)

120590

(1)

11(k) = 119868

11

2

[1 +

119892 (k)2119876 (k)

]

120590

(2)

11(k) = 119868

11

2

[1 minus

119892 (k)2119876 (k)

]

120590

(1)

12(k) = 119898

12(k)

2119876 (k)

120590

(2)

12(k) = minus

11989812(k)

2119876 (k)

120590

(1)

22(k) = 119868

22

2

[1 minus

119892 (k)2119876 (k)

]

120590

(2)

22(k) = 119868

22

2

[1 +

119892 (k)2119876 (k)

]

(26)

where

119877 (k) = minus 120583 minus 4119905120572 (k) + 1

2

119880 minus

12057612(k)

211986811

119876 (k) = 1

2

radic119892

2(k) +

4120576

2

12(k) 119868

22

11986811

119892 (k) = minus 119880 +

1 minus 119899

11986811

12057612(k)

(27)





119869 (i 119905) = sum

j119886 (i j 119905) 120595 (j 119905) (28)

where


120583119899120583(119894) 120572ij

11988612(i j 119905) = minus4119905120572ij minus 2119905120590

120583119899120583(119894) 120572ij

11988621(i j 119905) = 2119905120590

120583119899120583(119894) 120572ij

11988622(i j 119905) = (119880 minus 120583) 120575ij + 2119905120590

120583119899120583(119894) 120572ij

(29)



we approximate

120575119869 (i 119905) asymp sum

j[119886 (i j 119905) minus ⟨119886 (i j 119905)⟩] 120595 (j 119905) (30)

Therefore

119861irr (119894 119895) = 4119905

2119865 (119894 119895) (1 minus 120590

1) (31)

where we defined

119865 (119894 119895) = ⟨119877 [120590

120583120575119899

120583(119894) 119888

120572

(119894) 119888

dagger120572(119895) 120575119899

120582(119895) 120590

120582]⟩ (32)

with 120575119899120583(119894) = 119899

120583(119894) minus ⟨119899


as

Σ (k 120596) = 4119905

2119865 (k 120596)(

119868

minus2

11minus119868

minus1

11119868

minus1

22

minus119868

minus1

11119868

minus1

22119868

minus2

22

) (33)


119865 (119894 119895) =

119894

2120587

int

+infin

minusinfin

119889120596eminusi120596(119905119894minus119905119895)

times

1

2120587

int

+infin

minusinfin

119889120596

1015840 1 + eminus1205731205961015840

120596 minus 120596

1015840+ i120576

119862 (i minus j 1205961015840)

(34)


119862 (119894 119895) = ⟨120590

120583120575119899

120583(119894) 119888

120572

(119894) 119888

dagger120572(119895) 120575119899

120582(119895) 120590

120582⟩

=

1

2120587



⟨120590

120583120575119899

120583(119894) 119888

120572

(119894) 119888

dagger120572(119895) 120575119899

120582(119895) 120590

120582⟩

asymp ⟨120575119899120583(119894) 120575119899

120583(119895)⟩ ⟨119888

120572

(119894) 119888

dagger120572(119895)⟩

(36)



119865 (k 120596) = 1

120587

int

+infin

minusinfin

119889120596

1015840 1

120596 minus 120596

1015840+ i120575

119886

2

(2120587)

3int119889

2119901119889Ω120572

2(119901)

times [tanh120573Ω

2

+ coth120573 (120596

1015840minus Ω)

2

]


minus Ω)]

(37)


(13)]

119866119888119888(k 120596) =

2

sum

119886119887=1

119866119886119887(k 120596) (38)

120594 (k 120596) = sum

120583

F ⟨119877 [120575119899120583(119894) 120575119899

120583(119895)]⟩ (39)




119867

119877(119894 119895) = ⟨R [119861 (119894) 119865 (119894) 119865

dagger(119895) 119861

dagger(119895)]⟩ (40)


119867

119877

(k 120596)

= minus

1

120587

int

+infin

minusinfin

119889120596

1015840 1

120596 minus 120596

1015840+ i120575

coth120573120596

1015840

2

I [119867

119862(k 1205961015840

)]

(41)

where 119867119862(119894 119895) = ⟨T[119861(119894)119865(119894)119865

dagger(119895)119861



119867

119862(119894 119895) asymp 119891

119862(119894 119895) 119887

119862(119894 119895)

119891

119862(119894 119895) = ⟨T [119865 (119894) 119865

dagger(119895)]⟩

119887

119862(119894 119895) = ⟨T [119861 (119894) 119861

dagger(119895)]⟩

(42)


119891

119862

(k 120596) = minus

1

120587

int

+infin

minusinfin

119889120596

1015840[

1 minus 119891F (1205961015840)

120596 minus 120596

1015840+ i120575

+

119891F (1205961015840)

120596 minus 120596

1015840minus i120575

]

timesI [119891

119877(k 1205961015840

)]

119887

119862

(k 120596) = minus

1

120587

int

+infin

minusinfin

119889120596

1015840[

1 + 119891119861(120596

1015840)

120596 minus 120596

1015840+ i120575

minus

119891119861(120596

1015840)

120596 minus 120596

1015840minus i120575

]

timesI [119887

119877(k 1205961015840

)]

(43)

which leads to

119867

119877

(k 120596) = 1

120587

int

+infin

minusinfin

119889120596

1015840 1

120596 minus 120596

1015840+ i120575

119886

119889

(2120587)

119889+1

times int

Ω119861

119889

119889119901119889ΩI [119891

119877(119901 Ω)]

timesI [119887

119877(119896 minus 119901 120596

1015840minus Ω)]

times [tanh120573Ω

2

+ coth120573 (120596

1015840minus Ω)

2

]

(44)





119873

(120583)

(119894) = (

119899120583(119894)

120588120583(119894)

)

119899120583(119894) = 119888

dagger

(119894) 120590120583119888 (119894)

120588120583(119894) = 119888

dagger

(119894) 120590120583119888

120572

(119894) minus 119888

dagger120572

(119894) 120590120583119888 (119894)

(45)



i 120597120597119905

119873

(120583)

(119894) = 119869

(120583)

(119894) = (

119869

(120583)

1(119894)

119869

(120583)

2(119894)

)

119869

(120583)

1(119894) = minus4119905120588

120583(119894)

119869

(120583)

2(119894) = 119880120581

120583(119894) minus 4119905119897

120583(119894)

(46)


120583(119894) are

defined as

120581120583(119894) = 119888

dagger

(119894) 120590120583120578

120572

(119894) minus 120578

dagger

(119894) 120590120583119888

120572

(119894)

+ 120578

dagger120572

(119894) 120590120583119888 (119894) minus 119888

dagger120572

(119894) 120590120583120578 (119894)

119897120583(119894) = 119888

dagger

(119894) 120590120583119888

1205722

(119894) + 119888

dagger1205722

(119894) 120590120583119888 (119894)

minus 2119888

dagger120572

(119894) 120590120583119888

120572

(119894)

(47)


119888

1205722

(i 119905) = sum

j120572

2

ij119888 (j 119905) = sum

jl120572il120572lj119888 (j 119905) (48)




i 120597120597119905

119873

(120583)

(i 119905) = sum

j120576

(120583)(i j)119873(120583)

(j 119905) (49)


119898

(120583)(i j) = sum

l120576

(120583)

(i l) 119868(120583) (l j) (50)



119868

(120583)(i j) = ⟨[119873

(120583)

(i 119905) 119873(120583)dagger(j 119905)]⟩

119898

(120583)(i j) = ⟨[119869

(120583)

(i 119905) 119873(120583)dagger(j 119905)]⟩

(51)



119868

(120583)

(k) = (

0 119868

(120583)

12(k)

119868

(120583)

12(k) 0

)

119898

(120583)

(k) = (

119898

(120583)

11(k) 0

0 119898

(120583)

22(k)

)

(52)

where

119868

(120583)

12(k) = 4 [1 minus 120572 (k)] 119862120572

119888119888

119898

(120583)

11(k) = minus 4119905119868

(120583)

12(k)

119898

(120583)

22(k) = minus 4119905119868

119897120583120588120583(k) + 119880119868

120581120583120588120583(k)

(53)


120572= ⟨119888

120572(119894)119888


119897120583120588120583(k) and 119868

120581120583120588120583(k) are

defined as

119868119897120583120588120583

(k) = F ⟨[119897120583(i 119905) 120588dagger

120583(j 119905)]⟩

119868120581120583120588120583

(k) = F ⟨[120581120583(i 119905) 120588dagger

120583(j 119905)]⟩

(54)


119866

(120583)(119894 119895) = ⟨119879 [119873

(120583)

(119894)119873

(120583)dagger(119895)]⟩

=

i1198862

(2120587)

3int119889


119866

(120583)

(k 120596)

(55)



[120596 minus 120576

(120583)

(k)] 119866(120583)

(k 120596) = 119868

(120583)

(k) (56)


120576

(120583)

(k) = (

0 120576

(120583)

12(k)

120576

(120583)

21(k) 0

)

120576

(120583)

12(k) = minus4119905

120576

(120583)

21(k) =

119898

(120583)

22(k)

119868

(120583)

12(k)

(57)


119866

(120583)

(k 120596)

= Γ

(120583)

(k) [ 1

120596 + i120575minus

1

120596 minus i120575]

+

2

sum

119899=1

120590

(119899120583)

(k) [1 + 119891B (120596)

120596 minus 120596

(120583)

119899 (k) + i120575minus

119891B (120596)

120596 minus 120596

(120583)

119899 (k) minus i120575]

(58)



(120583)(k 120596) =

⟨119873

(120583)(119894)119873


119862

(120583)

(k 120596) = 2120587Γ

(120583)

(k) 120575 (120596)

+ 2120587

2

sum

119899=1

120575 [120596 minus 120596

(120583)

119899(k)] [1 + 119891B (120596)] 120590

(119899120583)

(k)

(59)



120596

(120583)

119899(k) = (minus)

119899120596

(120583)

(k)

120596

(120583)

(k) = radic120576

(120583)

12(k) 120576(120583)

21(k)

(60)


120590

(119899120583)

(k) =119868

(120583)

12(k)2

(

120576

(120583)

12(k)

120596

(120583)

119899 (k)1

1

120576

(120583)

21(k)

120596

(120583)

119899 (k)

) (61)


119868119897120583120588120583

(k) = 3

4

[1 minus 120572 (k)] (12119862120572+ 119862

120582+ 6119862

120583)

minus 3 [1 minus 120573 (k)] (119862120572+ 119862

120583)

minus

3

4

[1 minus 120578 (k)] (119862120572+ 119862

120582+ 2119862

120583)

+

1

4

[1 minus 120582 (k)] 119862120582+

3

2

[1 minus 120583 (k)] 119862120583

119868120581120583120588120583


120578)

+ 2120573 (k) 119864120573+ 120578 (k) 119864120578

+ [1 minus 2120572 (k)] 119886120583

+

1

4

[119887120583+ 2120573 (k) 119888

120583+ 120578 (k) 119889

120583]

(62)



119864 = ⟨119888 (119894) 120578

dagger

(119894)⟩ 119862

120572= ⟨119888

120572

(119894) 119888

dagger

(119894)⟩

119864

120573= ⟨119888

120573

(119894) 120578

dagger

(119894)⟩ 119862

120582= ⟨119888

120582

(119894) 119888

dagger

(119894)⟩

119864

120578= ⟨119888

120578

(119894) 120578

dagger

(119894)⟩ 119862

120583= ⟨119888

120583

(119894) 119888

dagger

(119894)⟩

(63)

119886120583= 2 ⟨119888

dagger

(119894) 120590120583119888

120572

(119894) 119888

dagger

(119894) 120590120583119888

120572

(119894)⟩

minus ⟨119888

dagger120572

(119894) 120590120583120590

120582120590120583119888

120572

(119894) 119899120582(119894)⟩

119887120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger

(119894) 120590120583[119888 (119894) 119888 (119894)]

120572⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894) 119899

120572

120582(119894)⟩

119888120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger(119894

120578) 120590

120583119888 (119894

120572) 119888 (119894

120572)⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894

120578) 119899

120582(119894

120572)⟩

119889120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger(119894

120573) 120590

120583119888 (119894

120572) 119888 (119894

120572)⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894

120573) 119899

120582(119894

120572)⟩

(64)


119894 = (119894119909 119894119910 119905) 119894

120573= (119894

119909+ 119886 119894

119910+ 119886 119905)

119894

120572= (119894

119909+ 119886 119894

119910 119905) 119894

120578= (119894

119909+ 2119886 119894

119910 119905)

(65)


(120583)(119894 119895) =

⟨119879[119873

(120583)(119894)119873



120583 119887

120583 119888

120583 and 119889


Γ



dagger(119895)⟩


119887120583= 119886

120583+ 3119863 + 2119864

120573+ 119864

120578minus 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

119888120583= 119886

120583minus 119863 minus 2119864

120573+ 119864

120578+ 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

119889120583= 119886

120583minus 119863 + 2119864

120573minus 3119864

120578minus 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

(66)


(120583)

11(k) are fixed by


⟨119899120583(119894) 119899

120583(119894)⟩ =

119899 + 2119863 for 120583 = 0

119899 minus 2119863 for 120583 = 1 2 3

(67)


B0(ac0 as0)

G0(1205830 Δ0 p0)

n = 0

n = n + 1

Yes No

Σn[Bn Gn]



120583n Δn pn ==


where 119863 = ⟨119899uarr(119894)119899


ergodic value

Γ

(120583)

11(k) = 120575

1205830

(2120587)

2

119886

2120575

(2)

(k) ⟨119899⟩2 (68)


⟨120575119899120583(119894) 120575119899

120583(119895)⟩

=

119886

2

2 (2120587)

2

2

sum

119899=1

int119889


(120583)

119899(k)(119905119894minus119905119895)


(120583)

119899(k)

2119896B119879]120590

(119899120583)

11(k)

(69)

⟨119877 [120575119899120583(119894) 120575119899

120583(119895)]⟩

=

i1198862

(2120587)

3

2

sum

119899=1

int119889


(70)

times

120590

(119899120583)

11(k)

120596 minus 120596

(120583)

119899 (k) + i120575 (71)



120594120583(k 120596) = minusF [⟨119877 [120575119899

120583(119894) 120575119899

120583(119895)]⟩]

=

16119905 [1 minus 120572 (k)] 119862120572

120596

2minus (120596

(120583)(k))2

(72)



119899 = 2 (1 minus 11986211minus 119862

22)

Δ = 119862

120572

11minus 119862

120572

22

11986212

= ⟨120585 (119894) 120578

dagger

(119894)⟩ = 0

(73)

where 119862119899119898

and 119862

120572


functions 119862119899119898

= ⟨120595119899(119894)120595

dagger

119898(119894)⟩ and 119862

120572

119899119898= ⟨120595

120572

119899(119894)120595

dagger

119898(119894)⟩


119888119888(k 120596) [cf (38)] and



120583(k 120596)


0and 119886



22


(k 120596) and 120594120583(k 120596) in the two-









119888119888(k 120596) = F⟨119879[119888(119894)119888

dagger(119895)]⟩


3 Results


119860 (k 120596) = minus

1

120587

I [119866119888119888(k 120596)] (74)


119899 (k) = int119889120596119891F (120596)119860 (k 120596) (75)


119873(120596) =

1

(2120587)

2int119889

2119896119860 (k 120596) (76)

where 119866119888119888(k 120596) = 119866

11(k 120596) + 119866

12(k 120596) + 119866

21(k 120596) +




119866119888119888(k 120596) = 1

120596 minus 1205980(k) minus Σ

119888119888(k 120596)

(77)



1015840

119888119888(k 120596 = 0)


120596rarr0Σ

10158401015840

119888119888(k 120596 119879 =

0) prop 120596


Σ

10158401015840

119888119888(k 120596 = 0 119879) prop 119879



120572

119911⟩ the



limkrarrQ

120594

(3)

(k 0) =120594

(3)(Q 0)

1 + 120585

2|k minusQ|

2 (78)

where 120594(3)(k 0) = minus119866














0(k 120596) and 120594






9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 07

T = 001

Σ998400998400(k 120596)

(a)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 078

T = 001

Σ998400998400(k 120596)

(b)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596A(k120596)

00

06

12

18

24

30

U = 8n = 085

T = 001

Σ998400998400(k 120596)

(c)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 092

T = 002

Σ998400998400(k 120596)

(d)










050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 07

T = 001

A(k120596)

Σ998400998400(k 120596)

120596

(a)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 078

T = 001

120596

(b)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 085

T = 001

120596

(c)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 092

T = 002

120596

(d)



(119879 = 002)








3(k 120596) (see Section 36)






00 20 40 60 80 100 120 120

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

5120587831205874

120587

120587

120587

120587

U = 8

T = 001

n = 070

A(k120596

=0)

31205878

31205878

31205878

51205878

51205878

51205878

31205874 31205874

31205874

31205874

71205878

71205878

71205878

71205878

000

0

12058781205878

4

120587444444444444444444444

1205874

1205874

1205872

1205871205871205871205871205871205871205871205871205871205872222222222222222222222222222222222222

1205872

1205872

55555555555555555555555555555555555555555555555555555555555555555552222212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444

120587

1205878

31205878

333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

51205878

51205878

51205878

31205874 31205874

3120587

33333333333333333333333331205871205871205871205871205871205871205871205871205874

71205878

71205878

7

74 1205878

kx

ky

(a)

00 20 40 60 80 100

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

3120587471205878

120587

120587

120587

120587

U = 8

T = 001

n = 078

kx

ky

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

3120587851205878

31205878

51205878

31205874

71205878

0

0

12058781205878

4

1205874 1205874

1205872

1205872

1205872

1205872

33333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444444444444444444447777777777777777777777777777777777777777777777777777777712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333333333333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

512058783120587

7

5120587831205874 71205878

3120587852212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888

1205878

51205878

31205874

71205878

(b)

00 10 20 30 40 50 60 70 80 90

00

0

0

123456789

1011

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

71205878

120587

120587

120587

120587

U = 8

T = 001n = 085

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

312058785120587831205874

00

000000000000000000000000000000000000000000

12058781205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

4

1205874 1205874

1205872

1205872

1205872

1205872

777777777777777777777777777777777777777777777777777777777771205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333331205878

512058783120587

7

3120587851205878

31205874 71205878

8

51205878

31205874

71205878

3120587852212058783333333333312058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058744444444444444444444444444444444444444444444444

kx

ky

(c)

00 10 20 30 40 50 60 70

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

120587

120587

120587

120587

U = 8

T = 002n = 092

A(k120596

=0)

31205878

5120587831205874

71205878

71205878

312058785120587831205874

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

0000

0

12058781205878

4

1205874

1205874

1205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874

1205872

1205872

1205872

1205872

120587

333331205878

512058783120587

7

77777777777777777777777120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587888888888888888888888888888888888888888

31205878522120587831205874

3120587851205878

31205874 71205878

1205878

51205878

31205874

71205878

kx

ky

(d)







119888119888(k 120596 = 0) (notice that



119888119888(k 120596 = 0) is quite


119888119888(k 120596 = 0) in those





00

25

50

75

100

125

r(k) = 0

n(k) = 05 U = 8

n = 070

T = 001

minus010 minus020

minus030

minus040minus050

Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

ky

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(a)

minus010


minus050

0

1205878

1205874

31205878

1205872

51205878

31205874

71205878

120587

0 1205878 1205874 31205878 1205872 51205878 31205874 71205878 120587

r(k) = 0

n(k) = 05

kx

k y

00

21

42

63

84

105

A(k120596

=0)

U = 8

n = 078

T = 001Σ998400998400cc(k 120596 = 0)

(b)

00

18

36

54

72

90

r(k) = 0

n(k) = 05 U = 8

n = 085

T = 001Σ998400998400cc(k 120596 = 0)

A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(c)

r(k) = 0

n(k) = 05 U = 8

n = 092

T = 002Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y00

14

28

42

56

70

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(d)








00

02

04

06

08

10

12

n = 070

n = 078n = 085

n = 092


120596

N(120596)

U = 8 T = 001 U = 8 T = 002

(a)

1

2

3

4

5

6

7

8


U = 8T = 002n = 092

A(k

=k120596)

120596

k = S

k = S

k = X

(b)









YX SS M

k

n(k)

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

n = 070

n = 075

n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

00

01

02

03

04

05

06

07

08

09

10

(a)

S M

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

kΓ

n = 070n = 075n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

(b)







and 119899 = 07 075 08 and 085 (119879 = 001) and for 119899 =





09

0807 06

05

0403

0202

00

01

02

03

04

05

06

07

08

09

10

n(k)

U = 8n = 092T = 001

Y

X

M

Γ














YXS M

U= 8[[

n = 092

T = 100

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrT= 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

(a)

YXS M

U = 01 [[

n = 092

T = 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrU = 7

(b)




















119911119899

120572






119911119899

120572

119911⟩ is






120596

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

minus08

minus06

minus04

minus02

00

02

04

06

08

(a)


120596

minus08

minus06

minus04

minus02

00

02

04

06

08

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

(b)

000 001 002 003 004 005minus012

minus010

minus008

minus006

minus004

minus002

000

T

Σ998400998400(k120596

=0)

n = 092

U = 8k = Sk = S

(c)

minus012

minus010

minus008

minus006

minus004

minus002

000

Σ998400998400(k120596

=0)

0000 0001 0002

T2

n = 092

U = 8k = Sk = S

(d)





guide to the eye






T

n

00 01 02 03 04 05 06 07 08 09 10minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009090 087 084 081 078 075 072

n = 092

U = 8

T = 001

⟨nzn120572 z⟩

8 7 6 5 4 3 2 1 0

U

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8T = 001

⟨nzn120572 z⟩


120572



0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

n

090 087 084 081 078 075 072

120585

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

78 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

120585

n = 092

U = 8

T = 001


00

02

04

06

08

10

12

14

16

00

02

04

06

08

10

12

14

16

n

090 087 084 081 078 075 072

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

00

04

08

12

16

20

24

28

32

00

04

08

12

16

20

24

28

32

8 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

120596(Q

)

120596(Q

)
















Acknowledgments


References












































































2minus119909Sr

119909CuO


























2CuO

6+120575









Na119909CuO

2Cl

2rdquo Science









2Sr

2minus119909LaxCuO






2Cu

4O

8rdquo Physical




119888



2Cu

3O

65rdquo








2Cu

3O

661and YBa

2Cu

3O

669



2Sr

2CaCu

2O




2Sr

2CaCu

2O





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119898(k) = 120576 (k) 119868 (k) (10)


119868 (i j) = ⟨120595 (i 119905) 120595dagger(j 119905)⟩

=

1

119873

sum

keiksdot(RiminusRj)

119868 (k) (11)


119898(i j) = ⟨119869 (i 119905) 120595dagger(j 119905)⟩

=

1

119873

sum

keiksdot(RiminusRj)

119898(k) (12)



119866 (119894 119895) = ⟨119877 [120595 (119894) 120595

dagger(119895)]⟩

= 120579 (119905119894minus 119905

119895) ⟨120595 (119894) 120595

dagger(119895)⟩

(13)


Λ (120597119894) 119866 (119894 119895) Λ

dagger(120597119895) = Λ (120597

119894) 119866

0(119894 119895) Λ

dagger(120597119895)

+ ⟨119877 [120575119869 (119894) 120575119869

dagger(119895)]⟩

(14)


Λ (120597119894) = i 120597

120597119905119894



Λ (120597119894) 119866

0(119894 119895) = i120575 (119905

119894minus 119905

119895) 119868 (119894 119895) (16)


119866 (119894 119895) =

1

119873

sum

k

i2120587



119866 (k 120596) = 119866

0

(k 120596) + 119866

0

(k 120596) 119868minus1 (k) Σ (k 120596) 119866 (k 120596) (18)


119866 (k 120596) = 1

120596 minus 120576 (k) minus Σ (k 120596)119868 (k) (19)


Σ (k 120596) = 119861irr (k 120596) 119868minus1

(k) (20)

with

119861 (k 120596) = F ⟨119877 [120575119869 (119894) 120575119869

dagger(119895)]⟩ (21)



0(k 120596) and




119866

0

(k 120596) = 1

120596 minus 120576 (k)119868 (k) (22)


119868 (k) = (

1 minus

119899

2

0

0

119899

2

) = (

11986811

0

0 11986822

)

12057611(k) = minus120583 minus 4119905119868

minus1

11[Δ + (1 minus 119899 + 119901) 120572 (k)]

12057612(k) = 4119905119868

minus1

22[Δ + (119901 minus 119868

22) 120572 (k)]

12057621(k) = 4119905119868

minus1

11[Δ + (119901 minus 119868

22) 120572 (k)]

12057622(k) = 119880 minus 120583 minus 4119905119868

minus1

22[Δ + 119901120572 (k)]

(23)



Δ = ⟨120585

120572

(119894) 120585

dagger

(119894)⟩ minus ⟨120578

120572

(119894) 120578

dagger

(119894)⟩

119901 =

1

4

⟨119899

120572

120583(119894) 119899

120583(119894)⟩ minus ⟨[119888

uarr(119894) 119888

darr(119894)]

120572

119888

dagger

darr(119894) 119888

dagger

uarr(119894)⟩

(24)


119866

0

(k 120596) =2

sum

119899=1

120590

(119899)(k)

120596 minus 119864119899(k) + i120575

(25)


120590


1198641(k) = 119877 (k) + 119876 (k)

1198642(k) = 119877 (k) minus 119876 (k)

120590

(1)

11(k) = 119868

11

2

[1 +

119892 (k)2119876 (k)

]

120590

(2)

11(k) = 119868

11

2

[1 minus

119892 (k)2119876 (k)

]

120590

(1)

12(k) = 119898

12(k)

2119876 (k)

120590

(2)

12(k) = minus

11989812(k)

2119876 (k)

120590

(1)

22(k) = 119868

22

2

[1 minus

119892 (k)2119876 (k)

]

120590

(2)

22(k) = 119868

22

2

[1 +

119892 (k)2119876 (k)

]

(26)

where

119877 (k) = minus 120583 minus 4119905120572 (k) + 1

2

119880 minus

12057612(k)

211986811

119876 (k) = 1

2

radic119892

2(k) +

4120576

2

12(k) 119868

22

11986811

119892 (k) = minus 119880 +

1 minus 119899

11986811

12057612(k)

(27)





119869 (i 119905) = sum

j119886 (i j 119905) 120595 (j 119905) (28)

where


120583119899120583(119894) 120572ij

11988612(i j 119905) = minus4119905120572ij minus 2119905120590

120583119899120583(119894) 120572ij

11988621(i j 119905) = 2119905120590

120583119899120583(119894) 120572ij

11988622(i j 119905) = (119880 minus 120583) 120575ij + 2119905120590

120583119899120583(119894) 120572ij

(29)



we approximate

120575119869 (i 119905) asymp sum

j[119886 (i j 119905) minus ⟨119886 (i j 119905)⟩] 120595 (j 119905) (30)

Therefore

119861irr (119894 119895) = 4119905

2119865 (119894 119895) (1 minus 120590

1) (31)

where we defined

119865 (119894 119895) = ⟨119877 [120590

120583120575119899

120583(119894) 119888

120572

(119894) 119888

dagger120572(119895) 120575119899

120582(119895) 120590

120582]⟩ (32)

with 120575119899120583(119894) = 119899

120583(119894) minus ⟨119899


as

Σ (k 120596) = 4119905

2119865 (k 120596)(

119868

minus2

11minus119868

minus1

11119868

minus1

22

minus119868

minus1

11119868

minus1

22119868

minus2

22

) (33)


119865 (119894 119895) =

119894

2120587

int

+infin

minusinfin

119889120596eminusi120596(119905119894minus119905119895)

times

1

2120587

int

+infin

minusinfin

119889120596

1015840 1 + eminus1205731205961015840

120596 minus 120596

1015840+ i120576

119862 (i minus j 1205961015840)

(34)


119862 (119894 119895) = ⟨120590

120583120575119899

120583(119894) 119888

120572

(119894) 119888

dagger120572(119895) 120575119899

120582(119895) 120590

120582⟩

=

1

2120587



⟨120590

120583120575119899

120583(119894) 119888

120572

(119894) 119888

dagger120572(119895) 120575119899

120582(119895) 120590

120582⟩

asymp ⟨120575119899120583(119894) 120575119899

120583(119895)⟩ ⟨119888

120572

(119894) 119888

dagger120572(119895)⟩

(36)



119865 (k 120596) = 1

120587

int

+infin

minusinfin

119889120596

1015840 1

120596 minus 120596

1015840+ i120575

119886

2

(2120587)

3int119889

2119901119889Ω120572

2(119901)

times [tanh120573Ω

2

+ coth120573 (120596

1015840minus Ω)

2

]


minus Ω)]

(37)


(13)]

119866119888119888(k 120596) =

2

sum

119886119887=1

119866119886119887(k 120596) (38)

120594 (k 120596) = sum

120583

F ⟨119877 [120575119899120583(119894) 120575119899

120583(119895)]⟩ (39)




119867

119877(119894 119895) = ⟨R [119861 (119894) 119865 (119894) 119865

dagger(119895) 119861

dagger(119895)]⟩ (40)


119867

119877

(k 120596)

= minus

1

120587

int

+infin

minusinfin

119889120596

1015840 1

120596 minus 120596

1015840+ i120575

coth120573120596

1015840

2

I [119867

119862(k 1205961015840

)]

(41)

where 119867119862(119894 119895) = ⟨T[119861(119894)119865(119894)119865

dagger(119895)119861



119867

119862(119894 119895) asymp 119891

119862(119894 119895) 119887

119862(119894 119895)

119891

119862(119894 119895) = ⟨T [119865 (119894) 119865

dagger(119895)]⟩

119887

119862(119894 119895) = ⟨T [119861 (119894) 119861

dagger(119895)]⟩

(42)


119891

119862

(k 120596) = minus

1

120587

int

+infin

minusinfin

119889120596

1015840[

1 minus 119891F (1205961015840)

120596 minus 120596

1015840+ i120575

+

119891F (1205961015840)

120596 minus 120596

1015840minus i120575

]

timesI [119891

119877(k 1205961015840

)]

119887

119862

(k 120596) = minus

1

120587

int

+infin

minusinfin

119889120596

1015840[

1 + 119891119861(120596

1015840)

120596 minus 120596

1015840+ i120575

minus

119891119861(120596

1015840)

120596 minus 120596

1015840minus i120575

]

timesI [119887

119877(k 1205961015840

)]

(43)

which leads to

119867

119877

(k 120596) = 1

120587

int

+infin

minusinfin

119889120596

1015840 1

120596 minus 120596

1015840+ i120575

119886

119889

(2120587)

119889+1

times int

Ω119861

119889

119889119901119889ΩI [119891

119877(119901 Ω)]

timesI [119887

119877(119896 minus 119901 120596

1015840minus Ω)]

times [tanh120573Ω

2

+ coth120573 (120596

1015840minus Ω)

2

]

(44)





119873

(120583)

(119894) = (

119899120583(119894)

120588120583(119894)

)

119899120583(119894) = 119888

dagger

(119894) 120590120583119888 (119894)

120588120583(119894) = 119888

dagger

(119894) 120590120583119888

120572

(119894) minus 119888

dagger120572

(119894) 120590120583119888 (119894)

(45)



i 120597120597119905

119873

(120583)

(119894) = 119869

(120583)

(119894) = (

119869

(120583)

1(119894)

119869

(120583)

2(119894)

)

119869

(120583)

1(119894) = minus4119905120588

120583(119894)

119869

(120583)

2(119894) = 119880120581

120583(119894) minus 4119905119897

120583(119894)

(46)


120583(119894) are

defined as

120581120583(119894) = 119888

dagger

(119894) 120590120583120578

120572

(119894) minus 120578

dagger

(119894) 120590120583119888

120572

(119894)

+ 120578

dagger120572

(119894) 120590120583119888 (119894) minus 119888

dagger120572

(119894) 120590120583120578 (119894)

119897120583(119894) = 119888

dagger

(119894) 120590120583119888

1205722

(119894) + 119888

dagger1205722

(119894) 120590120583119888 (119894)

minus 2119888

dagger120572

(119894) 120590120583119888

120572

(119894)

(47)


119888

1205722

(i 119905) = sum

j120572

2

ij119888 (j 119905) = sum

jl120572il120572lj119888 (j 119905) (48)




i 120597120597119905

119873

(120583)

(i 119905) = sum

j120576

(120583)(i j)119873(120583)

(j 119905) (49)


119898

(120583)(i j) = sum

l120576

(120583)

(i l) 119868(120583) (l j) (50)



119868

(120583)(i j) = ⟨[119873

(120583)

(i 119905) 119873(120583)dagger(j 119905)]⟩

119898

(120583)(i j) = ⟨[119869

(120583)

(i 119905) 119873(120583)dagger(j 119905)]⟩

(51)



119868

(120583)

(k) = (

0 119868

(120583)

12(k)

119868

(120583)

12(k) 0

)

119898

(120583)

(k) = (

119898

(120583)

11(k) 0

0 119898

(120583)

22(k)

)

(52)

where

119868

(120583)

12(k) = 4 [1 minus 120572 (k)] 119862120572

119888119888

119898

(120583)

11(k) = minus 4119905119868

(120583)

12(k)

119898

(120583)

22(k) = minus 4119905119868

119897120583120588120583(k) + 119880119868

120581120583120588120583(k)

(53)


120572= ⟨119888

120572(119894)119888


119897120583120588120583(k) and 119868

120581120583120588120583(k) are

defined as

119868119897120583120588120583

(k) = F ⟨[119897120583(i 119905) 120588dagger

120583(j 119905)]⟩

119868120581120583120588120583

(k) = F ⟨[120581120583(i 119905) 120588dagger

120583(j 119905)]⟩

(54)


119866

(120583)(119894 119895) = ⟨119879 [119873

(120583)

(119894)119873

(120583)dagger(119895)]⟩

=

i1198862

(2120587)

3int119889


119866

(120583)

(k 120596)

(55)



[120596 minus 120576

(120583)

(k)] 119866(120583)

(k 120596) = 119868

(120583)

(k) (56)


120576

(120583)

(k) = (

0 120576

(120583)

12(k)

120576

(120583)

21(k) 0

)

120576

(120583)

12(k) = minus4119905

120576

(120583)

21(k) =

119898

(120583)

22(k)

119868

(120583)

12(k)

(57)


119866

(120583)

(k 120596)

= Γ

(120583)

(k) [ 1

120596 + i120575minus

1

120596 minus i120575]

+

2

sum

119899=1

120590

(119899120583)

(k) [1 + 119891B (120596)

120596 minus 120596

(120583)

119899 (k) + i120575minus

119891B (120596)

120596 minus 120596

(120583)

119899 (k) minus i120575]

(58)



(120583)(k 120596) =

⟨119873

(120583)(119894)119873


119862

(120583)

(k 120596) = 2120587Γ

(120583)

(k) 120575 (120596)

+ 2120587

2

sum

119899=1

120575 [120596 minus 120596

(120583)

119899(k)] [1 + 119891B (120596)] 120590

(119899120583)

(k)

(59)



120596

(120583)

119899(k) = (minus)

119899120596

(120583)

(k)

120596

(120583)

(k) = radic120576

(120583)

12(k) 120576(120583)

21(k)

(60)


120590

(119899120583)

(k) =119868

(120583)

12(k)2

(

120576

(120583)

12(k)

120596

(120583)

119899 (k)1

1

120576

(120583)

21(k)

120596

(120583)

119899 (k)

) (61)


119868119897120583120588120583

(k) = 3

4

[1 minus 120572 (k)] (12119862120572+ 119862

120582+ 6119862

120583)

minus 3 [1 minus 120573 (k)] (119862120572+ 119862

120583)

minus

3

4

[1 minus 120578 (k)] (119862120572+ 119862

120582+ 2119862

120583)

+

1

4

[1 minus 120582 (k)] 119862120582+

3

2

[1 minus 120583 (k)] 119862120583

119868120581120583120588120583


120578)

+ 2120573 (k) 119864120573+ 120578 (k) 119864120578

+ [1 minus 2120572 (k)] 119886120583

+

1

4

[119887120583+ 2120573 (k) 119888

120583+ 120578 (k) 119889

120583]

(62)



119864 = ⟨119888 (119894) 120578

dagger

(119894)⟩ 119862

120572= ⟨119888

120572

(119894) 119888

dagger

(119894)⟩

119864

120573= ⟨119888

120573

(119894) 120578

dagger

(119894)⟩ 119862

120582= ⟨119888

120582

(119894) 119888

dagger

(119894)⟩

119864

120578= ⟨119888

120578

(119894) 120578

dagger

(119894)⟩ 119862

120583= ⟨119888

120583

(119894) 119888

dagger

(119894)⟩

(63)

119886120583= 2 ⟨119888

dagger

(119894) 120590120583119888

120572

(119894) 119888

dagger

(119894) 120590120583119888

120572

(119894)⟩

minus ⟨119888

dagger120572

(119894) 120590120583120590

120582120590120583119888

120572

(119894) 119899120582(119894)⟩

119887120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger

(119894) 120590120583[119888 (119894) 119888 (119894)]

120572⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894) 119899

120572

120582(119894)⟩

119888120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger(119894

120578) 120590

120583119888 (119894

120572) 119888 (119894

120572)⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894

120578) 119899

120582(119894

120572)⟩

119889120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger(119894

120573) 120590

120583119888 (119894

120572) 119888 (119894

120572)⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894

120573) 119899

120582(119894

120572)⟩

(64)


119894 = (119894119909 119894119910 119905) 119894

120573= (119894

119909+ 119886 119894

119910+ 119886 119905)

119894

120572= (119894

119909+ 119886 119894

119910 119905) 119894

120578= (119894

119909+ 2119886 119894

119910 119905)

(65)


(120583)(119894 119895) =

⟨119879[119873

(120583)(119894)119873



120583 119887

120583 119888

120583 and 119889


Γ



dagger(119895)⟩


119887120583= 119886

120583+ 3119863 + 2119864

120573+ 119864

120578minus 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

119888120583= 119886

120583minus 119863 minus 2119864

120573+ 119864

120578+ 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

119889120583= 119886

120583minus 119863 + 2119864

120573minus 3119864

120578minus 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

(66)


(120583)

11(k) are fixed by


⟨119899120583(119894) 119899

120583(119894)⟩ =

119899 + 2119863 for 120583 = 0

119899 minus 2119863 for 120583 = 1 2 3

(67)


B0(ac0 as0)

G0(1205830 Δ0 p0)

n = 0

n = n + 1

Yes No

Σn[Bn Gn]



120583n Δn pn ==


where 119863 = ⟨119899uarr(119894)119899


ergodic value

Γ

(120583)

11(k) = 120575

1205830

(2120587)

2

119886

2120575

(2)

(k) ⟨119899⟩2 (68)


⟨120575119899120583(119894) 120575119899

120583(119895)⟩

=

119886

2

2 (2120587)

2

2

sum

119899=1

int119889


(120583)

119899(k)(119905119894minus119905119895)


(120583)

119899(k)

2119896B119879]120590

(119899120583)

11(k)

(69)

⟨119877 [120575119899120583(119894) 120575119899

120583(119895)]⟩

=

i1198862

(2120587)

3

2

sum

119899=1

int119889


(70)

times

120590

(119899120583)

11(k)

120596 minus 120596

(120583)

119899 (k) + i120575 (71)



120594120583(k 120596) = minusF [⟨119877 [120575119899

120583(119894) 120575119899

120583(119895)]⟩]

=

16119905 [1 minus 120572 (k)] 119862120572

120596

2minus (120596

(120583)(k))2

(72)



119899 = 2 (1 minus 11986211minus 119862

22)

Δ = 119862

120572

11minus 119862

120572

22

11986212

= ⟨120585 (119894) 120578

dagger

(119894)⟩ = 0

(73)

where 119862119899119898

and 119862

120572


functions 119862119899119898

= ⟨120595119899(119894)120595

dagger

119898(119894)⟩ and 119862

120572

119899119898= ⟨120595

120572

119899(119894)120595

dagger

119898(119894)⟩


119888119888(k 120596) [cf (38)] and



120583(k 120596)


0and 119886



22


(k 120596) and 120594120583(k 120596) in the two-









119888119888(k 120596) = F⟨119879[119888(119894)119888

dagger(119895)]⟩


3 Results


119860 (k 120596) = minus

1

120587

I [119866119888119888(k 120596)] (74)


119899 (k) = int119889120596119891F (120596)119860 (k 120596) (75)


119873(120596) =

1

(2120587)

2int119889

2119896119860 (k 120596) (76)

where 119866119888119888(k 120596) = 119866

11(k 120596) + 119866

12(k 120596) + 119866

21(k 120596) +




119866119888119888(k 120596) = 1

120596 minus 1205980(k) minus Σ

119888119888(k 120596)

(77)



1015840

119888119888(k 120596 = 0)


120596rarr0Σ

10158401015840

119888119888(k 120596 119879 =

0) prop 120596


Σ

10158401015840

119888119888(k 120596 = 0 119879) prop 119879



120572

119911⟩ the



limkrarrQ

120594

(3)

(k 0) =120594

(3)(Q 0)

1 + 120585

2|k minusQ|

2 (78)

where 120594(3)(k 0) = minus119866














0(k 120596) and 120594






9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 07

T = 001

Σ998400998400(k 120596)

(a)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 078

T = 001

Σ998400998400(k 120596)

(b)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596A(k120596)

00

06

12

18

24

30

U = 8n = 085

T = 001

Σ998400998400(k 120596)

(c)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 092

T = 002

Σ998400998400(k 120596)

(d)










050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 07

T = 001

A(k120596)

Σ998400998400(k 120596)

120596

(a)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 078

T = 001

120596

(b)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 085

T = 001

120596

(c)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 092

T = 002

120596

(d)



(119879 = 002)








3(k 120596) (see Section 36)






00 20 40 60 80 100 120 120

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

5120587831205874

120587

120587

120587

120587

U = 8

T = 001

n = 070

A(k120596

=0)

31205878

31205878

31205878

51205878

51205878

51205878

31205874 31205874

31205874

31205874

71205878

71205878

71205878

71205878

000

0

12058781205878

4

120587444444444444444444444

1205874

1205874

1205872

1205871205871205871205871205871205871205871205871205871205872222222222222222222222222222222222222

1205872

1205872

55555555555555555555555555555555555555555555555555555555555555555552222212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444

120587

1205878

31205878

333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

51205878

51205878

51205878

31205874 31205874

3120587

33333333333333333333333331205871205871205871205871205871205871205871205871205874

71205878

71205878

7

74 1205878

kx

ky

(a)

00 20 40 60 80 100

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

3120587471205878

120587

120587

120587

120587

U = 8

T = 001

n = 078

kx

ky

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

3120587851205878

31205878

51205878

31205874

71205878

0

0

12058781205878

4

1205874 1205874

1205872

1205872

1205872

1205872

33333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444444444444444444447777777777777777777777777777777777777777777777777777777712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333333333333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

512058783120587

7

5120587831205874 71205878

3120587852212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888

1205878

51205878

31205874

71205878

(b)

00 10 20 30 40 50 60 70 80 90

00

0

0

123456789

1011

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

71205878

120587

120587

120587

120587

U = 8

T = 001n = 085

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

312058785120587831205874

00

000000000000000000000000000000000000000000

12058781205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

4

1205874 1205874

1205872

1205872

1205872

1205872

777777777777777777777777777777777777777777777777777777777771205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333331205878

512058783120587

7

3120587851205878

31205874 71205878

8

51205878

31205874

71205878

3120587852212058783333333333312058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058744444444444444444444444444444444444444444444444

kx

ky

(c)

00 10 20 30 40 50 60 70

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

120587

120587

120587

120587

U = 8

T = 002n = 092

A(k120596

=0)

31205878

5120587831205874

71205878

71205878

312058785120587831205874

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

0000

0

12058781205878

4

1205874

1205874

1205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874

1205872

1205872

1205872

1205872

120587

333331205878

512058783120587

7

77777777777777777777777120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587888888888888888888888888888888888888888

31205878522120587831205874

3120587851205878

31205874 71205878

1205878

51205878

31205874

71205878

kx

ky

(d)







119888119888(k 120596 = 0) (notice that



119888119888(k 120596 = 0) is quite


119888119888(k 120596 = 0) in those





00

25

50

75

100

125

r(k) = 0

n(k) = 05 U = 8

n = 070

T = 001

minus010 minus020

minus030

minus040minus050

Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

ky

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(a)

minus010


minus050

0

1205878

1205874

31205878

1205872

51205878

31205874

71205878

120587

0 1205878 1205874 31205878 1205872 51205878 31205874 71205878 120587

r(k) = 0

n(k) = 05

kx

k y

00

21

42

63

84

105

A(k120596

=0)

U = 8

n = 078

T = 001Σ998400998400cc(k 120596 = 0)

(b)

00

18

36

54

72

90

r(k) = 0

n(k) = 05 U = 8

n = 085

T = 001Σ998400998400cc(k 120596 = 0)

A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(c)

r(k) = 0

n(k) = 05 U = 8

n = 092

T = 002Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y00

14

28

42

56

70

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(d)








00

02

04

06

08

10

12

n = 070

n = 078n = 085

n = 092


120596

N(120596)

U = 8 T = 001 U = 8 T = 002

(a)

1

2

3

4

5

6

7

8


U = 8T = 002n = 092

A(k

=k120596)

120596

k = S

k = S

k = X

(b)









YX SS M

k

n(k)

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

n = 070

n = 075

n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

00

01

02

03

04

05

06

07

08

09

10

(a)

S M

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

kΓ

n = 070n = 075n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

(b)







and 119899 = 07 075 08 and 085 (119879 = 001) and for 119899 =





09

0807 06

05

0403

0202

00

01

02

03

04

05

06

07

08

09

10

n(k)

U = 8n = 092T = 001

Y

X

M

Γ














YXS M

U= 8[[

n = 092

T = 100

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrT= 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

(a)

YXS M

U = 01 [[

n = 092

T = 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrU = 7

(b)




















119911119899

120572






119911119899

120572

119911⟩ is






120596

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

minus08

minus06

minus04

minus02

00

02

04

06

08

(a)


120596

minus08

minus06

minus04

minus02

00

02

04

06

08

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

(b)

000 001 002 003 004 005minus012

minus010

minus008

minus006

minus004

minus002

000

T

Σ998400998400(k120596

=0)

n = 092

U = 8k = Sk = S

(c)

minus012

minus010

minus008

minus006

minus004

minus002

000

Σ998400998400(k120596

=0)

0000 0001 0002

T2

n = 092

U = 8k = Sk = S

(d)





guide to the eye






T

n

00 01 02 03 04 05 06 07 08 09 10minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009090 087 084 081 078 075 072

n = 092

U = 8

T = 001

⟨nzn120572 z⟩

8 7 6 5 4 3 2 1 0

U

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8T = 001

⟨nzn120572 z⟩


120572



0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

n

090 087 084 081 078 075 072

120585

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

78 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

120585

n = 092

U = 8

T = 001


00

02

04

06

08

10

12

14

16

00

02

04

06

08

10

12

14

16

n

090 087 084 081 078 075 072

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

00

04

08

12

16

20

24

28

32

00

04

08

12

16

20

24

28

32

8 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

120596(Q

)

120596(Q

)
















Acknowledgments


References












































































2minus119909Sr

119909CuO


























2CuO

6+120575









Na119909CuO

2Cl

2rdquo Science









2Sr

2minus119909LaxCuO






2Cu

4O

8rdquo Physical




119888



2Cu

3O

65rdquo








2Cu

3O

661and YBa

2Cu

3O

669



2Sr

2CaCu

2O




2Sr

2CaCu

2O





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





Δ = ⟨120585

120572

(119894) 120585

dagger

(119894)⟩ minus ⟨120578

120572

(119894) 120578

dagger

(119894)⟩

119901 =

1

4

⟨119899

120572

120583(119894) 119899

120583(119894)⟩ minus ⟨[119888

uarr(119894) 119888

darr(119894)]

120572

119888

dagger

darr(119894) 119888

dagger

uarr(119894)⟩

(24)


119866

0

(k 120596) =2

sum

119899=1

120590

(119899)(k)

120596 minus 119864119899(k) + i120575

(25)


120590


1198641(k) = 119877 (k) + 119876 (k)

1198642(k) = 119877 (k) minus 119876 (k)

120590

(1)

11(k) = 119868

11

2

[1 +

119892 (k)2119876 (k)

]

120590

(2)

11(k) = 119868

11

2

[1 minus

119892 (k)2119876 (k)

]

120590

(1)

12(k) = 119898

12(k)

2119876 (k)

120590

(2)

12(k) = minus

11989812(k)

2119876 (k)

120590

(1)

22(k) = 119868

22

2

[1 minus

119892 (k)2119876 (k)

]

120590

(2)

22(k) = 119868

22

2

[1 +

119892 (k)2119876 (k)

]

(26)

where

119877 (k) = minus 120583 minus 4119905120572 (k) + 1

2

119880 minus

12057612(k)

211986811

119876 (k) = 1

2

radic119892

2(k) +

4120576

2

12(k) 119868

22

11986811

119892 (k) = minus 119880 +

1 minus 119899

11986811

12057612(k)

(27)





119869 (i 119905) = sum

j119886 (i j 119905) 120595 (j 119905) (28)

where


120583119899120583(119894) 120572ij

11988612(i j 119905) = minus4119905120572ij minus 2119905120590

120583119899120583(119894) 120572ij

11988621(i j 119905) = 2119905120590

120583119899120583(119894) 120572ij

11988622(i j 119905) = (119880 minus 120583) 120575ij + 2119905120590

120583119899120583(119894) 120572ij

(29)



we approximate

120575119869 (i 119905) asymp sum

j[119886 (i j 119905) minus ⟨119886 (i j 119905)⟩] 120595 (j 119905) (30)

Therefore

119861irr (119894 119895) = 4119905

2119865 (119894 119895) (1 minus 120590

1) (31)

where we defined

119865 (119894 119895) = ⟨119877 [120590

120583120575119899

120583(119894) 119888

120572

(119894) 119888

dagger120572(119895) 120575119899

120582(119895) 120590

120582]⟩ (32)

with 120575119899120583(119894) = 119899

120583(119894) minus ⟨119899


as

Σ (k 120596) = 4119905

2119865 (k 120596)(

119868

minus2

11minus119868

minus1

11119868

minus1

22

minus119868

minus1

11119868

minus1

22119868

minus2

22

) (33)


119865 (119894 119895) =

119894

2120587

int

+infin

minusinfin

119889120596eminusi120596(119905119894minus119905119895)

times

1

2120587

int

+infin

minusinfin

119889120596

1015840 1 + eminus1205731205961015840

120596 minus 120596

1015840+ i120576

119862 (i minus j 1205961015840)

(34)


119862 (119894 119895) = ⟨120590

120583120575119899

120583(119894) 119888

120572

(119894) 119888

dagger120572(119895) 120575119899

120582(119895) 120590

120582⟩

=

1

2120587



⟨120590

120583120575119899

120583(119894) 119888

120572

(119894) 119888

dagger120572(119895) 120575119899

120582(119895) 120590

120582⟩

asymp ⟨120575119899120583(119894) 120575119899

120583(119895)⟩ ⟨119888

120572

(119894) 119888

dagger120572(119895)⟩

(36)



119865 (k 120596) = 1

120587

int

+infin

minusinfin

119889120596

1015840 1

120596 minus 120596

1015840+ i120575

119886

2

(2120587)

3int119889

2119901119889Ω120572

2(119901)

times [tanh120573Ω

2

+ coth120573 (120596

1015840minus Ω)

2

]


minus Ω)]

(37)


(13)]

119866119888119888(k 120596) =

2

sum

119886119887=1

119866119886119887(k 120596) (38)

120594 (k 120596) = sum

120583

F ⟨119877 [120575119899120583(119894) 120575119899

120583(119895)]⟩ (39)




119867

119877(119894 119895) = ⟨R [119861 (119894) 119865 (119894) 119865

dagger(119895) 119861

dagger(119895)]⟩ (40)


119867

119877

(k 120596)

= minus

1

120587

int

+infin

minusinfin

119889120596

1015840 1

120596 minus 120596

1015840+ i120575

coth120573120596

1015840

2

I [119867

119862(k 1205961015840

)]

(41)

where 119867119862(119894 119895) = ⟨T[119861(119894)119865(119894)119865

dagger(119895)119861



119867

119862(119894 119895) asymp 119891

119862(119894 119895) 119887

119862(119894 119895)

119891

119862(119894 119895) = ⟨T [119865 (119894) 119865

dagger(119895)]⟩

119887

119862(119894 119895) = ⟨T [119861 (119894) 119861

dagger(119895)]⟩

(42)


119891

119862

(k 120596) = minus

1

120587

int

+infin

minusinfin

119889120596

1015840[

1 minus 119891F (1205961015840)

120596 minus 120596

1015840+ i120575

+

119891F (1205961015840)

120596 minus 120596

1015840minus i120575

]

timesI [119891

119877(k 1205961015840

)]

119887

119862

(k 120596) = minus

1

120587

int

+infin

minusinfin

119889120596

1015840[

1 + 119891119861(120596

1015840)

120596 minus 120596

1015840+ i120575

minus

119891119861(120596

1015840)

120596 minus 120596

1015840minus i120575

]

timesI [119887

119877(k 1205961015840

)]

(43)

which leads to

119867

119877

(k 120596) = 1

120587

int

+infin

minusinfin

119889120596

1015840 1

120596 minus 120596

1015840+ i120575

119886

119889

(2120587)

119889+1

times int

Ω119861

119889

119889119901119889ΩI [119891

119877(119901 Ω)]

timesI [119887

119877(119896 minus 119901 120596

1015840minus Ω)]

times [tanh120573Ω

2

+ coth120573 (120596

1015840minus Ω)

2

]

(44)





119873

(120583)

(119894) = (

119899120583(119894)

120588120583(119894)

)

119899120583(119894) = 119888

dagger

(119894) 120590120583119888 (119894)

120588120583(119894) = 119888

dagger

(119894) 120590120583119888

120572

(119894) minus 119888

dagger120572

(119894) 120590120583119888 (119894)

(45)



i 120597120597119905

119873

(120583)

(119894) = 119869

(120583)

(119894) = (

119869

(120583)

1(119894)

119869

(120583)

2(119894)

)

119869

(120583)

1(119894) = minus4119905120588

120583(119894)

119869

(120583)

2(119894) = 119880120581

120583(119894) minus 4119905119897

120583(119894)

(46)


120583(119894) are

defined as

120581120583(119894) = 119888

dagger

(119894) 120590120583120578

120572

(119894) minus 120578

dagger

(119894) 120590120583119888

120572

(119894)

+ 120578

dagger120572

(119894) 120590120583119888 (119894) minus 119888

dagger120572

(119894) 120590120583120578 (119894)

119897120583(119894) = 119888

dagger

(119894) 120590120583119888

1205722

(119894) + 119888

dagger1205722

(119894) 120590120583119888 (119894)

minus 2119888

dagger120572

(119894) 120590120583119888

120572

(119894)

(47)


119888

1205722

(i 119905) = sum

j120572

2

ij119888 (j 119905) = sum

jl120572il120572lj119888 (j 119905) (48)




i 120597120597119905

119873

(120583)

(i 119905) = sum

j120576

(120583)(i j)119873(120583)

(j 119905) (49)


119898

(120583)(i j) = sum

l120576

(120583)

(i l) 119868(120583) (l j) (50)



119868

(120583)(i j) = ⟨[119873

(120583)

(i 119905) 119873(120583)dagger(j 119905)]⟩

119898

(120583)(i j) = ⟨[119869

(120583)

(i 119905) 119873(120583)dagger(j 119905)]⟩

(51)



119868

(120583)

(k) = (

0 119868

(120583)

12(k)

119868

(120583)

12(k) 0

)

119898

(120583)

(k) = (

119898

(120583)

11(k) 0

0 119898

(120583)

22(k)

)

(52)

where

119868

(120583)

12(k) = 4 [1 minus 120572 (k)] 119862120572

119888119888

119898

(120583)

11(k) = minus 4119905119868

(120583)

12(k)

119898

(120583)

22(k) = minus 4119905119868

119897120583120588120583(k) + 119880119868

120581120583120588120583(k)

(53)


120572= ⟨119888

120572(119894)119888


119897120583120588120583(k) and 119868

120581120583120588120583(k) are

defined as

119868119897120583120588120583

(k) = F ⟨[119897120583(i 119905) 120588dagger

120583(j 119905)]⟩

119868120581120583120588120583

(k) = F ⟨[120581120583(i 119905) 120588dagger

120583(j 119905)]⟩

(54)


119866

(120583)(119894 119895) = ⟨119879 [119873

(120583)

(119894)119873

(120583)dagger(119895)]⟩

=

i1198862

(2120587)

3int119889


119866

(120583)

(k 120596)

(55)



[120596 minus 120576

(120583)

(k)] 119866(120583)

(k 120596) = 119868

(120583)

(k) (56)


120576

(120583)

(k) = (

0 120576

(120583)

12(k)

120576

(120583)

21(k) 0

)

120576

(120583)

12(k) = minus4119905

120576

(120583)

21(k) =

119898

(120583)

22(k)

119868

(120583)

12(k)

(57)


119866

(120583)

(k 120596)

= Γ

(120583)

(k) [ 1

120596 + i120575minus

1

120596 minus i120575]

+

2

sum

119899=1

120590

(119899120583)

(k) [1 + 119891B (120596)

120596 minus 120596

(120583)

119899 (k) + i120575minus

119891B (120596)

120596 minus 120596

(120583)

119899 (k) minus i120575]

(58)



(120583)(k 120596) =

⟨119873

(120583)(119894)119873


119862

(120583)

(k 120596) = 2120587Γ

(120583)

(k) 120575 (120596)

+ 2120587

2

sum

119899=1

120575 [120596 minus 120596

(120583)

119899(k)] [1 + 119891B (120596)] 120590

(119899120583)

(k)

(59)



120596

(120583)

119899(k) = (minus)

119899120596

(120583)

(k)

120596

(120583)

(k) = radic120576

(120583)

12(k) 120576(120583)

21(k)

(60)


120590

(119899120583)

(k) =119868

(120583)

12(k)2

(

120576

(120583)

12(k)

120596

(120583)

119899 (k)1

1

120576

(120583)

21(k)

120596

(120583)

119899 (k)

) (61)


119868119897120583120588120583

(k) = 3

4

[1 minus 120572 (k)] (12119862120572+ 119862

120582+ 6119862

120583)

minus 3 [1 minus 120573 (k)] (119862120572+ 119862

120583)

minus

3

4

[1 minus 120578 (k)] (119862120572+ 119862

120582+ 2119862

120583)

+

1

4

[1 minus 120582 (k)] 119862120582+

3

2

[1 minus 120583 (k)] 119862120583

119868120581120583120588120583


120578)

+ 2120573 (k) 119864120573+ 120578 (k) 119864120578

+ [1 minus 2120572 (k)] 119886120583

+

1

4

[119887120583+ 2120573 (k) 119888

120583+ 120578 (k) 119889

120583]

(62)



119864 = ⟨119888 (119894) 120578

dagger

(119894)⟩ 119862

120572= ⟨119888

120572

(119894) 119888

dagger

(119894)⟩

119864

120573= ⟨119888

120573

(119894) 120578

dagger

(119894)⟩ 119862

120582= ⟨119888

120582

(119894) 119888

dagger

(119894)⟩

119864

120578= ⟨119888

120578

(119894) 120578

dagger

(119894)⟩ 119862

120583= ⟨119888

120583

(119894) 119888

dagger

(119894)⟩

(63)

119886120583= 2 ⟨119888

dagger

(119894) 120590120583119888

120572

(119894) 119888

dagger

(119894) 120590120583119888

120572

(119894)⟩

minus ⟨119888

dagger120572

(119894) 120590120583120590

120582120590120583119888

120572

(119894) 119899120582(119894)⟩

119887120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger

(119894) 120590120583[119888 (119894) 119888 (119894)]

120572⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894) 119899

120572

120582(119894)⟩

119888120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger(119894

120578) 120590

120583119888 (119894

120572) 119888 (119894

120572)⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894

120578) 119899

120582(119894

120572)⟩

119889120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger(119894

120573) 120590

120583119888 (119894

120572) 119888 (119894

120572)⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894

120573) 119899

120582(119894

120572)⟩

(64)


119894 = (119894119909 119894119910 119905) 119894

120573= (119894

119909+ 119886 119894

119910+ 119886 119905)

119894

120572= (119894

119909+ 119886 119894

119910 119905) 119894

120578= (119894

119909+ 2119886 119894

119910 119905)

(65)


(120583)(119894 119895) =

⟨119879[119873

(120583)(119894)119873



120583 119887

120583 119888

120583 and 119889


Γ



dagger(119895)⟩


119887120583= 119886

120583+ 3119863 + 2119864

120573+ 119864

120578minus 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

119888120583= 119886

120583minus 119863 minus 2119864

120573+ 119864

120578+ 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

119889120583= 119886

120583minus 119863 + 2119864

120573minus 3119864

120578minus 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

(66)


(120583)

11(k) are fixed by


⟨119899120583(119894) 119899

120583(119894)⟩ =

119899 + 2119863 for 120583 = 0

119899 minus 2119863 for 120583 = 1 2 3

(67)


B0(ac0 as0)

G0(1205830 Δ0 p0)

n = 0

n = n + 1

Yes No

Σn[Bn Gn]



120583n Δn pn ==


where 119863 = ⟨119899uarr(119894)119899


ergodic value

Γ

(120583)

11(k) = 120575

1205830

(2120587)

2

119886

2120575

(2)

(k) ⟨119899⟩2 (68)


⟨120575119899120583(119894) 120575119899

120583(119895)⟩

=

119886

2

2 (2120587)

2

2

sum

119899=1

int119889


(120583)

119899(k)(119905119894minus119905119895)


(120583)

119899(k)

2119896B119879]120590

(119899120583)

11(k)

(69)

⟨119877 [120575119899120583(119894) 120575119899

120583(119895)]⟩

=

i1198862

(2120587)

3

2

sum

119899=1

int119889


(70)

times

120590

(119899120583)

11(k)

120596 minus 120596

(120583)

119899 (k) + i120575 (71)



120594120583(k 120596) = minusF [⟨119877 [120575119899

120583(119894) 120575119899

120583(119895)]⟩]

=

16119905 [1 minus 120572 (k)] 119862120572

120596

2minus (120596

(120583)(k))2

(72)



119899 = 2 (1 minus 11986211minus 119862

22)

Δ = 119862

120572

11minus 119862

120572

22

11986212

= ⟨120585 (119894) 120578

dagger

(119894)⟩ = 0

(73)

where 119862119899119898

and 119862

120572


functions 119862119899119898

= ⟨120595119899(119894)120595

dagger

119898(119894)⟩ and 119862

120572

119899119898= ⟨120595

120572

119899(119894)120595

dagger

119898(119894)⟩


119888119888(k 120596) [cf (38)] and



120583(k 120596)


0and 119886



22


(k 120596) and 120594120583(k 120596) in the two-









119888119888(k 120596) = F⟨119879[119888(119894)119888

dagger(119895)]⟩


3 Results


119860 (k 120596) = minus

1

120587

I [119866119888119888(k 120596)] (74)


119899 (k) = int119889120596119891F (120596)119860 (k 120596) (75)


119873(120596) =

1

(2120587)

2int119889

2119896119860 (k 120596) (76)

where 119866119888119888(k 120596) = 119866

11(k 120596) + 119866

12(k 120596) + 119866

21(k 120596) +




119866119888119888(k 120596) = 1

120596 minus 1205980(k) minus Σ

119888119888(k 120596)

(77)



1015840

119888119888(k 120596 = 0)


120596rarr0Σ

10158401015840

119888119888(k 120596 119879 =

0) prop 120596


Σ

10158401015840

119888119888(k 120596 = 0 119879) prop 119879



120572

119911⟩ the



limkrarrQ

120594

(3)

(k 0) =120594

(3)(Q 0)

1 + 120585

2|k minusQ|

2 (78)

where 120594(3)(k 0) = minus119866














0(k 120596) and 120594






9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 07

T = 001

Σ998400998400(k 120596)

(a)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 078

T = 001

Σ998400998400(k 120596)

(b)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596A(k120596)

00

06

12

18

24

30

U = 8n = 085

T = 001

Σ998400998400(k 120596)

(c)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 092

T = 002

Σ998400998400(k 120596)

(d)










050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 07

T = 001

A(k120596)

Σ998400998400(k 120596)

120596

(a)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 078

T = 001

120596

(b)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 085

T = 001

120596

(c)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 092

T = 002

120596

(d)



(119879 = 002)








3(k 120596) (see Section 36)






00 20 40 60 80 100 120 120

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

5120587831205874

120587

120587

120587

120587

U = 8

T = 001

n = 070

A(k120596

=0)

31205878

31205878

31205878

51205878

51205878

51205878

31205874 31205874

31205874

31205874

71205878

71205878

71205878

71205878

000

0

12058781205878

4

120587444444444444444444444

1205874

1205874

1205872

1205871205871205871205871205871205871205871205871205871205872222222222222222222222222222222222222

1205872

1205872

55555555555555555555555555555555555555555555555555555555555555555552222212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444

120587

1205878

31205878

333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

51205878

51205878

51205878

31205874 31205874

3120587

33333333333333333333333331205871205871205871205871205871205871205871205871205874

71205878

71205878

7

74 1205878

kx

ky

(a)

00 20 40 60 80 100

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

3120587471205878

120587

120587

120587

120587

U = 8

T = 001

n = 078

kx

ky

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

3120587851205878

31205878

51205878

31205874

71205878

0

0

12058781205878

4

1205874 1205874

1205872

1205872

1205872

1205872

33333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444444444444444444447777777777777777777777777777777777777777777777777777777712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333333333333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

512058783120587

7

5120587831205874 71205878

3120587852212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888

1205878

51205878

31205874

71205878

(b)

00 10 20 30 40 50 60 70 80 90

00

0

0

123456789

1011

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

71205878

120587

120587

120587

120587

U = 8

T = 001n = 085

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

312058785120587831205874

00

000000000000000000000000000000000000000000

12058781205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

4

1205874 1205874

1205872

1205872

1205872

1205872

777777777777777777777777777777777777777777777777777777777771205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333331205878

512058783120587

7

3120587851205878

31205874 71205878

8

51205878

31205874

71205878

3120587852212058783333333333312058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058744444444444444444444444444444444444444444444444

kx

ky

(c)

00 10 20 30 40 50 60 70

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

120587

120587

120587

120587

U = 8

T = 002n = 092

A(k120596

=0)

31205878

5120587831205874

71205878

71205878

312058785120587831205874

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

0000

0

12058781205878

4

1205874

1205874

1205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874

1205872

1205872

1205872

1205872

120587

333331205878

512058783120587

7

77777777777777777777777120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587888888888888888888888888888888888888888

31205878522120587831205874

3120587851205878

31205874 71205878

1205878

51205878

31205874

71205878

kx

ky

(d)







119888119888(k 120596 = 0) (notice that



119888119888(k 120596 = 0) is quite


119888119888(k 120596 = 0) in those





00

25

50

75

100

125

r(k) = 0

n(k) = 05 U = 8

n = 070

T = 001

minus010 minus020

minus030

minus040minus050

Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

ky

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(a)

minus010


minus050

0

1205878

1205874

31205878

1205872

51205878

31205874

71205878

120587

0 1205878 1205874 31205878 1205872 51205878 31205874 71205878 120587

r(k) = 0

n(k) = 05

kx

k y

00

21

42

63

84

105

A(k120596

=0)

U = 8

n = 078

T = 001Σ998400998400cc(k 120596 = 0)

(b)

00

18

36

54

72

90

r(k) = 0

n(k) = 05 U = 8

n = 085

T = 001Σ998400998400cc(k 120596 = 0)

A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(c)

r(k) = 0

n(k) = 05 U = 8

n = 092

T = 002Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y00

14

28

42

56

70

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(d)








00

02

04

06

08

10

12

n = 070

n = 078n = 085

n = 092


120596

N(120596)

U = 8 T = 001 U = 8 T = 002

(a)

1

2

3

4

5

6

7

8


U = 8T = 002n = 092

A(k

=k120596)

120596

k = S

k = S

k = X

(b)









YX SS M

k

n(k)

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

n = 070

n = 075

n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

00

01

02

03

04

05

06

07

08

09

10

(a)

S M

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

kΓ

n = 070n = 075n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

(b)







and 119899 = 07 075 08 and 085 (119879 = 001) and for 119899 =





09

0807 06

05

0403

0202

00

01

02

03

04

05

06

07

08

09

10

n(k)

U = 8n = 092T = 001

Y

X

M

Γ














YXS M

U= 8[[

n = 092

T = 100

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrT= 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

(a)

YXS M

U = 01 [[

n = 092

T = 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrU = 7

(b)




















119911119899

120572






119911119899

120572

119911⟩ is






120596

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

minus08

minus06

minus04

minus02

00

02

04

06

08

(a)


120596

minus08

minus06

minus04

minus02

00

02

04

06

08

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

(b)

000 001 002 003 004 005minus012

minus010

minus008

minus006

minus004

minus002

000

T

Σ998400998400(k120596

=0)

n = 092

U = 8k = Sk = S

(c)

minus012

minus010

minus008

minus006

minus004

minus002

000

Σ998400998400(k120596

=0)

0000 0001 0002

T2

n = 092

U = 8k = Sk = S

(d)





guide to the eye






T

n

00 01 02 03 04 05 06 07 08 09 10minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009090 087 084 081 078 075 072

n = 092

U = 8

T = 001

⟨nzn120572 z⟩

8 7 6 5 4 3 2 1 0

U

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8T = 001

⟨nzn120572 z⟩


120572



0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

n

090 087 084 081 078 075 072

120585

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

78 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

120585

n = 092

U = 8

T = 001


00

02

04

06

08

10

12

14

16

00

02

04

06

08

10

12

14

16

n

090 087 084 081 078 075 072

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

00

04

08

12

16

20

24

28

32

00

04

08

12

16

20

24

28

32

8 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

120596(Q

)

120596(Q

)
















Acknowledgments


References












































































2minus119909Sr

119909CuO


























2CuO

6+120575









Na119909CuO

2Cl

2rdquo Science









2Sr

2minus119909LaxCuO






2Cu

4O

8rdquo Physical




119888



2Cu

3O

65rdquo








2Cu

3O

661and YBa

2Cu

3O

669



2Sr

2CaCu

2O




2Sr

2CaCu

2O





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119865 (k 120596) = 1

120587

int

+infin

minusinfin

119889120596

1015840 1

120596 minus 120596

1015840+ i120575

119886

2

(2120587)

3int119889

2119901119889Ω120572

2(119901)

times [tanh120573Ω

2

+ coth120573 (120596

1015840minus Ω)

2

]


minus Ω)]

(37)


(13)]

119866119888119888(k 120596) =

2

sum

119886119887=1

119866119886119887(k 120596) (38)

120594 (k 120596) = sum

120583

F ⟨119877 [120575119899120583(119894) 120575119899

120583(119895)]⟩ (39)




119867

119877(119894 119895) = ⟨R [119861 (119894) 119865 (119894) 119865

dagger(119895) 119861

dagger(119895)]⟩ (40)


119867

119877

(k 120596)

= minus

1

120587

int

+infin

minusinfin

119889120596

1015840 1

120596 minus 120596

1015840+ i120575

coth120573120596

1015840

2

I [119867

119862(k 1205961015840

)]

(41)

where 119867119862(119894 119895) = ⟨T[119861(119894)119865(119894)119865

dagger(119895)119861



119867

119862(119894 119895) asymp 119891

119862(119894 119895) 119887

119862(119894 119895)

119891

119862(119894 119895) = ⟨T [119865 (119894) 119865

dagger(119895)]⟩

119887

119862(119894 119895) = ⟨T [119861 (119894) 119861

dagger(119895)]⟩

(42)


119891

119862

(k 120596) = minus

1

120587

int

+infin

minusinfin

119889120596

1015840[

1 minus 119891F (1205961015840)

120596 minus 120596

1015840+ i120575

+

119891F (1205961015840)

120596 minus 120596

1015840minus i120575

]

timesI [119891

119877(k 1205961015840

)]

119887

119862

(k 120596) = minus

1

120587

int

+infin

minusinfin

119889120596

1015840[

1 + 119891119861(120596

1015840)

120596 minus 120596

1015840+ i120575

minus

119891119861(120596

1015840)

120596 minus 120596

1015840minus i120575

]

timesI [119887

119877(k 1205961015840

)]

(43)

which leads to

119867

119877

(k 120596) = 1

120587

int

+infin

minusinfin

119889120596

1015840 1

120596 minus 120596

1015840+ i120575

119886

119889

(2120587)

119889+1

times int

Ω119861

119889

119889119901119889ΩI [119891

119877(119901 Ω)]

timesI [119887

119877(119896 minus 119901 120596

1015840minus Ω)]

times [tanh120573Ω

2

+ coth120573 (120596

1015840minus Ω)

2

]

(44)





119873

(120583)

(119894) = (

119899120583(119894)

120588120583(119894)

)

119899120583(119894) = 119888

dagger

(119894) 120590120583119888 (119894)

120588120583(119894) = 119888

dagger

(119894) 120590120583119888

120572

(119894) minus 119888

dagger120572

(119894) 120590120583119888 (119894)

(45)



i 120597120597119905

119873

(120583)

(119894) = 119869

(120583)

(119894) = (

119869

(120583)

1(119894)

119869

(120583)

2(119894)

)

119869

(120583)

1(119894) = minus4119905120588

120583(119894)

119869

(120583)

2(119894) = 119880120581

120583(119894) minus 4119905119897

120583(119894)

(46)


120583(119894) are

defined as

120581120583(119894) = 119888

dagger

(119894) 120590120583120578

120572

(119894) minus 120578

dagger

(119894) 120590120583119888

120572

(119894)

+ 120578

dagger120572

(119894) 120590120583119888 (119894) minus 119888

dagger120572

(119894) 120590120583120578 (119894)

119897120583(119894) = 119888

dagger

(119894) 120590120583119888

1205722

(119894) + 119888

dagger1205722

(119894) 120590120583119888 (119894)

minus 2119888

dagger120572

(119894) 120590120583119888

120572

(119894)

(47)


119888

1205722

(i 119905) = sum

j120572

2

ij119888 (j 119905) = sum

jl120572il120572lj119888 (j 119905) (48)




i 120597120597119905

119873

(120583)

(i 119905) = sum

j120576

(120583)(i j)119873(120583)

(j 119905) (49)


119898

(120583)(i j) = sum

l120576

(120583)

(i l) 119868(120583) (l j) (50)



119868

(120583)(i j) = ⟨[119873

(120583)

(i 119905) 119873(120583)dagger(j 119905)]⟩

119898

(120583)(i j) = ⟨[119869

(120583)

(i 119905) 119873(120583)dagger(j 119905)]⟩

(51)



119868

(120583)

(k) = (

0 119868

(120583)

12(k)

119868

(120583)

12(k) 0

)

119898

(120583)

(k) = (

119898

(120583)

11(k) 0

0 119898

(120583)

22(k)

)

(52)

where

119868

(120583)

12(k) = 4 [1 minus 120572 (k)] 119862120572

119888119888

119898

(120583)

11(k) = minus 4119905119868

(120583)

12(k)

119898

(120583)

22(k) = minus 4119905119868

119897120583120588120583(k) + 119880119868

120581120583120588120583(k)

(53)


120572= ⟨119888

120572(119894)119888


119897120583120588120583(k) and 119868

120581120583120588120583(k) are

defined as

119868119897120583120588120583

(k) = F ⟨[119897120583(i 119905) 120588dagger

120583(j 119905)]⟩

119868120581120583120588120583

(k) = F ⟨[120581120583(i 119905) 120588dagger

120583(j 119905)]⟩

(54)


119866

(120583)(119894 119895) = ⟨119879 [119873

(120583)

(119894)119873

(120583)dagger(119895)]⟩

=

i1198862

(2120587)

3int119889


119866

(120583)

(k 120596)

(55)



[120596 minus 120576

(120583)

(k)] 119866(120583)

(k 120596) = 119868

(120583)

(k) (56)


120576

(120583)

(k) = (

0 120576

(120583)

12(k)

120576

(120583)

21(k) 0

)

120576

(120583)

12(k) = minus4119905

120576

(120583)

21(k) =

119898

(120583)

22(k)

119868

(120583)

12(k)

(57)


119866

(120583)

(k 120596)

= Γ

(120583)

(k) [ 1

120596 + i120575minus

1

120596 minus i120575]

+

2

sum

119899=1

120590

(119899120583)

(k) [1 + 119891B (120596)

120596 minus 120596

(120583)

119899 (k) + i120575minus

119891B (120596)

120596 minus 120596

(120583)

119899 (k) minus i120575]

(58)



(120583)(k 120596) =

⟨119873

(120583)(119894)119873


119862

(120583)

(k 120596) = 2120587Γ

(120583)

(k) 120575 (120596)

+ 2120587

2

sum

119899=1

120575 [120596 minus 120596

(120583)

119899(k)] [1 + 119891B (120596)] 120590

(119899120583)

(k)

(59)



120596

(120583)

119899(k) = (minus)

119899120596

(120583)

(k)

120596

(120583)

(k) = radic120576

(120583)

12(k) 120576(120583)

21(k)

(60)


120590

(119899120583)

(k) =119868

(120583)

12(k)2

(

120576

(120583)

12(k)

120596

(120583)

119899 (k)1

1

120576

(120583)

21(k)

120596

(120583)

119899 (k)

) (61)


119868119897120583120588120583

(k) = 3

4

[1 minus 120572 (k)] (12119862120572+ 119862

120582+ 6119862

120583)

minus 3 [1 minus 120573 (k)] (119862120572+ 119862

120583)

minus

3

4

[1 minus 120578 (k)] (119862120572+ 119862

120582+ 2119862

120583)

+

1

4

[1 minus 120582 (k)] 119862120582+

3

2

[1 minus 120583 (k)] 119862120583

119868120581120583120588120583


120578)

+ 2120573 (k) 119864120573+ 120578 (k) 119864120578

+ [1 minus 2120572 (k)] 119886120583

+

1

4

[119887120583+ 2120573 (k) 119888

120583+ 120578 (k) 119889

120583]

(62)



119864 = ⟨119888 (119894) 120578

dagger

(119894)⟩ 119862

120572= ⟨119888

120572

(119894) 119888

dagger

(119894)⟩

119864

120573= ⟨119888

120573

(119894) 120578

dagger

(119894)⟩ 119862

120582= ⟨119888

120582

(119894) 119888

dagger

(119894)⟩

119864

120578= ⟨119888

120578

(119894) 120578

dagger

(119894)⟩ 119862

120583= ⟨119888

120583

(119894) 119888

dagger

(119894)⟩

(63)

119886120583= 2 ⟨119888

dagger

(119894) 120590120583119888

120572

(119894) 119888

dagger

(119894) 120590120583119888

120572

(119894)⟩

minus ⟨119888

dagger120572

(119894) 120590120583120590

120582120590120583119888

120572

(119894) 119899120582(119894)⟩

119887120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger

(119894) 120590120583[119888 (119894) 119888 (119894)]

120572⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894) 119899

120572

120582(119894)⟩

119888120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger(119894

120578) 120590

120583119888 (119894

120572) 119888 (119894

120572)⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894

120578) 119899

120582(119894

120572)⟩

119889120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger(119894

120573) 120590

120583119888 (119894

120572) 119888 (119894

120572)⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894

120573) 119899

120582(119894

120572)⟩

(64)


119894 = (119894119909 119894119910 119905) 119894

120573= (119894

119909+ 119886 119894

119910+ 119886 119905)

119894

120572= (119894

119909+ 119886 119894

119910 119905) 119894

120578= (119894

119909+ 2119886 119894

119910 119905)

(65)


(120583)(119894 119895) =

⟨119879[119873

(120583)(119894)119873



120583 119887

120583 119888

120583 and 119889


Γ



dagger(119895)⟩


119887120583= 119886

120583+ 3119863 + 2119864

120573+ 119864

120578minus 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

119888120583= 119886

120583minus 119863 minus 2119864

120573+ 119864

120578+ 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

119889120583= 119886

120583minus 119863 + 2119864

120573minus 3119864

120578minus 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

(66)


(120583)

11(k) are fixed by


⟨119899120583(119894) 119899

120583(119894)⟩ =

119899 + 2119863 for 120583 = 0

119899 minus 2119863 for 120583 = 1 2 3

(67)


B0(ac0 as0)

G0(1205830 Δ0 p0)

n = 0

n = n + 1

Yes No

Σn[Bn Gn]



120583n Δn pn ==


where 119863 = ⟨119899uarr(119894)119899


ergodic value

Γ

(120583)

11(k) = 120575

1205830

(2120587)

2

119886

2120575

(2)

(k) ⟨119899⟩2 (68)


⟨120575119899120583(119894) 120575119899

120583(119895)⟩

=

119886

2

2 (2120587)

2

2

sum

119899=1

int119889


(120583)

119899(k)(119905119894minus119905119895)


(120583)

119899(k)

2119896B119879]120590

(119899120583)

11(k)

(69)

⟨119877 [120575119899120583(119894) 120575119899

120583(119895)]⟩

=

i1198862

(2120587)

3

2

sum

119899=1

int119889


(70)

times

120590

(119899120583)

11(k)

120596 minus 120596

(120583)

119899 (k) + i120575 (71)



120594120583(k 120596) = minusF [⟨119877 [120575119899

120583(119894) 120575119899

120583(119895)]⟩]

=

16119905 [1 minus 120572 (k)] 119862120572

120596

2minus (120596

(120583)(k))2

(72)



119899 = 2 (1 minus 11986211minus 119862

22)

Δ = 119862

120572

11minus 119862

120572

22

11986212

= ⟨120585 (119894) 120578

dagger

(119894)⟩ = 0

(73)

where 119862119899119898

and 119862

120572


functions 119862119899119898

= ⟨120595119899(119894)120595

dagger

119898(119894)⟩ and 119862

120572

119899119898= ⟨120595

120572

119899(119894)120595

dagger

119898(119894)⟩


119888119888(k 120596) [cf (38)] and



120583(k 120596)


0and 119886



22


(k 120596) and 120594120583(k 120596) in the two-









119888119888(k 120596) = F⟨119879[119888(119894)119888

dagger(119895)]⟩


3 Results


119860 (k 120596) = minus

1

120587

I [119866119888119888(k 120596)] (74)


119899 (k) = int119889120596119891F (120596)119860 (k 120596) (75)


119873(120596) =

1

(2120587)

2int119889

2119896119860 (k 120596) (76)

where 119866119888119888(k 120596) = 119866

11(k 120596) + 119866

12(k 120596) + 119866

21(k 120596) +




119866119888119888(k 120596) = 1

120596 minus 1205980(k) minus Σ

119888119888(k 120596)

(77)



1015840

119888119888(k 120596 = 0)


120596rarr0Σ

10158401015840

119888119888(k 120596 119879 =

0) prop 120596


Σ

10158401015840

119888119888(k 120596 = 0 119879) prop 119879



120572

119911⟩ the



limkrarrQ

120594

(3)

(k 0) =120594

(3)(Q 0)

1 + 120585

2|k minusQ|

2 (78)

where 120594(3)(k 0) = minus119866














0(k 120596) and 120594






9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 07

T = 001

Σ998400998400(k 120596)

(a)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 078

T = 001

Σ998400998400(k 120596)

(b)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596A(k120596)

00

06

12

18

24

30

U = 8n = 085

T = 001

Σ998400998400(k 120596)

(c)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 092

T = 002

Σ998400998400(k 120596)

(d)










050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 07

T = 001

A(k120596)

Σ998400998400(k 120596)

120596

(a)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 078

T = 001

120596

(b)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 085

T = 001

120596

(c)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 092

T = 002

120596

(d)



(119879 = 002)








3(k 120596) (see Section 36)






00 20 40 60 80 100 120 120

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

5120587831205874

120587

120587

120587

120587

U = 8

T = 001

n = 070

A(k120596

=0)

31205878

31205878

31205878

51205878

51205878

51205878

31205874 31205874

31205874

31205874

71205878

71205878

71205878

71205878

000

0

12058781205878

4

120587444444444444444444444

1205874

1205874

1205872

1205871205871205871205871205871205871205871205871205871205872222222222222222222222222222222222222

1205872

1205872

55555555555555555555555555555555555555555555555555555555555555555552222212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444

120587

1205878

31205878

333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

51205878

51205878

51205878

31205874 31205874

3120587

33333333333333333333333331205871205871205871205871205871205871205871205871205874

71205878

71205878

7

74 1205878

kx

ky

(a)

00 20 40 60 80 100

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

3120587471205878

120587

120587

120587

120587

U = 8

T = 001

n = 078

kx

ky

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

3120587851205878

31205878

51205878

31205874

71205878

0

0

12058781205878

4

1205874 1205874

1205872

1205872

1205872

1205872

33333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444444444444444444447777777777777777777777777777777777777777777777777777777712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333333333333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

512058783120587

7

5120587831205874 71205878

3120587852212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888

1205878

51205878

31205874

71205878

(b)

00 10 20 30 40 50 60 70 80 90

00

0

0

123456789

1011

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

71205878

120587

120587

120587

120587

U = 8

T = 001n = 085

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

312058785120587831205874

00

000000000000000000000000000000000000000000

12058781205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

4

1205874 1205874

1205872

1205872

1205872

1205872

777777777777777777777777777777777777777777777777777777777771205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333331205878

512058783120587

7

3120587851205878

31205874 71205878

8

51205878

31205874

71205878

3120587852212058783333333333312058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058744444444444444444444444444444444444444444444444

kx

ky

(c)

00 10 20 30 40 50 60 70

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

120587

120587

120587

120587

U = 8

T = 002n = 092

A(k120596

=0)

31205878

5120587831205874

71205878

71205878

312058785120587831205874

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

0000

0

12058781205878

4

1205874

1205874

1205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874

1205872

1205872

1205872

1205872

120587

333331205878

512058783120587

7

77777777777777777777777120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587888888888888888888888888888888888888888

31205878522120587831205874

3120587851205878

31205874 71205878

1205878

51205878

31205874

71205878

kx

ky

(d)







119888119888(k 120596 = 0) (notice that



119888119888(k 120596 = 0) is quite


119888119888(k 120596 = 0) in those





00

25

50

75

100

125

r(k) = 0

n(k) = 05 U = 8

n = 070

T = 001

minus010 minus020

minus030

minus040minus050

Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

ky

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(a)

minus010


minus050

0

1205878

1205874

31205878

1205872

51205878

31205874

71205878

120587

0 1205878 1205874 31205878 1205872 51205878 31205874 71205878 120587

r(k) = 0

n(k) = 05

kx

k y

00

21

42

63

84

105

A(k120596

=0)

U = 8

n = 078

T = 001Σ998400998400cc(k 120596 = 0)

(b)

00

18

36

54

72

90

r(k) = 0

n(k) = 05 U = 8

n = 085

T = 001Σ998400998400cc(k 120596 = 0)

A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(c)

r(k) = 0

n(k) = 05 U = 8

n = 092

T = 002Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y00

14

28

42

56

70

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(d)








00

02

04

06

08

10

12

n = 070

n = 078n = 085

n = 092


120596

N(120596)

U = 8 T = 001 U = 8 T = 002

(a)

1

2

3

4

5

6

7

8


U = 8T = 002n = 092

A(k

=k120596)

120596

k = S

k = S

k = X

(b)









YX SS M

k

n(k)

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

n = 070

n = 075

n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

00

01

02

03

04

05

06

07

08

09

10

(a)

S M

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

kΓ

n = 070n = 075n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

(b)







and 119899 = 07 075 08 and 085 (119879 = 001) and for 119899 =





09

0807 06

05

0403

0202

00

01

02

03

04

05

06

07

08

09

10

n(k)

U = 8n = 092T = 001

Y

X

M

Γ














YXS M

U= 8[[

n = 092

T = 100

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrT= 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

(a)

YXS M

U = 01 [[

n = 092

T = 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrU = 7

(b)




















119911119899

120572






119911119899

120572

119911⟩ is






120596

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

minus08

minus06

minus04

minus02

00

02

04

06

08

(a)


120596

minus08

minus06

minus04

minus02

00

02

04

06

08

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

(b)

000 001 002 003 004 005minus012

minus010

minus008

minus006

minus004

minus002

000

T

Σ998400998400(k120596

=0)

n = 092

U = 8k = Sk = S

(c)

minus012

minus010

minus008

minus006

minus004

minus002

000

Σ998400998400(k120596

=0)

0000 0001 0002

T2

n = 092

U = 8k = Sk = S

(d)





guide to the eye






T

n

00 01 02 03 04 05 06 07 08 09 10minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009090 087 084 081 078 075 072

n = 092

U = 8

T = 001

⟨nzn120572 z⟩

8 7 6 5 4 3 2 1 0

U

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8T = 001

⟨nzn120572 z⟩


120572



0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

n

090 087 084 081 078 075 072

120585

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

78 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

120585

n = 092

U = 8

T = 001


00

02

04

06

08

10

12

14

16

00

02

04

06

08

10

12

14

16

n

090 087 084 081 078 075 072

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

00

04

08

12

16

20

24

28

32

00

04

08

12

16

20

24

28

32

8 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

120596(Q

)

120596(Q

)
















Acknowledgments


References












































































2minus119909Sr

119909CuO


























2CuO

6+120575









Na119909CuO

2Cl

2rdquo Science









2Sr

2minus119909LaxCuO






2Cu

4O

8rdquo Physical




119888



2Cu

3O

65rdquo








2Cu

3O

661and YBa

2Cu

3O

669



2Sr

2CaCu

2O




2Sr

2CaCu

2O





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





i 120597120597119905

119873

(120583)

(119894) = 119869

(120583)

(119894) = (

119869

(120583)

1(119894)

119869

(120583)

2(119894)

)

119869

(120583)

1(119894) = minus4119905120588

120583(119894)

119869

(120583)

2(119894) = 119880120581

120583(119894) minus 4119905119897

120583(119894)

(46)


120583(119894) are

defined as

120581120583(119894) = 119888

dagger

(119894) 120590120583120578

120572

(119894) minus 120578

dagger

(119894) 120590120583119888

120572

(119894)

+ 120578

dagger120572

(119894) 120590120583119888 (119894) minus 119888

dagger120572

(119894) 120590120583120578 (119894)

119897120583(119894) = 119888

dagger

(119894) 120590120583119888

1205722

(119894) + 119888

dagger1205722

(119894) 120590120583119888 (119894)

minus 2119888

dagger120572

(119894) 120590120583119888

120572

(119894)

(47)


119888

1205722

(i 119905) = sum

j120572

2

ij119888 (j 119905) = sum

jl120572il120572lj119888 (j 119905) (48)




i 120597120597119905

119873

(120583)

(i 119905) = sum

j120576

(120583)(i j)119873(120583)

(j 119905) (49)


119898

(120583)(i j) = sum

l120576

(120583)

(i l) 119868(120583) (l j) (50)



119868

(120583)(i j) = ⟨[119873

(120583)

(i 119905) 119873(120583)dagger(j 119905)]⟩

119898

(120583)(i j) = ⟨[119869

(120583)

(i 119905) 119873(120583)dagger(j 119905)]⟩

(51)



119868

(120583)

(k) = (

0 119868

(120583)

12(k)

119868

(120583)

12(k) 0

)

119898

(120583)

(k) = (

119898

(120583)

11(k) 0

0 119898

(120583)

22(k)

)

(52)

where

119868

(120583)

12(k) = 4 [1 minus 120572 (k)] 119862120572

119888119888

119898

(120583)

11(k) = minus 4119905119868

(120583)

12(k)

119898

(120583)

22(k) = minus 4119905119868

119897120583120588120583(k) + 119880119868

120581120583120588120583(k)

(53)


120572= ⟨119888

120572(119894)119888


119897120583120588120583(k) and 119868

120581120583120588120583(k) are

defined as

119868119897120583120588120583

(k) = F ⟨[119897120583(i 119905) 120588dagger

120583(j 119905)]⟩

119868120581120583120588120583

(k) = F ⟨[120581120583(i 119905) 120588dagger

120583(j 119905)]⟩

(54)


119866

(120583)(119894 119895) = ⟨119879 [119873

(120583)

(119894)119873

(120583)dagger(119895)]⟩

=

i1198862

(2120587)

3int119889


119866

(120583)

(k 120596)

(55)



[120596 minus 120576

(120583)

(k)] 119866(120583)

(k 120596) = 119868

(120583)

(k) (56)


120576

(120583)

(k) = (

0 120576

(120583)

12(k)

120576

(120583)

21(k) 0

)

120576

(120583)

12(k) = minus4119905

120576

(120583)

21(k) =

119898

(120583)

22(k)

119868

(120583)

12(k)

(57)


119866

(120583)

(k 120596)

= Γ

(120583)

(k) [ 1

120596 + i120575minus

1

120596 minus i120575]

+

2

sum

119899=1

120590

(119899120583)

(k) [1 + 119891B (120596)

120596 minus 120596

(120583)

119899 (k) + i120575minus

119891B (120596)

120596 minus 120596

(120583)

119899 (k) minus i120575]

(58)



(120583)(k 120596) =

⟨119873

(120583)(119894)119873


119862

(120583)

(k 120596) = 2120587Γ

(120583)

(k) 120575 (120596)

+ 2120587

2

sum

119899=1

120575 [120596 minus 120596

(120583)

119899(k)] [1 + 119891B (120596)] 120590

(119899120583)

(k)

(59)



120596

(120583)

119899(k) = (minus)

119899120596

(120583)

(k)

120596

(120583)

(k) = radic120576

(120583)

12(k) 120576(120583)

21(k)

(60)


120590

(119899120583)

(k) =119868

(120583)

12(k)2

(

120576

(120583)

12(k)

120596

(120583)

119899 (k)1

1

120576

(120583)

21(k)

120596

(120583)

119899 (k)

) (61)


119868119897120583120588120583

(k) = 3

4

[1 minus 120572 (k)] (12119862120572+ 119862

120582+ 6119862

120583)

minus 3 [1 minus 120573 (k)] (119862120572+ 119862

120583)

minus

3

4

[1 minus 120578 (k)] (119862120572+ 119862

120582+ 2119862

120583)

+

1

4

[1 minus 120582 (k)] 119862120582+

3

2

[1 minus 120583 (k)] 119862120583

119868120581120583120588120583


120578)

+ 2120573 (k) 119864120573+ 120578 (k) 119864120578

+ [1 minus 2120572 (k)] 119886120583

+

1

4

[119887120583+ 2120573 (k) 119888

120583+ 120578 (k) 119889

120583]

(62)



119864 = ⟨119888 (119894) 120578

dagger

(119894)⟩ 119862

120572= ⟨119888

120572

(119894) 119888

dagger

(119894)⟩

119864

120573= ⟨119888

120573

(119894) 120578

dagger

(119894)⟩ 119862

120582= ⟨119888

120582

(119894) 119888

dagger

(119894)⟩

119864

120578= ⟨119888

120578

(119894) 120578

dagger

(119894)⟩ 119862

120583= ⟨119888

120583

(119894) 119888

dagger

(119894)⟩

(63)

119886120583= 2 ⟨119888

dagger

(119894) 120590120583119888

120572

(119894) 119888

dagger

(119894) 120590120583119888

120572

(119894)⟩

minus ⟨119888

dagger120572

(119894) 120590120583120590

120582120590120583119888

120572

(119894) 119899120582(119894)⟩

119887120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger

(119894) 120590120583[119888 (119894) 119888 (119894)]

120572⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894) 119899

120572

120582(119894)⟩

119888120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger(119894

120578) 120590

120583119888 (119894

120572) 119888 (119894

120572)⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894

120578) 119899

120582(119894

120572)⟩

119889120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger(119894

120573) 120590

120583119888 (119894

120572) 119888 (119894

120572)⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894

120573) 119899

120582(119894

120572)⟩

(64)


119894 = (119894119909 119894119910 119905) 119894

120573= (119894

119909+ 119886 119894

119910+ 119886 119905)

119894

120572= (119894

119909+ 119886 119894

119910 119905) 119894

120578= (119894

119909+ 2119886 119894

119910 119905)

(65)


(120583)(119894 119895) =

⟨119879[119873

(120583)(119894)119873



120583 119887

120583 119888

120583 and 119889


Γ



dagger(119895)⟩


119887120583= 119886

120583+ 3119863 + 2119864

120573+ 119864

120578minus 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

119888120583= 119886

120583minus 119863 minus 2119864

120573+ 119864

120578+ 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

119889120583= 119886

120583minus 119863 + 2119864

120573minus 3119864

120578minus 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

(66)


(120583)

11(k) are fixed by


⟨119899120583(119894) 119899

120583(119894)⟩ =

119899 + 2119863 for 120583 = 0

119899 minus 2119863 for 120583 = 1 2 3

(67)


B0(ac0 as0)

G0(1205830 Δ0 p0)

n = 0

n = n + 1

Yes No

Σn[Bn Gn]



120583n Δn pn ==


where 119863 = ⟨119899uarr(119894)119899


ergodic value

Γ

(120583)

11(k) = 120575

1205830

(2120587)

2

119886

2120575

(2)

(k) ⟨119899⟩2 (68)


⟨120575119899120583(119894) 120575119899

120583(119895)⟩

=

119886

2

2 (2120587)

2

2

sum

119899=1

int119889


(120583)

119899(k)(119905119894minus119905119895)


(120583)

119899(k)

2119896B119879]120590

(119899120583)

11(k)

(69)

⟨119877 [120575119899120583(119894) 120575119899

120583(119895)]⟩

=

i1198862

(2120587)

3

2

sum

119899=1

int119889


(70)

times

120590

(119899120583)

11(k)

120596 minus 120596

(120583)

119899 (k) + i120575 (71)



120594120583(k 120596) = minusF [⟨119877 [120575119899

120583(119894) 120575119899

120583(119895)]⟩]

=

16119905 [1 minus 120572 (k)] 119862120572

120596

2minus (120596

(120583)(k))2

(72)



119899 = 2 (1 minus 11986211minus 119862

22)

Δ = 119862

120572

11minus 119862

120572

22

11986212

= ⟨120585 (119894) 120578

dagger

(119894)⟩ = 0

(73)

where 119862119899119898

and 119862

120572


functions 119862119899119898

= ⟨120595119899(119894)120595

dagger

119898(119894)⟩ and 119862

120572

119899119898= ⟨120595

120572

119899(119894)120595

dagger

119898(119894)⟩


119888119888(k 120596) [cf (38)] and



120583(k 120596)


0and 119886



22


(k 120596) and 120594120583(k 120596) in the two-









119888119888(k 120596) = F⟨119879[119888(119894)119888

dagger(119895)]⟩


3 Results


119860 (k 120596) = minus

1

120587

I [119866119888119888(k 120596)] (74)


119899 (k) = int119889120596119891F (120596)119860 (k 120596) (75)


119873(120596) =

1

(2120587)

2int119889

2119896119860 (k 120596) (76)

where 119866119888119888(k 120596) = 119866

11(k 120596) + 119866

12(k 120596) + 119866

21(k 120596) +




119866119888119888(k 120596) = 1

120596 minus 1205980(k) minus Σ

119888119888(k 120596)

(77)



1015840

119888119888(k 120596 = 0)


120596rarr0Σ

10158401015840

119888119888(k 120596 119879 =

0) prop 120596


Σ

10158401015840

119888119888(k 120596 = 0 119879) prop 119879



120572

119911⟩ the



limkrarrQ

120594

(3)

(k 0) =120594

(3)(Q 0)

1 + 120585

2|k minusQ|

2 (78)

where 120594(3)(k 0) = minus119866














0(k 120596) and 120594






9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 07

T = 001

Σ998400998400(k 120596)

(a)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 078

T = 001

Σ998400998400(k 120596)

(b)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596A(k120596)

00

06

12

18

24

30

U = 8n = 085

T = 001

Σ998400998400(k 120596)

(c)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 092

T = 002

Σ998400998400(k 120596)

(d)










050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 07

T = 001

A(k120596)

Σ998400998400(k 120596)

120596

(a)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 078

T = 001

120596

(b)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 085

T = 001

120596

(c)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 092

T = 002

120596

(d)



(119879 = 002)








3(k 120596) (see Section 36)






00 20 40 60 80 100 120 120

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

5120587831205874

120587

120587

120587

120587

U = 8

T = 001

n = 070

A(k120596

=0)

31205878

31205878

31205878

51205878

51205878

51205878

31205874 31205874

31205874

31205874

71205878

71205878

71205878

71205878

000

0

12058781205878

4

120587444444444444444444444

1205874

1205874

1205872

1205871205871205871205871205871205871205871205871205871205872222222222222222222222222222222222222

1205872

1205872

55555555555555555555555555555555555555555555555555555555555555555552222212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444

120587

1205878

31205878

333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

51205878

51205878

51205878

31205874 31205874

3120587

33333333333333333333333331205871205871205871205871205871205871205871205871205874

71205878

71205878

7

74 1205878

kx

ky

(a)

00 20 40 60 80 100

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

3120587471205878

120587

120587

120587

120587

U = 8

T = 001

n = 078

kx

ky

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

3120587851205878

31205878

51205878

31205874

71205878

0

0

12058781205878

4

1205874 1205874

1205872

1205872

1205872

1205872

33333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444444444444444444447777777777777777777777777777777777777777777777777777777712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333333333333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

512058783120587

7

5120587831205874 71205878

3120587852212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888

1205878

51205878

31205874

71205878

(b)

00 10 20 30 40 50 60 70 80 90

00

0

0

123456789

1011

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

71205878

120587

120587

120587

120587

U = 8

T = 001n = 085

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

312058785120587831205874

00

000000000000000000000000000000000000000000

12058781205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

4

1205874 1205874

1205872

1205872

1205872

1205872

777777777777777777777777777777777777777777777777777777777771205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333331205878

512058783120587

7

3120587851205878

31205874 71205878

8

51205878

31205874

71205878

3120587852212058783333333333312058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058744444444444444444444444444444444444444444444444

kx

ky

(c)

00 10 20 30 40 50 60 70

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

120587

120587

120587

120587

U = 8

T = 002n = 092

A(k120596

=0)

31205878

5120587831205874

71205878

71205878

312058785120587831205874

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

0000

0

12058781205878

4

1205874

1205874

1205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874

1205872

1205872

1205872

1205872

120587

333331205878

512058783120587

7

77777777777777777777777120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587888888888888888888888888888888888888888

31205878522120587831205874

3120587851205878

31205874 71205878

1205878

51205878

31205874

71205878

kx

ky

(d)







119888119888(k 120596 = 0) (notice that



119888119888(k 120596 = 0) is quite


119888119888(k 120596 = 0) in those





00

25

50

75

100

125

r(k) = 0

n(k) = 05 U = 8

n = 070

T = 001

minus010 minus020

minus030

minus040minus050

Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

ky

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(a)

minus010


minus050

0

1205878

1205874

31205878

1205872

51205878

31205874

71205878

120587

0 1205878 1205874 31205878 1205872 51205878 31205874 71205878 120587

r(k) = 0

n(k) = 05

kx

k y

00

21

42

63

84

105

A(k120596

=0)

U = 8

n = 078

T = 001Σ998400998400cc(k 120596 = 0)

(b)

00

18

36

54

72

90

r(k) = 0

n(k) = 05 U = 8

n = 085

T = 001Σ998400998400cc(k 120596 = 0)

A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(c)

r(k) = 0

n(k) = 05 U = 8

n = 092

T = 002Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y00

14

28

42

56

70

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(d)








00

02

04

06

08

10

12

n = 070

n = 078n = 085

n = 092


120596

N(120596)

U = 8 T = 001 U = 8 T = 002

(a)

1

2

3

4

5

6

7

8


U = 8T = 002n = 092

A(k

=k120596)

120596

k = S

k = S

k = X

(b)









YX SS M

k

n(k)

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

n = 070

n = 075

n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

00

01

02

03

04

05

06

07

08

09

10

(a)

S M

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

kΓ

n = 070n = 075n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

(b)







and 119899 = 07 075 08 and 085 (119879 = 001) and for 119899 =





09

0807 06

05

0403

0202

00

01

02

03

04

05

06

07

08

09

10

n(k)

U = 8n = 092T = 001

Y

X

M

Γ














YXS M

U= 8[[

n = 092

T = 100

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrT= 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

(a)

YXS M

U = 01 [[

n = 092

T = 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrU = 7

(b)




















119911119899

120572






119911119899

120572

119911⟩ is






120596

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

minus08

minus06

minus04

minus02

00

02

04

06

08

(a)


120596

minus08

minus06

minus04

minus02

00

02

04

06

08

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

(b)

000 001 002 003 004 005minus012

minus010

minus008

minus006

minus004

minus002

000

T

Σ998400998400(k120596

=0)

n = 092

U = 8k = Sk = S

(c)

minus012

minus010

minus008

minus006

minus004

minus002

000

Σ998400998400(k120596

=0)

0000 0001 0002

T2

n = 092

U = 8k = Sk = S

(d)





guide to the eye






T

n

00 01 02 03 04 05 06 07 08 09 10minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009090 087 084 081 078 075 072

n = 092

U = 8

T = 001

⟨nzn120572 z⟩

8 7 6 5 4 3 2 1 0

U

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8T = 001

⟨nzn120572 z⟩


120572



0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

n

090 087 084 081 078 075 072

120585

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

78 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

120585

n = 092

U = 8

T = 001


00

02

04

06

08

10

12

14

16

00

02

04

06

08

10

12

14

16

n

090 087 084 081 078 075 072

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

00

04

08

12

16

20

24

28

32

00

04

08

12

16

20

24

28

32

8 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

120596(Q

)

120596(Q

)
















Acknowledgments


References












































































2minus119909Sr

119909CuO


























2CuO

6+120575









Na119909CuO

2Cl

2rdquo Science









2Sr

2minus119909LaxCuO






2Cu

4O

8rdquo Physical




119888



2Cu

3O

65rdquo








2Cu

3O

661and YBa

2Cu

3O

669



2Sr

2CaCu

2O




2Sr

2CaCu

2O





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





(120583)(k 120596) =

⟨119873

(120583)(119894)119873


119862

(120583)

(k 120596) = 2120587Γ

(120583)

(k) 120575 (120596)

+ 2120587

2

sum

119899=1

120575 [120596 minus 120596

(120583)

119899(k)] [1 + 119891B (120596)] 120590

(119899120583)

(k)

(59)



120596

(120583)

119899(k) = (minus)

119899120596

(120583)

(k)

120596

(120583)

(k) = radic120576

(120583)

12(k) 120576(120583)

21(k)

(60)


120590

(119899120583)

(k) =119868

(120583)

12(k)2

(

120576

(120583)

12(k)

120596

(120583)

119899 (k)1

1

120576

(120583)

21(k)

120596

(120583)

119899 (k)

) (61)


119868119897120583120588120583

(k) = 3

4

[1 minus 120572 (k)] (12119862120572+ 119862

120582+ 6119862

120583)

minus 3 [1 minus 120573 (k)] (119862120572+ 119862

120583)

minus

3

4

[1 minus 120578 (k)] (119862120572+ 119862

120582+ 2119862

120583)

+

1

4

[1 minus 120582 (k)] 119862120582+

3

2

[1 minus 120583 (k)] 119862120583

119868120581120583120588120583


120578)

+ 2120573 (k) 119864120573+ 120578 (k) 119864120578

+ [1 minus 2120572 (k)] 119886120583

+

1

4

[119887120583+ 2120573 (k) 119888

120583+ 120578 (k) 119889

120583]

(62)



119864 = ⟨119888 (119894) 120578

dagger

(119894)⟩ 119862

120572= ⟨119888

120572

(119894) 119888

dagger

(119894)⟩

119864

120573= ⟨119888

120573

(119894) 120578

dagger

(119894)⟩ 119862

120582= ⟨119888

120582

(119894) 119888

dagger

(119894)⟩

119864

120578= ⟨119888

120578

(119894) 120578

dagger

(119894)⟩ 119862

120583= ⟨119888

120583

(119894) 119888

dagger

(119894)⟩

(63)

119886120583= 2 ⟨119888

dagger

(119894) 120590120583119888

120572

(119894) 119888

dagger

(119894) 120590120583119888

120572

(119894)⟩

minus ⟨119888

dagger120572

(119894) 120590120583120590

120582120590120583119888

120572

(119894) 119899120582(119894)⟩

119887120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger

(119894) 120590120583[119888 (119894) 119888 (119894)]

120572⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894) 119899

120572

120582(119894)⟩

119888120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger(119894

120578) 120590

120583119888 (119894

120572) 119888 (119894

120572)⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894

120578) 119899

120582(119894

120572)⟩

119889120583= 2 ⟨119888

dagger

(119894) 120590120583119888

dagger(119894

120573) 120590

120583119888 (119894

120572) 119888 (119894

120572)⟩

minus ⟨119888

dagger

(119894) 120590120583120590

120582120590120583119888 (119894

120573) 119899

120582(119894

120572)⟩

(64)


119894 = (119894119909 119894119910 119905) 119894

120573= (119894

119909+ 119886 119894

119910+ 119886 119905)

119894

120572= (119894

119909+ 119886 119894

119910 119905) 119894

120578= (119894

119909+ 2119886 119894

119910 119905)

(65)


(120583)(119894 119895) =

⟨119879[119873

(120583)(119894)119873



120583 119887

120583 119888

120583 and 119889


Γ



dagger(119895)⟩


119887120583= 119886

120583+ 3119863 + 2119864

120573+ 119864

120578minus 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

119888120583= 119886

120583minus 119863 minus 2119864

120573+ 119864

120578+ 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

119889120583= 119886

120583minus 119863 + 2119864

120573minus 3119864

120578minus 6

119905

119880

(119862

120572+ 119862

120582minus 2119862

120583)

(66)


(120583)

11(k) are fixed by


⟨119899120583(119894) 119899

120583(119894)⟩ =

119899 + 2119863 for 120583 = 0

119899 minus 2119863 for 120583 = 1 2 3

(67)


B0(ac0 as0)

G0(1205830 Δ0 p0)

n = 0

n = n + 1

Yes No

Σn[Bn Gn]



120583n Δn pn ==


where 119863 = ⟨119899uarr(119894)119899


ergodic value

Γ

(120583)

11(k) = 120575

1205830

(2120587)

2

119886

2120575

(2)

(k) ⟨119899⟩2 (68)


⟨120575119899120583(119894) 120575119899

120583(119895)⟩

=

119886

2

2 (2120587)

2

2

sum

119899=1

int119889


(120583)

119899(k)(119905119894minus119905119895)


(120583)

119899(k)

2119896B119879]120590

(119899120583)

11(k)

(69)

⟨119877 [120575119899120583(119894) 120575119899

120583(119895)]⟩

=

i1198862

(2120587)

3

2

sum

119899=1

int119889


(70)

times

120590

(119899120583)

11(k)

120596 minus 120596

(120583)

119899 (k) + i120575 (71)



120594120583(k 120596) = minusF [⟨119877 [120575119899

120583(119894) 120575119899

120583(119895)]⟩]

=

16119905 [1 minus 120572 (k)] 119862120572

120596

2minus (120596

(120583)(k))2

(72)



119899 = 2 (1 minus 11986211minus 119862

22)

Δ = 119862

120572

11minus 119862

120572

22

11986212

= ⟨120585 (119894) 120578

dagger

(119894)⟩ = 0

(73)

where 119862119899119898

and 119862

120572


functions 119862119899119898

= ⟨120595119899(119894)120595

dagger

119898(119894)⟩ and 119862

120572

119899119898= ⟨120595

120572

119899(119894)120595

dagger

119898(119894)⟩


119888119888(k 120596) [cf (38)] and



120583(k 120596)


0and 119886



22


(k 120596) and 120594120583(k 120596) in the two-









119888119888(k 120596) = F⟨119879[119888(119894)119888

dagger(119895)]⟩


3 Results


119860 (k 120596) = minus

1

120587

I [119866119888119888(k 120596)] (74)


119899 (k) = int119889120596119891F (120596)119860 (k 120596) (75)


119873(120596) =

1

(2120587)

2int119889

2119896119860 (k 120596) (76)

where 119866119888119888(k 120596) = 119866

11(k 120596) + 119866

12(k 120596) + 119866

21(k 120596) +




119866119888119888(k 120596) = 1

120596 minus 1205980(k) minus Σ

119888119888(k 120596)

(77)



1015840

119888119888(k 120596 = 0)


120596rarr0Σ

10158401015840

119888119888(k 120596 119879 =

0) prop 120596


Σ

10158401015840

119888119888(k 120596 = 0 119879) prop 119879



120572

119911⟩ the



limkrarrQ

120594

(3)

(k 0) =120594

(3)(Q 0)

1 + 120585

2|k minusQ|

2 (78)

where 120594(3)(k 0) = minus119866














0(k 120596) and 120594






9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 07

T = 001

Σ998400998400(k 120596)

(a)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 078

T = 001

Σ998400998400(k 120596)

(b)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596A(k120596)

00

06

12

18

24

30

U = 8n = 085

T = 001

Σ998400998400(k 120596)

(c)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 092

T = 002

Σ998400998400(k 120596)

(d)










050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 07

T = 001

A(k120596)

Σ998400998400(k 120596)

120596

(a)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 078

T = 001

120596

(b)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 085

T = 001

120596

(c)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 092

T = 002

120596

(d)



(119879 = 002)








3(k 120596) (see Section 36)






00 20 40 60 80 100 120 120

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

5120587831205874

120587

120587

120587

120587

U = 8

T = 001

n = 070

A(k120596

=0)

31205878

31205878

31205878

51205878

51205878

51205878

31205874 31205874

31205874

31205874

71205878

71205878

71205878

71205878

000

0

12058781205878

4

120587444444444444444444444

1205874

1205874

1205872

1205871205871205871205871205871205871205871205871205871205872222222222222222222222222222222222222

1205872

1205872

55555555555555555555555555555555555555555555555555555555555555555552222212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444

120587

1205878

31205878

333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

51205878

51205878

51205878

31205874 31205874

3120587

33333333333333333333333331205871205871205871205871205871205871205871205871205874

71205878

71205878

7

74 1205878

kx

ky

(a)

00 20 40 60 80 100

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

3120587471205878

120587

120587

120587

120587

U = 8

T = 001

n = 078

kx

ky

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

3120587851205878

31205878

51205878

31205874

71205878

0

0

12058781205878

4

1205874 1205874

1205872

1205872

1205872

1205872

33333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444444444444444444447777777777777777777777777777777777777777777777777777777712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333333333333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

512058783120587

7

5120587831205874 71205878

3120587852212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888

1205878

51205878

31205874

71205878

(b)

00 10 20 30 40 50 60 70 80 90

00

0

0

123456789

1011

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

71205878

120587

120587

120587

120587

U = 8

T = 001n = 085

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

312058785120587831205874

00

000000000000000000000000000000000000000000

12058781205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

4

1205874 1205874

1205872

1205872

1205872

1205872

777777777777777777777777777777777777777777777777777777777771205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333331205878

512058783120587

7

3120587851205878

31205874 71205878

8

51205878

31205874

71205878

3120587852212058783333333333312058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058744444444444444444444444444444444444444444444444

kx

ky

(c)

00 10 20 30 40 50 60 70

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

120587

120587

120587

120587

U = 8

T = 002n = 092

A(k120596

=0)

31205878

5120587831205874

71205878

71205878

312058785120587831205874

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

0000

0

12058781205878

4

1205874

1205874

1205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874

1205872

1205872

1205872

1205872

120587

333331205878

512058783120587

7

77777777777777777777777120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587888888888888888888888888888888888888888

31205878522120587831205874

3120587851205878

31205874 71205878

1205878

51205878

31205874

71205878

kx

ky

(d)







119888119888(k 120596 = 0) (notice that



119888119888(k 120596 = 0) is quite


119888119888(k 120596 = 0) in those





00

25

50

75

100

125

r(k) = 0

n(k) = 05 U = 8

n = 070

T = 001

minus010 minus020

minus030

minus040minus050

Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

ky

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(a)

minus010


minus050

0

1205878

1205874

31205878

1205872

51205878

31205874

71205878

120587

0 1205878 1205874 31205878 1205872 51205878 31205874 71205878 120587

r(k) = 0

n(k) = 05

kx

k y

00

21

42

63

84

105

A(k120596

=0)

U = 8

n = 078

T = 001Σ998400998400cc(k 120596 = 0)

(b)

00

18

36

54

72

90

r(k) = 0

n(k) = 05 U = 8

n = 085

T = 001Σ998400998400cc(k 120596 = 0)

A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(c)

r(k) = 0

n(k) = 05 U = 8

n = 092

T = 002Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y00

14

28

42

56

70

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(d)








00

02

04

06

08

10

12

n = 070

n = 078n = 085

n = 092


120596

N(120596)

U = 8 T = 001 U = 8 T = 002

(a)

1

2

3

4

5

6

7

8


U = 8T = 002n = 092

A(k

=k120596)

120596

k = S

k = S

k = X

(b)









YX SS M

k

n(k)

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

n = 070

n = 075

n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

00

01

02

03

04

05

06

07

08

09

10

(a)

S M

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

kΓ

n = 070n = 075n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

(b)







and 119899 = 07 075 08 and 085 (119879 = 001) and for 119899 =





09

0807 06

05

0403

0202

00

01

02

03

04

05

06

07

08

09

10

n(k)

U = 8n = 092T = 001

Y

X

M

Γ














YXS M

U= 8[[

n = 092

T = 100

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrT= 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

(a)

YXS M

U = 01 [[

n = 092

T = 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrU = 7

(b)




















119911119899

120572






119911119899

120572

119911⟩ is






120596

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

minus08

minus06

minus04

minus02

00

02

04

06

08

(a)


120596

minus08

minus06

minus04

minus02

00

02

04

06

08

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

(b)

000 001 002 003 004 005minus012

minus010

minus008

minus006

minus004

minus002

000

T

Σ998400998400(k120596

=0)

n = 092

U = 8k = Sk = S

(c)

minus012

minus010

minus008

minus006

minus004

minus002

000

Σ998400998400(k120596

=0)

0000 0001 0002

T2

n = 092

U = 8k = Sk = S

(d)





guide to the eye






T

n

00 01 02 03 04 05 06 07 08 09 10minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009090 087 084 081 078 075 072

n = 092

U = 8

T = 001

⟨nzn120572 z⟩

8 7 6 5 4 3 2 1 0

U

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8T = 001

⟨nzn120572 z⟩


120572



0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

n

090 087 084 081 078 075 072

120585

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

78 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

120585

n = 092

U = 8

T = 001


00

02

04

06

08

10

12

14

16

00

02

04

06

08

10

12

14

16

n

090 087 084 081 078 075 072

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

00

04

08

12

16

20

24

28

32

00

04

08

12

16

20

24

28

32

8 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

120596(Q

)

120596(Q

)
















Acknowledgments


References












































































2minus119909Sr

119909CuO


























2CuO

6+120575









Na119909CuO

2Cl

2rdquo Science









2Sr

2minus119909LaxCuO






2Cu

4O

8rdquo Physical




119888



2Cu

3O

65rdquo








2Cu

3O

661and YBa

2Cu

3O

669



2Sr

2CaCu

2O




2Sr

2CaCu

2O





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




B0(ac0 as0)

G0(1205830 Δ0 p0)

n = 0

n = n + 1

Yes No

Σn[Bn Gn]



120583n Δn pn ==


where 119863 = ⟨119899uarr(119894)119899


ergodic value

Γ

(120583)

11(k) = 120575

1205830

(2120587)

2

119886

2120575

(2)

(k) ⟨119899⟩2 (68)


⟨120575119899120583(119894) 120575119899

120583(119895)⟩

=

119886

2

2 (2120587)

2

2

sum

119899=1

int119889


(120583)

119899(k)(119905119894minus119905119895)


(120583)

119899(k)

2119896B119879]120590

(119899120583)

11(k)

(69)

⟨119877 [120575119899120583(119894) 120575119899

120583(119895)]⟩

=

i1198862

(2120587)

3

2

sum

119899=1

int119889


(70)

times

120590

(119899120583)

11(k)

120596 minus 120596

(120583)

119899 (k) + i120575 (71)



120594120583(k 120596) = minusF [⟨119877 [120575119899

120583(119894) 120575119899

120583(119895)]⟩]

=

16119905 [1 minus 120572 (k)] 119862120572

120596

2minus (120596

(120583)(k))2

(72)



119899 = 2 (1 minus 11986211minus 119862

22)

Δ = 119862

120572

11minus 119862

120572

22

11986212

= ⟨120585 (119894) 120578

dagger

(119894)⟩ = 0

(73)

where 119862119899119898

and 119862

120572


functions 119862119899119898

= ⟨120595119899(119894)120595

dagger

119898(119894)⟩ and 119862

120572

119899119898= ⟨120595

120572

119899(119894)120595

dagger

119898(119894)⟩


119888119888(k 120596) [cf (38)] and



120583(k 120596)


0and 119886



22


(k 120596) and 120594120583(k 120596) in the two-









119888119888(k 120596) = F⟨119879[119888(119894)119888

dagger(119895)]⟩


3 Results


119860 (k 120596) = minus

1

120587

I [119866119888119888(k 120596)] (74)


119899 (k) = int119889120596119891F (120596)119860 (k 120596) (75)


119873(120596) =

1

(2120587)

2int119889

2119896119860 (k 120596) (76)

where 119866119888119888(k 120596) = 119866

11(k 120596) + 119866

12(k 120596) + 119866

21(k 120596) +




119866119888119888(k 120596) = 1

120596 minus 1205980(k) minus Σ

119888119888(k 120596)

(77)



1015840

119888119888(k 120596 = 0)


120596rarr0Σ

10158401015840

119888119888(k 120596 119879 =

0) prop 120596


Σ

10158401015840

119888119888(k 120596 = 0 119879) prop 119879



120572

119911⟩ the



limkrarrQ

120594

(3)

(k 0) =120594

(3)(Q 0)

1 + 120585

2|k minusQ|

2 (78)

where 120594(3)(k 0) = minus119866














0(k 120596) and 120594






9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 07

T = 001

Σ998400998400(k 120596)

(a)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 078

T = 001

Σ998400998400(k 120596)

(b)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596A(k120596)

00

06

12

18

24

30

U = 8n = 085

T = 001

Σ998400998400(k 120596)

(c)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 092

T = 002

Σ998400998400(k 120596)

(d)










050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 07

T = 001

A(k120596)

Σ998400998400(k 120596)

120596

(a)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 078

T = 001

120596

(b)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 085

T = 001

120596

(c)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 092

T = 002

120596

(d)



(119879 = 002)








3(k 120596) (see Section 36)






00 20 40 60 80 100 120 120

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

5120587831205874

120587

120587

120587

120587

U = 8

T = 001

n = 070

A(k120596

=0)

31205878

31205878

31205878

51205878

51205878

51205878

31205874 31205874

31205874

31205874

71205878

71205878

71205878

71205878

000

0

12058781205878

4

120587444444444444444444444

1205874

1205874

1205872

1205871205871205871205871205871205871205871205871205871205872222222222222222222222222222222222222

1205872

1205872

55555555555555555555555555555555555555555555555555555555555555555552222212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444

120587

1205878

31205878

333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

51205878

51205878

51205878

31205874 31205874

3120587

33333333333333333333333331205871205871205871205871205871205871205871205871205874

71205878

71205878

7

74 1205878

kx

ky

(a)

00 20 40 60 80 100

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

3120587471205878

120587

120587

120587

120587

U = 8

T = 001

n = 078

kx

ky

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

3120587851205878

31205878

51205878

31205874

71205878

0

0

12058781205878

4

1205874 1205874

1205872

1205872

1205872

1205872

33333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444444444444444444447777777777777777777777777777777777777777777777777777777712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333333333333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

512058783120587

7

5120587831205874 71205878

3120587852212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888

1205878

51205878

31205874

71205878

(b)

00 10 20 30 40 50 60 70 80 90

00

0

0

123456789

1011

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

71205878

120587

120587

120587

120587

U = 8

T = 001n = 085

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

312058785120587831205874

00

000000000000000000000000000000000000000000

12058781205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

4

1205874 1205874

1205872

1205872

1205872

1205872

777777777777777777777777777777777777777777777777777777777771205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333331205878

512058783120587

7

3120587851205878

31205874 71205878

8

51205878

31205874

71205878

3120587852212058783333333333312058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058744444444444444444444444444444444444444444444444

kx

ky

(c)

00 10 20 30 40 50 60 70

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

120587

120587

120587

120587

U = 8

T = 002n = 092

A(k120596

=0)

31205878

5120587831205874

71205878

71205878

312058785120587831205874

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

0000

0

12058781205878

4

1205874

1205874

1205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874

1205872

1205872

1205872

1205872

120587

333331205878

512058783120587

7

77777777777777777777777120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587888888888888888888888888888888888888888

31205878522120587831205874

3120587851205878

31205874 71205878

1205878

51205878

31205874

71205878

kx

ky

(d)







119888119888(k 120596 = 0) (notice that



119888119888(k 120596 = 0) is quite


119888119888(k 120596 = 0) in those





00

25

50

75

100

125

r(k) = 0

n(k) = 05 U = 8

n = 070

T = 001

minus010 minus020

minus030

minus040minus050

Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

ky

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(a)

minus010


minus050

0

1205878

1205874

31205878

1205872

51205878

31205874

71205878

120587

0 1205878 1205874 31205878 1205872 51205878 31205874 71205878 120587

r(k) = 0

n(k) = 05

kx

k y

00

21

42

63

84

105

A(k120596

=0)

U = 8

n = 078

T = 001Σ998400998400cc(k 120596 = 0)

(b)

00

18

36

54

72

90

r(k) = 0

n(k) = 05 U = 8

n = 085

T = 001Σ998400998400cc(k 120596 = 0)

A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(c)

r(k) = 0

n(k) = 05 U = 8

n = 092

T = 002Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y00

14

28

42

56

70

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(d)








00

02

04

06

08

10

12

n = 070

n = 078n = 085

n = 092


120596

N(120596)

U = 8 T = 001 U = 8 T = 002

(a)

1

2

3

4

5

6

7

8


U = 8T = 002n = 092

A(k

=k120596)

120596

k = S

k = S

k = X

(b)









YX SS M

k

n(k)

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

n = 070

n = 075

n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

00

01

02

03

04

05

06

07

08

09

10

(a)

S M

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

kΓ

n = 070n = 075n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

(b)







and 119899 = 07 075 08 and 085 (119879 = 001) and for 119899 =





09

0807 06

05

0403

0202

00

01

02

03

04

05

06

07

08

09

10

n(k)

U = 8n = 092T = 001

Y

X

M

Γ














YXS M

U= 8[[

n = 092

T = 100

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrT= 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

(a)

YXS M

U = 01 [[

n = 092

T = 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrU = 7

(b)




















119911119899

120572






119911119899

120572

119911⟩ is






120596

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

minus08

minus06

minus04

minus02

00

02

04

06

08

(a)


120596

minus08

minus06

minus04

minus02

00

02

04

06

08

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

(b)

000 001 002 003 004 005minus012

minus010

minus008

minus006

minus004

minus002

000

T

Σ998400998400(k120596

=0)

n = 092

U = 8k = Sk = S

(c)

minus012

minus010

minus008

minus006

minus004

minus002

000

Σ998400998400(k120596

=0)

0000 0001 0002

T2

n = 092

U = 8k = Sk = S

(d)





guide to the eye






T

n

00 01 02 03 04 05 06 07 08 09 10minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009090 087 084 081 078 075 072

n = 092

U = 8

T = 001

⟨nzn120572 z⟩

8 7 6 5 4 3 2 1 0

U

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8T = 001

⟨nzn120572 z⟩


120572



0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

n

090 087 084 081 078 075 072

120585

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

78 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

120585

n = 092

U = 8

T = 001


00

02

04

06

08

10

12

14

16

00

02

04

06

08

10

12

14

16

n

090 087 084 081 078 075 072

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

00

04

08

12

16

20

24

28

32

00

04

08

12

16

20

24

28

32

8 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

120596(Q

)

120596(Q

)
















Acknowledgments


References












































































2minus119909Sr

119909CuO


























2CuO

6+120575









Na119909CuO

2Cl

2rdquo Science









2Sr

2minus119909LaxCuO






2Cu

4O

8rdquo Physical




119888



2Cu

3O

65rdquo








2Cu

3O

661and YBa

2Cu

3O

669



2Sr

2CaCu

2O




2Sr

2CaCu

2O





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








119888119888(k 120596) = F⟨119879[119888(119894)119888

dagger(119895)]⟩


3 Results


119860 (k 120596) = minus

1

120587

I [119866119888119888(k 120596)] (74)


119899 (k) = int119889120596119891F (120596)119860 (k 120596) (75)


119873(120596) =

1

(2120587)

2int119889

2119896119860 (k 120596) (76)

where 119866119888119888(k 120596) = 119866

11(k 120596) + 119866

12(k 120596) + 119866

21(k 120596) +




119866119888119888(k 120596) = 1

120596 minus 1205980(k) minus Σ

119888119888(k 120596)

(77)



1015840

119888119888(k 120596 = 0)


120596rarr0Σ

10158401015840

119888119888(k 120596 119879 =

0) prop 120596


Σ

10158401015840

119888119888(k 120596 = 0 119879) prop 119879



120572

119911⟩ the



limkrarrQ

120594

(3)

(k 0) =120594

(3)(Q 0)

1 + 120585

2|k minusQ|

2 (78)

where 120594(3)(k 0) = minus119866














0(k 120596) and 120594






9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 07

T = 001

Σ998400998400(k 120596)

(a)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 078

T = 001

Σ998400998400(k 120596)

(b)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596A(k120596)

00

06

12

18

24

30

U = 8n = 085

T = 001

Σ998400998400(k 120596)

(c)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 092

T = 002

Σ998400998400(k 120596)

(d)










050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 07

T = 001

A(k120596)

Σ998400998400(k 120596)

120596

(a)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 078

T = 001

120596

(b)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 085

T = 001

120596

(c)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 092

T = 002

120596

(d)



(119879 = 002)








3(k 120596) (see Section 36)






00 20 40 60 80 100 120 120

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

5120587831205874

120587

120587

120587

120587

U = 8

T = 001

n = 070

A(k120596

=0)

31205878

31205878

31205878

51205878

51205878

51205878

31205874 31205874

31205874

31205874

71205878

71205878

71205878

71205878

000

0

12058781205878

4

120587444444444444444444444

1205874

1205874

1205872

1205871205871205871205871205871205871205871205871205871205872222222222222222222222222222222222222

1205872

1205872

55555555555555555555555555555555555555555555555555555555555555555552222212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444

120587

1205878

31205878

333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

51205878

51205878

51205878

31205874 31205874

3120587

33333333333333333333333331205871205871205871205871205871205871205871205871205874

71205878

71205878

7

74 1205878

kx

ky

(a)

00 20 40 60 80 100

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

3120587471205878

120587

120587

120587

120587

U = 8

T = 001

n = 078

kx

ky

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

3120587851205878

31205878

51205878

31205874

71205878

0

0

12058781205878

4

1205874 1205874

1205872

1205872

1205872

1205872

33333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444444444444444444447777777777777777777777777777777777777777777777777777777712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333333333333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

512058783120587

7

5120587831205874 71205878

3120587852212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888

1205878

51205878

31205874

71205878

(b)

00 10 20 30 40 50 60 70 80 90

00

0

0

123456789

1011

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

71205878

120587

120587

120587

120587

U = 8

T = 001n = 085

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

312058785120587831205874

00

000000000000000000000000000000000000000000

12058781205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

4

1205874 1205874

1205872

1205872

1205872

1205872

777777777777777777777777777777777777777777777777777777777771205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333331205878

512058783120587

7

3120587851205878

31205874 71205878

8

51205878

31205874

71205878

3120587852212058783333333333312058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058744444444444444444444444444444444444444444444444

kx

ky

(c)

00 10 20 30 40 50 60 70

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

120587

120587

120587

120587

U = 8

T = 002n = 092

A(k120596

=0)

31205878

5120587831205874

71205878

71205878

312058785120587831205874

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

0000

0

12058781205878

4

1205874

1205874

1205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874

1205872

1205872

1205872

1205872

120587

333331205878

512058783120587

7

77777777777777777777777120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587888888888888888888888888888888888888888

31205878522120587831205874

3120587851205878

31205874 71205878

1205878

51205878

31205874

71205878

kx

ky

(d)







119888119888(k 120596 = 0) (notice that



119888119888(k 120596 = 0) is quite


119888119888(k 120596 = 0) in those





00

25

50

75

100

125

r(k) = 0

n(k) = 05 U = 8

n = 070

T = 001

minus010 minus020

minus030

minus040minus050

Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

ky

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(a)

minus010


minus050

0

1205878

1205874

31205878

1205872

51205878

31205874

71205878

120587

0 1205878 1205874 31205878 1205872 51205878 31205874 71205878 120587

r(k) = 0

n(k) = 05

kx

k y

00

21

42

63

84

105

A(k120596

=0)

U = 8

n = 078

T = 001Σ998400998400cc(k 120596 = 0)

(b)

00

18

36

54

72

90

r(k) = 0

n(k) = 05 U = 8

n = 085

T = 001Σ998400998400cc(k 120596 = 0)

A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(c)

r(k) = 0

n(k) = 05 U = 8

n = 092

T = 002Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y00

14

28

42

56

70

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(d)








00

02

04

06

08

10

12

n = 070

n = 078n = 085

n = 092


120596

N(120596)

U = 8 T = 001 U = 8 T = 002

(a)

1

2

3

4

5

6

7

8


U = 8T = 002n = 092

A(k

=k120596)

120596

k = S

k = S

k = X

(b)









YX SS M

k

n(k)

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

n = 070

n = 075

n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

00

01

02

03

04

05

06

07

08

09

10

(a)

S M

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

kΓ

n = 070n = 075n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

(b)







and 119899 = 07 075 08 and 085 (119879 = 001) and for 119899 =





09

0807 06

05

0403

0202

00

01

02

03

04

05

06

07

08

09

10

n(k)

U = 8n = 092T = 001

Y

X

M

Γ














YXS M

U= 8[[

n = 092

T = 100

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrT= 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

(a)

YXS M

U = 01 [[

n = 092

T = 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrU = 7

(b)




















119911119899

120572






119911119899

120572

119911⟩ is






120596

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

minus08

minus06

minus04

minus02

00

02

04

06

08

(a)


120596

minus08

minus06

minus04

minus02

00

02

04

06

08

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

(b)

000 001 002 003 004 005minus012

minus010

minus008

minus006

minus004

minus002

000

T

Σ998400998400(k120596

=0)

n = 092

U = 8k = Sk = S

(c)

minus012

minus010

minus008

minus006

minus004

minus002

000

Σ998400998400(k120596

=0)

0000 0001 0002

T2

n = 092

U = 8k = Sk = S

(d)





guide to the eye






T

n

00 01 02 03 04 05 06 07 08 09 10minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009090 087 084 081 078 075 072

n = 092

U = 8

T = 001

⟨nzn120572 z⟩

8 7 6 5 4 3 2 1 0

U

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8T = 001

⟨nzn120572 z⟩


120572



0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

n

090 087 084 081 078 075 072

120585

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

78 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

120585

n = 092

U = 8

T = 001


00

02

04

06

08

10

12

14

16

00

02

04

06

08

10

12

14

16

n

090 087 084 081 078 075 072

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

00

04

08

12

16

20

24

28

32

00

04

08

12

16

20

24

28

32

8 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

120596(Q

)

120596(Q

)
















Acknowledgments


References












































































2minus119909Sr

119909CuO


























2CuO

6+120575









Na119909CuO

2Cl

2rdquo Science









2Sr

2minus119909LaxCuO






2Cu

4O

8rdquo Physical




119888



2Cu

3O

65rdquo








2Cu

3O

661and YBa

2Cu

3O

669



2Sr

2CaCu

2O




2Sr

2CaCu

2O





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 07

T = 001

Σ998400998400(k 120596)

(a)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 078

T = 001

Σ998400998400(k 120596)

(b)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596A(k120596)

00

06

12

18

24

30

U = 8n = 085

T = 001

Σ998400998400(k 120596)

(c)

9876543210

minus1minus2minus3

Γ ΓS M X S Y

k

120596

A(k120596)

00

06

12

18

24

30

U = 8n = 092

T = 002

Σ998400998400(k 120596)

(d)










050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 07

T = 001

A(k120596)

Σ998400998400(k 120596)

120596

(a)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 078

T = 001

120596

(b)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 085

T = 001

120596

(c)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 092

T = 002

120596

(d)



(119879 = 002)








3(k 120596) (see Section 36)






00 20 40 60 80 100 120 120

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

5120587831205874

120587

120587

120587

120587

U = 8

T = 001

n = 070

A(k120596

=0)

31205878

31205878

31205878

51205878

51205878

51205878

31205874 31205874

31205874

31205874

71205878

71205878

71205878

71205878

000

0

12058781205878

4

120587444444444444444444444

1205874

1205874

1205872

1205871205871205871205871205871205871205871205871205871205872222222222222222222222222222222222222

1205872

1205872

55555555555555555555555555555555555555555555555555555555555555555552222212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444

120587

1205878

31205878

333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

51205878

51205878

51205878

31205874 31205874

3120587

33333333333333333333333331205871205871205871205871205871205871205871205871205874

71205878

71205878

7

74 1205878

kx

ky

(a)

00 20 40 60 80 100

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

3120587471205878

120587

120587

120587

120587

U = 8

T = 001

n = 078

kx

ky

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

3120587851205878

31205878

51205878

31205874

71205878

0

0

12058781205878

4

1205874 1205874

1205872

1205872

1205872

1205872

33333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444444444444444444447777777777777777777777777777777777777777777777777777777712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333333333333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

512058783120587

7

5120587831205874 71205878

3120587852212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888

1205878

51205878

31205874

71205878

(b)

00 10 20 30 40 50 60 70 80 90

00

0

0

123456789

1011

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

71205878

120587

120587

120587

120587

U = 8

T = 001n = 085

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

312058785120587831205874

00

000000000000000000000000000000000000000000

12058781205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

4

1205874 1205874

1205872

1205872

1205872

1205872

777777777777777777777777777777777777777777777777777777777771205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333331205878

512058783120587

7

3120587851205878

31205874 71205878

8

51205878

31205874

71205878

3120587852212058783333333333312058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058744444444444444444444444444444444444444444444444

kx

ky

(c)

00 10 20 30 40 50 60 70

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

120587

120587

120587

120587

U = 8

T = 002n = 092

A(k120596

=0)

31205878

5120587831205874

71205878

71205878

312058785120587831205874

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

0000

0

12058781205878

4

1205874

1205874

1205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874

1205872

1205872

1205872

1205872

120587

333331205878

512058783120587

7

77777777777777777777777120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587888888888888888888888888888888888888888

31205878522120587831205874

3120587851205878

31205874 71205878

1205878

51205878

31205874

71205878

kx

ky

(d)







119888119888(k 120596 = 0) (notice that



119888119888(k 120596 = 0) is quite


119888119888(k 120596 = 0) in those





00

25

50

75

100

125

r(k) = 0

n(k) = 05 U = 8

n = 070

T = 001

minus010 minus020

minus030

minus040minus050

Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

ky

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(a)

minus010


minus050

0

1205878

1205874

31205878

1205872

51205878

31205874

71205878

120587

0 1205878 1205874 31205878 1205872 51205878 31205874 71205878 120587

r(k) = 0

n(k) = 05

kx

k y

00

21

42

63

84

105

A(k120596

=0)

U = 8

n = 078

T = 001Σ998400998400cc(k 120596 = 0)

(b)

00

18

36

54

72

90

r(k) = 0

n(k) = 05 U = 8

n = 085

T = 001Σ998400998400cc(k 120596 = 0)

A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(c)

r(k) = 0

n(k) = 05 U = 8

n = 092

T = 002Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y00

14

28

42

56

70

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(d)








00

02

04

06

08

10

12

n = 070

n = 078n = 085

n = 092


120596

N(120596)

U = 8 T = 001 U = 8 T = 002

(a)

1

2

3

4

5

6

7

8


U = 8T = 002n = 092

A(k

=k120596)

120596

k = S

k = S

k = X

(b)









YX SS M

k

n(k)

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

n = 070

n = 075

n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

00

01

02

03

04

05

06

07

08

09

10

(a)

S M

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

kΓ

n = 070n = 075n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

(b)







and 119899 = 07 075 08 and 085 (119879 = 001) and for 119899 =





09

0807 06

05

0403

0202

00

01

02

03

04

05

06

07

08

09

10

n(k)

U = 8n = 092T = 001

Y

X

M

Γ














YXS M

U= 8[[

n = 092

T = 100

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrT= 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

(a)

YXS M

U = 01 [[

n = 092

T = 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrU = 7

(b)




















119911119899

120572






119911119899

120572

119911⟩ is






120596

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

minus08

minus06

minus04

minus02

00

02

04

06

08

(a)


120596

minus08

minus06

minus04

minus02

00

02

04

06

08

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

(b)

000 001 002 003 004 005minus012

minus010

minus008

minus006

minus004

minus002

000

T

Σ998400998400(k120596

=0)

n = 092

U = 8k = Sk = S

(c)

minus012

minus010

minus008

minus006

minus004

minus002

000

Σ998400998400(k120596

=0)

0000 0001 0002

T2

n = 092

U = 8k = Sk = S

(d)





guide to the eye






T

n

00 01 02 03 04 05 06 07 08 09 10minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009090 087 084 081 078 075 072

n = 092

U = 8

T = 001

⟨nzn120572 z⟩

8 7 6 5 4 3 2 1 0

U

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8T = 001

⟨nzn120572 z⟩


120572



0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

n

090 087 084 081 078 075 072

120585

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

78 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

120585

n = 092

U = 8

T = 001


00

02

04

06

08

10

12

14

16

00

02

04

06

08

10

12

14

16

n

090 087 084 081 078 075 072

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

00

04

08

12

16

20

24

28

32

00

04

08

12

16

20

24

28

32

8 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

120596(Q

)

120596(Q

)
















Acknowledgments


References












































































2minus119909Sr

119909CuO


























2CuO

6+120575









Na119909CuO

2Cl

2rdquo Science









2Sr

2minus119909LaxCuO






2Cu

4O

8rdquo Physical




119888



2Cu

3O

65rdquo








2Cu

3O

661and YBa

2Cu

3O

669



2Sr

2CaCu

2O




2Sr

2CaCu

2O





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 07

T = 001

A(k120596)

Σ998400998400(k 120596)

120596

(a)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 078

T = 001

120596

(b)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 085

T = 001

120596

(c)

A(k120596)

Σ998400998400(k 120596)

050

025

000

minus025

minus050

minus075

minus100

minus125


k

00

10

20

30

40

50

U = 8n = 092

T = 002

120596

(d)



(119879 = 002)








3(k 120596) (see Section 36)






00 20 40 60 80 100 120 120

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

5120587831205874

120587

120587

120587

120587

U = 8

T = 001

n = 070

A(k120596

=0)

31205878

31205878

31205878

51205878

51205878

51205878

31205874 31205874

31205874

31205874

71205878

71205878

71205878

71205878

000

0

12058781205878

4

120587444444444444444444444

1205874

1205874

1205872

1205871205871205871205871205871205871205871205871205871205872222222222222222222222222222222222222

1205872

1205872

55555555555555555555555555555555555555555555555555555555555555555552222212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444

120587

1205878

31205878

333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

51205878

51205878

51205878

31205874 31205874

3120587

33333333333333333333333331205871205871205871205871205871205871205871205871205874

71205878

71205878

7

74 1205878

kx

ky

(a)

00 20 40 60 80 100

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

3120587471205878

120587

120587

120587

120587

U = 8

T = 001

n = 078

kx

ky

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

3120587851205878

31205878

51205878

31205874

71205878

0

0

12058781205878

4

1205874 1205874

1205872

1205872

1205872

1205872

33333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444444444444444444447777777777777777777777777777777777777777777777777777777712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333333333333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

512058783120587

7

5120587831205874 71205878

3120587852212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888

1205878

51205878

31205874

71205878

(b)

00 10 20 30 40 50 60 70 80 90

00

0

0

123456789

1011

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

71205878

120587

120587

120587

120587

U = 8

T = 001n = 085

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

312058785120587831205874

00

000000000000000000000000000000000000000000

12058781205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

4

1205874 1205874

1205872

1205872

1205872

1205872

777777777777777777777777777777777777777777777777777777777771205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333331205878

512058783120587

7

3120587851205878

31205874 71205878

8

51205878

31205874

71205878

3120587852212058783333333333312058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058744444444444444444444444444444444444444444444444

kx

ky

(c)

00 10 20 30 40 50 60 70

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

120587

120587

120587

120587

U = 8

T = 002n = 092

A(k120596

=0)

31205878

5120587831205874

71205878

71205878

312058785120587831205874

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

0000

0

12058781205878

4

1205874

1205874

1205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874

1205872

1205872

1205872

1205872

120587

333331205878

512058783120587

7

77777777777777777777777120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587888888888888888888888888888888888888888

31205878522120587831205874

3120587851205878

31205874 71205878

1205878

51205878

31205874

71205878

kx

ky

(d)







119888119888(k 120596 = 0) (notice that



119888119888(k 120596 = 0) is quite


119888119888(k 120596 = 0) in those





00

25

50

75

100

125

r(k) = 0

n(k) = 05 U = 8

n = 070

T = 001

minus010 minus020

minus030

minus040minus050

Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

ky

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(a)

minus010


minus050

0

1205878

1205874

31205878

1205872

51205878

31205874

71205878

120587

0 1205878 1205874 31205878 1205872 51205878 31205874 71205878 120587

r(k) = 0

n(k) = 05

kx

k y

00

21

42

63

84

105

A(k120596

=0)

U = 8

n = 078

T = 001Σ998400998400cc(k 120596 = 0)

(b)

00

18

36

54

72

90

r(k) = 0

n(k) = 05 U = 8

n = 085

T = 001Σ998400998400cc(k 120596 = 0)

A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(c)

r(k) = 0

n(k) = 05 U = 8

n = 092

T = 002Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y00

14

28

42

56

70

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(d)








00

02

04

06

08

10

12

n = 070

n = 078n = 085

n = 092


120596

N(120596)

U = 8 T = 001 U = 8 T = 002

(a)

1

2

3

4

5

6

7

8


U = 8T = 002n = 092

A(k

=k120596)

120596

k = S

k = S

k = X

(b)









YX SS M

k

n(k)

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

n = 070

n = 075

n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

00

01

02

03

04

05

06

07

08

09

10

(a)

S M

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

kΓ

n = 070n = 075n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

(b)







and 119899 = 07 075 08 and 085 (119879 = 001) and for 119899 =





09

0807 06

05

0403

0202

00

01

02

03

04

05

06

07

08

09

10

n(k)

U = 8n = 092T = 001

Y

X

M

Γ














YXS M

U= 8[[

n = 092

T = 100

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrT= 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

(a)

YXS M

U = 01 [[

n = 092

T = 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrU = 7

(b)




















119911119899

120572






119911119899

120572

119911⟩ is






120596

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

minus08

minus06

minus04

minus02

00

02

04

06

08

(a)


120596

minus08

minus06

minus04

minus02

00

02

04

06

08

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

(b)

000 001 002 003 004 005minus012

minus010

minus008

minus006

minus004

minus002

000

T

Σ998400998400(k120596

=0)

n = 092

U = 8k = Sk = S

(c)

minus012

minus010

minus008

minus006

minus004

minus002

000

Σ998400998400(k120596

=0)

0000 0001 0002

T2

n = 092

U = 8k = Sk = S

(d)





guide to the eye






T

n

00 01 02 03 04 05 06 07 08 09 10minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009090 087 084 081 078 075 072

n = 092

U = 8

T = 001

⟨nzn120572 z⟩

8 7 6 5 4 3 2 1 0

U

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8T = 001

⟨nzn120572 z⟩


120572



0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

n

090 087 084 081 078 075 072

120585

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

78 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

120585

n = 092

U = 8

T = 001


00

02

04

06

08

10

12

14

16

00

02

04

06

08

10

12

14

16

n

090 087 084 081 078 075 072

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

00

04

08

12

16

20

24

28

32

00

04

08

12

16

20

24

28

32

8 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

120596(Q

)

120596(Q

)
















Acknowledgments


References












































































2minus119909Sr

119909CuO


























2CuO

6+120575









Na119909CuO

2Cl

2rdquo Science









2Sr

2minus119909LaxCuO






2Cu

4O

8rdquo Physical




119888



2Cu

3O

65rdquo








2Cu

3O

661and YBa

2Cu

3O

669



2Sr

2CaCu

2O




2Sr

2CaCu

2O





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




00 20 40 60 80 100 120 120

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

5120587831205874

120587

120587

120587

120587

U = 8

T = 001

n = 070

A(k120596

=0)

31205878

31205878

31205878

51205878

51205878

51205878

31205874 31205874

31205874

31205874

71205878

71205878

71205878

71205878

000

0

12058781205878

4

120587444444444444444444444

1205874

1205874

1205872

1205871205871205871205871205871205871205871205871205871205872222222222222222222222222222222222222

1205872

1205872

55555555555555555555555555555555555555555555555555555555555555555552222212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444

120587

1205878

31205878

333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

51205878

51205878

51205878

31205874 31205874

3120587

33333333333333333333333331205871205871205871205871205871205871205871205871205874

71205878

71205878

7

74 1205878

kx

ky

(a)

00 20 40 60 80 100

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

3120587471205878

120587

120587

120587

120587

U = 8

T = 001

n = 078

kx

ky

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

3120587851205878

31205878

51205878

31205874

71205878

0

0

12058781205878

4

1205874 1205874

1205872

1205872

1205872

1205872

33333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874444444444444444444444444444444444444444447777777777777777777777777777777777777777777777777777777712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333333333333333333333333333333333333333333331205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

512058783120587

7

5120587831205874 71205878

3120587852212058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058788888888888888888888888888888888888888

1205878

51205878

31205874

71205878

(b)

00 10 20 30 40 50 60 70 80 90

00

0

0

123456789

1011

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

71205878

120587

120587

120587

120587

U = 8

T = 001n = 085

A(k120596

=0)

31205878

5120587831205874

71205878

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

312058785120587831205874

00

000000000000000000000000000000000000000000

12058781205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878

4

1205874 1205874

1205872

1205872

1205872

1205872

777777777777777777777777777777777777777777777777777777777771205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205878888888888888888888888888120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587

3333333333333333333333331205878

512058783120587

7

3120587851205878

31205874 71205878

8

51205878

31205874

71205878

3120587852212058783333333333312058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058712058744444444444444444444444444444444444444444444444

kx

ky

(c)

00 10 20 30 40 50 60 70

00

0

0

123456789

10

11

12

1205878

1205878

1205878

1205878

1205874

1205874

1205874

1205874

1205872

1205872

1205872

1205872

120587

120587

120587

120587

U = 8

T = 002n = 092

A(k120596

=0)

31205878

5120587831205874

71205878

71205878

312058785120587831205874

3120587851205878

31205874 71205878

31205878

51205878

31205874

71205878

0000

0

12058781205878

4

1205874

1205874

1205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205871205874

1205872

1205872

1205872

1205872

120587

333331205878

512058783120587

7

77777777777777777777777120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587120587888888888888888888888888888888888888888

31205878522120587831205874

3120587851205878

31205874 71205878

1205878

51205878

31205874

71205878

kx

ky

(d)







119888119888(k 120596 = 0) (notice that



119888119888(k 120596 = 0) is quite


119888119888(k 120596 = 0) in those





00

25

50

75

100

125

r(k) = 0

n(k) = 05 U = 8

n = 070

T = 001

minus010 minus020

minus030

minus040minus050

Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

ky

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(a)

minus010


minus050

0

1205878

1205874

31205878

1205872

51205878

31205874

71205878

120587

0 1205878 1205874 31205878 1205872 51205878 31205874 71205878 120587

r(k) = 0

n(k) = 05

kx

k y

00

21

42

63

84

105

A(k120596

=0)

U = 8

n = 078

T = 001Σ998400998400cc(k 120596 = 0)

(b)

00

18

36

54

72

90

r(k) = 0

n(k) = 05 U = 8

n = 085

T = 001Σ998400998400cc(k 120596 = 0)

A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(c)

r(k) = 0

n(k) = 05 U = 8

n = 092

T = 002Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y00

14

28

42

56

70

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(d)








00

02

04

06

08

10

12

n = 070

n = 078n = 085

n = 092


120596

N(120596)

U = 8 T = 001 U = 8 T = 002

(a)

1

2

3

4

5

6

7

8


U = 8T = 002n = 092

A(k

=k120596)

120596

k = S

k = S

k = X

(b)









YX SS M

k

n(k)

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

n = 070

n = 075

n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

00

01

02

03

04

05

06

07

08

09

10

(a)

S M

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

kΓ

n = 070n = 075n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

(b)







and 119899 = 07 075 08 and 085 (119879 = 001) and for 119899 =





09

0807 06

05

0403

0202

00

01

02

03

04

05

06

07

08

09

10

n(k)

U = 8n = 092T = 001

Y

X

M

Γ














YXS M

U= 8[[

n = 092

T = 100

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrT= 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

(a)

YXS M

U = 01 [[

n = 092

T = 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrU = 7

(b)




















119911119899

120572






119911119899

120572

119911⟩ is






120596

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

minus08

minus06

minus04

minus02

00

02

04

06

08

(a)


120596

minus08

minus06

minus04

minus02

00

02

04

06

08

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

(b)

000 001 002 003 004 005minus012

minus010

minus008

minus006

minus004

minus002

000

T

Σ998400998400(k120596

=0)

n = 092

U = 8k = Sk = S

(c)

minus012

minus010

minus008

minus006

minus004

minus002

000

Σ998400998400(k120596

=0)

0000 0001 0002

T2

n = 092

U = 8k = Sk = S

(d)





guide to the eye






T

n

00 01 02 03 04 05 06 07 08 09 10minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009090 087 084 081 078 075 072

n = 092

U = 8

T = 001

⟨nzn120572 z⟩

8 7 6 5 4 3 2 1 0

U

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8T = 001

⟨nzn120572 z⟩


120572



0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

n

090 087 084 081 078 075 072

120585

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

78 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

120585

n = 092

U = 8

T = 001


00

02

04

06

08

10

12

14

16

00

02

04

06

08

10

12

14

16

n

090 087 084 081 078 075 072

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

00

04

08

12

16

20

24

28

32

00

04

08

12

16

20

24

28

32

8 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

120596(Q

)

120596(Q

)
















Acknowledgments


References












































































2minus119909Sr

119909CuO


























2CuO

6+120575









Na119909CuO

2Cl

2rdquo Science









2Sr

2minus119909LaxCuO






2Cu

4O

8rdquo Physical




119888



2Cu

3O

65rdquo








2Cu

3O

661and YBa

2Cu

3O

669



2Sr

2CaCu

2O




2Sr

2CaCu

2O





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








119888119888(k 120596 = 0) (notice that



119888119888(k 120596 = 0) is quite


119888119888(k 120596 = 0) in those





00

25

50

75

100

125

r(k) = 0

n(k) = 05 U = 8

n = 070

T = 001

minus010 minus020

minus030

minus040minus050

Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

ky

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(a)

minus010


minus050

0

1205878

1205874

31205878

1205872

51205878

31205874

71205878

120587

0 1205878 1205874 31205878 1205872 51205878 31205874 71205878 120587

r(k) = 0

n(k) = 05

kx

k y

00

21

42

63

84

105

A(k120596

=0)

U = 8

n = 078

T = 001Σ998400998400cc(k 120596 = 0)

(b)

00

18

36

54

72

90

r(k) = 0

n(k) = 05 U = 8

n = 085

T = 001Σ998400998400cc(k 120596 = 0)

A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(c)

r(k) = 0

n(k) = 05 U = 8

n = 092

T = 002Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y00

14

28

42

56

70

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(d)








00

02

04

06

08

10

12

n = 070

n = 078n = 085

n = 092


120596

N(120596)

U = 8 T = 001 U = 8 T = 002

(a)

1

2

3

4

5

6

7

8


U = 8T = 002n = 092

A(k

=k120596)

120596

k = S

k = S

k = X

(b)









YX SS M

k

n(k)

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

n = 070

n = 075

n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

00

01

02

03

04

05

06

07

08

09

10

(a)

S M

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

kΓ

n = 070n = 075n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

(b)







and 119899 = 07 075 08 and 085 (119879 = 001) and for 119899 =





09

0807 06

05

0403

0202

00

01

02

03

04

05

06

07

08

09

10

n(k)

U = 8n = 092T = 001

Y

X

M

Γ














YXS M

U= 8[[

n = 092

T = 100

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrT= 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

(a)

YXS M

U = 01 [[

n = 092

T = 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrU = 7

(b)




















119911119899

120572






119911119899

120572

119911⟩ is






120596

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

minus08

minus06

minus04

minus02

00

02

04

06

08

(a)


120596

minus08

minus06

minus04

minus02

00

02

04

06

08

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

(b)

000 001 002 003 004 005minus012

minus010

minus008

minus006

minus004

minus002

000

T

Σ998400998400(k120596

=0)

n = 092

U = 8k = Sk = S

(c)

minus012

minus010

minus008

minus006

minus004

minus002

000

Σ998400998400(k120596

=0)

0000 0001 0002

T2

n = 092

U = 8k = Sk = S

(d)





guide to the eye






T

n

00 01 02 03 04 05 06 07 08 09 10minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009090 087 084 081 078 075 072

n = 092

U = 8

T = 001

⟨nzn120572 z⟩

8 7 6 5 4 3 2 1 0

U

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8T = 001

⟨nzn120572 z⟩


120572



0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

n

090 087 084 081 078 075 072

120585

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

78 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

120585

n = 092

U = 8

T = 001


00

02

04

06

08

10

12

14

16

00

02

04

06

08

10

12

14

16

n

090 087 084 081 078 075 072

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

00

04

08

12

16

20

24

28

32

00

04

08

12

16

20

24

28

32

8 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

120596(Q

)

120596(Q

)
















Acknowledgments


References












































































2minus119909Sr

119909CuO


























2CuO

6+120575









Na119909CuO

2Cl

2rdquo Science









2Sr

2minus119909LaxCuO






2Cu

4O

8rdquo Physical




119888



2Cu

3O

65rdquo








2Cu

3O

661and YBa

2Cu

3O

669



2Sr

2CaCu

2O




2Sr

2CaCu

2O





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




00

25

50

75

100

125

r(k) = 0

n(k) = 05 U = 8

n = 070

T = 001

minus010 minus020

minus030

minus040minus050

Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

ky

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(a)

minus010


minus050

0

1205878

1205874

31205878

1205872

51205878

31205874

71205878

120587

0 1205878 1205874 31205878 1205872 51205878 31205874 71205878 120587

r(k) = 0

n(k) = 05

kx

k y

00

21

42

63

84

105

A(k120596

=0)

U = 8

n = 078

T = 001Σ998400998400cc(k 120596 = 0)

(b)

00

18

36

54

72

90

r(k) = 0

n(k) = 05 U = 8

n = 085

T = 001Σ998400998400cc(k 120596 = 0)

A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(c)

r(k) = 0

n(k) = 05 U = 8

n = 092

T = 002Σ998400998400cc(k 120596 = 0)A(k120596

=0)

00

1205878

1205878

1205874

1205874

1205872

1205872

120587

120587

kx

k y00

14

28

42

56

70

minus010

minus020

minus030minus040

minus050

31205878

51205878

31205874

71205878

31205878 51205878 31205874 71205878

(d)








00

02

04

06

08

10

12

n = 070

n = 078n = 085

n = 092


120596

N(120596)

U = 8 T = 001 U = 8 T = 002

(a)

1

2

3

4

5

6

7

8


U = 8T = 002n = 092

A(k

=k120596)

120596

k = S

k = S

k = X

(b)









YX SS M

k

n(k)

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

n = 070

n = 075

n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

00

01

02

03

04

05

06

07

08

09

10

(a)

S M

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

kΓ

n = 070n = 075n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

(b)







and 119899 = 07 075 08 and 085 (119879 = 001) and for 119899 =





09

0807 06

05

0403

0202

00

01

02

03

04

05

06

07

08

09

10

n(k)

U = 8n = 092T = 001

Y

X

M

Γ














YXS M

U= 8[[

n = 092

T = 100

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrT= 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

(a)

YXS M

U = 01 [[

n = 092

T = 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrU = 7

(b)




















119911119899

120572






119911119899

120572

119911⟩ is






120596

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

minus08

minus06

minus04

minus02

00

02

04

06

08

(a)


120596

minus08

minus06

minus04

minus02

00

02

04

06

08

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

(b)

000 001 002 003 004 005minus012

minus010

minus008

minus006

minus004

minus002

000

T

Σ998400998400(k120596

=0)

n = 092

U = 8k = Sk = S

(c)

minus012

minus010

minus008

minus006

minus004

minus002

000

Σ998400998400(k120596

=0)

0000 0001 0002

T2

n = 092

U = 8k = Sk = S

(d)





guide to the eye






T

n

00 01 02 03 04 05 06 07 08 09 10minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009090 087 084 081 078 075 072

n = 092

U = 8

T = 001

⟨nzn120572 z⟩

8 7 6 5 4 3 2 1 0

U

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8T = 001

⟨nzn120572 z⟩


120572



0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

n

090 087 084 081 078 075 072

120585

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

78 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

120585

n = 092

U = 8

T = 001


00

02

04

06

08

10

12

14

16

00

02

04

06

08

10

12

14

16

n

090 087 084 081 078 075 072

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

00

04

08

12

16

20

24

28

32

00

04

08

12

16

20

24

28

32

8 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

120596(Q

)

120596(Q

)
















Acknowledgments


References












































































2minus119909Sr

119909CuO


























2CuO

6+120575









Na119909CuO

2Cl

2rdquo Science









2Sr

2minus119909LaxCuO






2Cu

4O

8rdquo Physical




119888



2Cu

3O

65rdquo








2Cu

3O

661and YBa

2Cu

3O

669



2Sr

2CaCu

2O




2Sr

2CaCu

2O





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




00

02

04

06

08

10

12

n = 070

n = 078n = 085

n = 092


120596

N(120596)

U = 8 T = 001 U = 8 T = 002

(a)

1

2

3

4

5

6

7

8


U = 8T = 002n = 092

A(k

=k120596)

120596

k = S

k = S

k = X

(b)









YX SS M

k

n(k)

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

n = 070

n = 075

n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

00

01

02

03

04

05

06

07

08

09

10

(a)

S M

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

kΓ

n = 070n = 075n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

(b)







and 119899 = 07 075 08 and 085 (119879 = 001) and for 119899 =





09

0807 06

05

0403

0202

00

01

02

03

04

05

06

07

08

09

10

n(k)

U = 8n = 092T = 001

Y

X

M

Γ














YXS M

U= 8[[

n = 092

T = 100

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrT= 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

(a)

YXS M

U = 01 [[

n = 092

T = 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrU = 7

(b)




















119911119899

120572






119911119899

120572

119911⟩ is






120596

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

minus08

minus06

minus04

minus02

00

02

04

06

08

(a)


120596

minus08

minus06

minus04

minus02

00

02

04

06

08

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

(b)

000 001 002 003 004 005minus012

minus010

minus008

minus006

minus004

minus002

000

T

Σ998400998400(k120596

=0)

n = 092

U = 8k = Sk = S

(c)

minus012

minus010

minus008

minus006

minus004

minus002

000

Σ998400998400(k120596

=0)

0000 0001 0002

T2

n = 092

U = 8k = Sk = S

(d)





guide to the eye






T

n

00 01 02 03 04 05 06 07 08 09 10minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009090 087 084 081 078 075 072

n = 092

U = 8

T = 001

⟨nzn120572 z⟩

8 7 6 5 4 3 2 1 0

U

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8T = 001

⟨nzn120572 z⟩


120572



0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

n

090 087 084 081 078 075 072

120585

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

78 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

120585

n = 092

U = 8

T = 001


00

02

04

06

08

10

12

14

16

00

02

04

06

08

10

12

14

16

n

090 087 084 081 078 075 072

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

00

04

08

12

16

20

24

28

32

00

04

08

12

16

20

24

28

32

8 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

120596(Q

)

120596(Q

)
















Acknowledgments


References












































































2minus119909Sr

119909CuO


























2CuO

6+120575









Na119909CuO

2Cl

2rdquo Science









2Sr

2minus119909LaxCuO






2Cu

4O

8rdquo Physical




119888



2Cu

3O

65rdquo








2Cu

3O

661and YBa

2Cu

3O

669



2Sr

2CaCu

2O




2Sr

2CaCu

2O





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




YX SS M

k

n(k)

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

n = 070

n = 075

n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

00

01

02

03

04

05

06

07

08

09

10

(a)

S M

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

kΓ

n = 070n = 075n = 080

n = 085

n = 092U = 8 T = 001 U = 8 T = 002

(b)







and 119899 = 07 075 08 and 085 (119879 = 001) and for 119899 =





09

0807 06

05

0403

0202

00

01

02

03

04

05

06

07

08

09

10

n(k)

U = 8n = 092T = 001

Y

X

M

Γ














YXS M

U= 8[[

n = 092

T = 100

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrT= 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

(a)

YXS M

U = 01 [[

n = 092

T = 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrU = 7

(b)




















119911119899

120572






119911119899

120572

119911⟩ is






120596

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

minus08

minus06

minus04

minus02

00

02

04

06

08

(a)


120596

minus08

minus06

minus04

minus02

00

02

04

06

08

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

(b)

000 001 002 003 004 005minus012

minus010

minus008

minus006

minus004

minus002

000

T

Σ998400998400(k120596

=0)

n = 092

U = 8k = Sk = S

(c)

minus012

minus010

minus008

minus006

minus004

minus002

000

Σ998400998400(k120596

=0)

0000 0001 0002

T2

n = 092

U = 8k = Sk = S

(d)





guide to the eye






T

n

00 01 02 03 04 05 06 07 08 09 10minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009090 087 084 081 078 075 072

n = 092

U = 8

T = 001

⟨nzn120572 z⟩

8 7 6 5 4 3 2 1 0

U

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8T = 001

⟨nzn120572 z⟩


120572



0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

n

090 087 084 081 078 075 072

120585

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

78 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

120585

n = 092

U = 8

T = 001


00

02

04

06

08

10

12

14

16

00

02

04

06

08

10

12

14

16

n

090 087 084 081 078 075 072

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

00

04

08

12

16

20

24

28

32

00

04

08

12

16

20

24

28

32

8 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

120596(Q

)

120596(Q

)
















Acknowledgments


References












































































2minus119909Sr

119909CuO


























2CuO

6+120575









Na119909CuO

2Cl

2rdquo Science









2Sr

2minus119909LaxCuO






2Cu

4O

8rdquo Physical




119888



2Cu

3O

65rdquo








2Cu

3O

661and YBa

2Cu

3O

669



2Sr

2CaCu

2O




2Sr

2CaCu

2O





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




09

0807 06

05

0403

0202

00

01

02

03

04

05

06

07

08

09

10

n(k)

U = 8n = 092T = 001

Y

X

M

Γ














YXS M

U= 8[[

n = 092

T = 100

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrT= 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

(a)

YXS M

U = 01 [[

n = 092

T = 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrU = 7

(b)




















119911119899

120572






119911119899

120572

119911⟩ is






120596

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

minus08

minus06

minus04

minus02

00

02

04

06

08

(a)


120596

minus08

minus06

minus04

minus02

00

02

04

06

08

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

(b)

000 001 002 003 004 005minus012

minus010

minus008

minus006

minus004

minus002

000

T

Σ998400998400(k120596

=0)

n = 092

U = 8k = Sk = S

(c)

minus012

minus010

minus008

minus006

minus004

minus002

000

Σ998400998400(k120596

=0)

0000 0001 0002

T2

n = 092

U = 8k = Sk = S

(d)





guide to the eye






T

n

00 01 02 03 04 05 06 07 08 09 10minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009090 087 084 081 078 075 072

n = 092

U = 8

T = 001

⟨nzn120572 z⟩

8 7 6 5 4 3 2 1 0

U

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8T = 001

⟨nzn120572 z⟩


120572



0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

n

090 087 084 081 078 075 072

120585

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

78 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

120585

n = 092

U = 8

T = 001


00

02

04

06

08

10

12

14

16

00

02

04

06

08

10

12

14

16

n

090 087 084 081 078 075 072

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

00

04

08

12

16

20

24

28

32

00

04

08

12

16

20

24

28

32

8 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

120596(Q

)

120596(Q

)
















Acknowledgments


References












































































2minus119909Sr

119909CuO


























2CuO

6+120575









Na119909CuO

2Cl

2rdquo Science









2Sr

2minus119909LaxCuO






2Cu

4O

8rdquo Physical




119888



2Cu

3O

65rdquo








2Cu

3O

661and YBa

2Cu

3O

669



2Sr

2CaCu

2O




2Sr

2CaCu

2O





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




YXS M

U= 8[[

n = 092

T = 100

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrT= 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

(a)

YXS M

U = 01 [[

n = 092

T = 001

n(k)

00

01

02

03

04

05

06

07

08

09

10

00

01

02

03

04

05

06

07

08

09

10

Γ Γ

] ]rarrU = 7

(b)




















119911119899

120572






119911119899

120572

119911⟩ is






120596

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

minus08

minus06

minus04

minus02

00

02

04

06

08

(a)


120596

minus08

minus06

minus04

minus02

00

02

04

06

08

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

(b)

000 001 002 003 004 005minus012

minus010

minus008

minus006

minus004

minus002

000

T

Σ998400998400(k120596

=0)

n = 092

U = 8k = Sk = S

(c)

minus012

minus010

minus008

minus006

minus004

minus002

000

Σ998400998400(k120596

=0)

0000 0001 0002

T2

n = 092

U = 8k = Sk = S

(d)





guide to the eye






T

n

00 01 02 03 04 05 06 07 08 09 10minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009090 087 084 081 078 075 072

n = 092

U = 8

T = 001

⟨nzn120572 z⟩

8 7 6 5 4 3 2 1 0

U

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8T = 001

⟨nzn120572 z⟩


120572



0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

n

090 087 084 081 078 075 072

120585

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

78 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

120585

n = 092

U = 8

T = 001


00

02

04

06

08

10

12

14

16

00

02

04

06

08

10

12

14

16

n

090 087 084 081 078 075 072

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

00

04

08

12

16

20

24

28

32

00

04

08

12

16

20

24

28

32

8 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

120596(Q

)

120596(Q

)
















Acknowledgments


References












































































2minus119909Sr

119909CuO


























2CuO

6+120575









Na119909CuO

2Cl

2rdquo Science









2Sr

2minus119909LaxCuO






2Cu

4O

8rdquo Physical




119888



2Cu

3O

65rdquo








2Cu

3O

661and YBa

2Cu

3O

669



2Sr

2CaCu

2O




2Sr

2CaCu

2O





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





120596

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

minus08

minus06

minus04

minus02

00

02

04

06

08

(a)


120596

minus08

minus06

minus04

minus02

00

02

04

06

08

A(k 120596)Σ998400(k 120596)Σ998400998400(k 120596)1205760(k 120596)

n = 092T = 002U = 8k = S

(b)

000 001 002 003 004 005minus012

minus010

minus008

minus006

minus004

minus002

000

T

Σ998400998400(k120596

=0)

n = 092

U = 8k = Sk = S

(c)

minus012

minus010

minus008

minus006

minus004

minus002

000

Σ998400998400(k120596

=0)

0000 0001 0002

T2

n = 092

U = 8k = Sk = S

(d)





guide to the eye






T

n

00 01 02 03 04 05 06 07 08 09 10minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009090 087 084 081 078 075 072

n = 092

U = 8

T = 001

⟨nzn120572 z⟩

8 7 6 5 4 3 2 1 0

U

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8T = 001

⟨nzn120572 z⟩


120572



0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

n

090 087 084 081 078 075 072

120585

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

78 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

120585

n = 092

U = 8

T = 001


00

02

04

06

08

10

12

14

16

00

02

04

06

08

10

12

14

16

n

090 087 084 081 078 075 072

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

00

04

08

12

16

20

24

28

32

00

04

08

12

16

20

24

28

32

8 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

120596(Q

)

120596(Q

)
















Acknowledgments


References












































































2minus119909Sr

119909CuO


























2CuO

6+120575









Na119909CuO

2Cl

2rdquo Science









2Sr

2minus119909LaxCuO






2Cu

4O

8rdquo Physical




119888



2Cu

3O

65rdquo








2Cu

3O

661and YBa

2Cu

3O

669



2Sr

2CaCu

2O




2Sr

2CaCu

2O





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




T

n

00 01 02 03 04 05 06 07 08 09 10minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009090 087 084 081 078 075 072

n = 092

U = 8

T = 001

⟨nzn120572 z⟩

8 7 6 5 4 3 2 1 0

U

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

minus033

minus030

minus027

minus024

minus021

minus018

minus015

minus012

minus009

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8T = 001

⟨nzn120572 z⟩


120572



0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

n

090 087 084 081 078 075 072

120585

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

78 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

120585

n = 092

U = 8

T = 001


00

02

04

06

08

10

12

14

16

00

02

04

06

08

10

12

14

16

n

090 087 084 081 078 075 072

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

00

04

08

12

16

20

24

28

32

00

04

08

12

16

20

24

28

32

8 7 6 5 4 3 2 1 0

U

T

00 01 02 03 04 05 06 07 08 09 10

n = 092

U = 8

T = 001

120596(Q

)

120596(Q

)
















Acknowledgments


References












































































2minus119909Sr

119909CuO


























2CuO

6+120575









Na119909CuO

2Cl

2rdquo Science









2Sr

2minus119909LaxCuO






2Cu

4O

8rdquo Physical




119888



2Cu

3O

65rdquo








2Cu

3O

661and YBa

2Cu

3O

669



2Sr

2CaCu

2O




2Sr

2CaCu

2O





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics
















Acknowledgments


References












































































2minus119909Sr

119909CuO


























2CuO

6+120575









Na119909CuO

2Cl

2rdquo Science









2Sr

2minus119909LaxCuO






2Cu

4O

8rdquo Physical




119888



2Cu

3O

65rdquo








2Cu

3O

661and YBa

2Cu

3O

669



2Sr

2CaCu

2O




2Sr

2CaCu

2O





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics









































































2minus119909Sr

119909CuO


























2CuO

6+120575









Na119909CuO

2Cl

2rdquo Science









2Sr

2minus119909LaxCuO






2Cu

4O

8rdquo Physical




119888



2Cu

3O

65rdquo








2Cu

3O

661and YBa

2Cu

3O

669



2Sr

2CaCu

2O




2Sr

2CaCu

2O





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





































2minus119909Sr

119909CuO


























2CuO

6+120575









Na119909CuO

2Cl

2rdquo Science









2Sr

2minus119909LaxCuO






2Cu

4O

8rdquo Physical




119888



2Cu

3O

65rdquo








2Cu

3O

661and YBa

2Cu

3O

669



2Sr

2CaCu

2O




2Sr

2CaCu

2O





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





















2CuO

6+120575









Na119909CuO

2Cl

2rdquo Science









2Sr

2minus119909LaxCuO






2Cu

4O

8rdquo Physical




119888



2Cu

3O

65rdquo








2Cu

3O

661and YBa

2Cu

3O

669



2Sr

2CaCu

2O




2Sr

2CaCu

2O





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





2Cu

3O

659shows free-spin



119888supercon-



2Cu

3

O6+119909








2Cu

3O

6+119909rdquo




2Cu

3O








2Cu

3O

65rdquo Physical


magnetic YBa2Cu

3O











2Cu

3O

6+119909rdquo



2Cu

3O










2Cu

3O

65rdquo









2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





2Cu

3O




3O

7minus120575charge



2Sr

2CaCu

2O




Na119909CuO

2Cl

2rdquo



2Sr

2Ca Cu

2O

8+120575rdquo



2Cu

3O




2Cu

3O






3O

6+119909rdquo



3O



YBa2Cu

3O




Cu3O






4











2Ca

2SrCu

2O

8 results of

























2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics














2CuO




178Sr

022CuO







2minus119909Sr

119909CuO

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Documents

Review Article Composite Operator Method Analysis of …downloads.hindawi.com/journals/acmp/2014/515698.pdf · Review Article Composite Operator Method Analysis of ... that strong