Spin waves and magnons

Consider an almost perfectly ordered ferromagnet at low temperatures T<<Tc

x

y

z

nr

nrAt magnetic moment characterized by spin: , ,x y z

n n n nS S S S

Interaction between spins and magnetic field given by Hamiltonian

0( , )

n m nBn m n

H J S S g H S

J

Spin waves:Thermal properties of ferromagnet determined at T<<Tc by low energy excitations, quantized spin waves

Strategy similar to photons and phononsPhonons: classical dynamical problem provides correct eigenfrequencies of eigenmodes

classical spin wave dynamical problem provides correct eigenfrequencies of eigenmodes With the dispersion relation for spin waves

Thermadynamics using quantized spin waves: magnons

0: Bh g H

Derivation of spin waves in the classical limitFor simplicity let’s consider classical Heisenberg spin chain

JClassical spin vectors S of length

S S

nS 1nS 1nS

J

S

Ground state : all spins parallel with energy

20E NJS NhS

Deviations from ground state are spin wave excitations which can be pictured as

Torque changes angular momentum

Deriving the spin wave dispersion relation

Spin is an angular momentum

Classical mechanics dLT

dt

Here: nAn

dST S H

dt

1 1A n nH J S S Exchange field, exchange interaction with neighbors can effectively be considered as a magnetic field acting on spin at position n

1 1n n nS J S S nS 1nS 1nS

J J

1 1n

n n n n

dSJ S S S S

dt

1 1 1 1 1 1

xn

x y z x y zy

x y z x y znn n n n n n

x y z x y zz n n n n n nn

dS

dt e e e e e edS

J S S S S S Sdt

S S S S S SdS

dt

Takes care of the fact that spins are at discrete lattice positions xn=n a

Let’s write down the x-component the rest follows from cyclic permutation(be careful with the signs though!)

1 1 1 1 1 1

xn

x y z x y zy

x y z x y znn n n n n n

x y z x y zz n n n n n nn

dS

dt e e e e e edS

J S S S S S Sdt

S S S S S SdS

dt

1 1 1 1 1 1 1 1

xy z z y y z z y y z z z y ynn n n n n n n n n n n n n n

dSJ S S S S S S S S J S S S S S S

dt

We consider excitations with small amplitude

,,z x yn nS S S S

1 1 1 12 2x

y y y y y ynn n n n n n

dSJ S S S S S JS S S S

dt

1 12y

x x xnn n n

dSJS S S S

dt

0zndS

dt

Solution with plane wave ansatz:

i nka txn

i nka tyn

S uSe

S vSe

With

xi nka tn

yi nka tn

dSi uSe

dt

dSi vSe

dt

into1 12

xy y ynn n n

dSJS S S S

dt

1 12y

x x xnn n n

dSJS S S S

dt

i nka txn

i nka tyn

S uSe

S vSe

and

2

2

i nka t i nka t i nka t i nka tika ika

i nka t i nka t i nka t i nka tika ika

i uSe JS vSe vSe e vSe e

i vSe JS uSe uSe e uSe e

2 1 cos

2 1 cos

i u vJS ka

i v uJS ka

2 1 cos

02 1 cos

i JS ka u

JS ka i v

Non-trivial solution meaning other than u=v=0 for:

2 1 cos0

2 1 cos

i JS ka

JS ka i

2 1 cosJS ka

Magnon dispersion relation

Thermodynamics of magnons

Calculation of the internal energy:

1

2k kk

E n

1

2k kk

U E n

01k

k

k

Ee

in complete analogy to the photons and phonons

33

... ...(2 )k

Vd k

We consider the limit T->0:

Only low energy magnons near k=0 excited

2 22 1 cosJS ka JSa k

With

2 2

2 22

0 34

(2 ) 1JSa k

V JSa kU E k dk

e

With2

B B

JSa Dx k k

k T k T

and hence

B

Ddx dk

k T

2

2

24

0 3

5/ 23/ 2 40 2

4(2 ) 1

2 1

B B

x

B x

V k T D k TU E x dx

D De

V dxE D k T x

e

Just a number which becomes with integration to infinity

2

4

0

3 3(5 / 2) 1.3419

8 81x

dxxe

3/ 2

BV

V

U k TC

T D

Exponent different than for phonons due to difference in dispersion

1

1( )

sk

sk

Z. Physik 61, 206 (1930):

0 ( ) kk

U E k n

The internal energy

can alternatively be expressed as

2 2

0

1( )2

x yk k

k

U E k S SS

where nik rn k

k

S S e2 21

2x yk k kS S n

S

Intuitive/hand-waving interpretation:# of excitations in a mode <-> average of classical amplitude squared

2 2 21 1

2 2E m x m x

2 2 2 2 21 1

2 2E m x m x m x n

2x n

Magnetization and its deviation from full alignment in z-direction is determined as ( ) zB

nn

gM T S

V

2 22 x yB

n nn

gS S S

V

Magnetization and the celebrated T3/2 Bloch law:

Let’s closer inspect

2 22 x yn nS S S and

remember 2 2 2x y

n nS S S for T->0

2 2 2 2

2 222 2

1 12

x y x yn n n nx y

n n

S S S SS S S S S

S S

2 2 2 2

2( ) 1

2 2

x y x yn n n nB B

n n

S S S Sg gM T S NS

V S V S

with nik rn k

k

S S e

2 21( )

2x yBk k

k

gM T NS S S

V S

using( )

,ni k k r

k kn

e and k kS S

( ) Bk

k

gM T NS n

V

Intuitive interpretation:Excitation of spin waves (magnons) means spins point on average less in z-direction -> magnetization goes down

Now let’s calculate M(T) with magnon dispersion at T->0

( ) Bk

k

gM T NS n

V

3

3... ...

(2 )k

Vd k

2 2JSa k with and

2 2

23

( ) 4(2 ) 1

B

JSa k

g V dkM T NS k

V e

Again with2

B B

JSa Dx k k

k T k T

and hence

B

Ddx dk

k T

2

3/ 22

2( )

2 1B B

x

g V k T dxM T NS x

V D e

3/ 2

2

(3 / 2)( ) ( 0) 1

2 4BV k T

M T M TNS D

Bg NS

V

Felix Bloch(1905 - 1983)Nobel Prize in 1952 for NMR

Modern research example:Bloch’s T3/2-law widely applicable also in exotic systems

Spin waves and phase transitions: Goldstone excitations A stability analysis against long wavelength fluctuations gives hints for the possible existence of a long range ordered phase

( , )n m

n m

H J S S Heisenberg Hamiltonian example for continuous rotational symmetry which can be spontaneously broken depending on the dimension, d

d=1

d=2Let’s have a look to spin wave approach for

( 0) ( )M M T M T in various spatial dimensions d

( ) Bk

k

gM T NS n

V

... ...

(2 )dk

ddLd k

From

and

1

2

ddk kM

k

2 2 2 2

01JSa k

ke JSa k

When a continuous symmetry is broken there must exist a Goldstone mode (boson) with 0 for k0

In low dimensions d=1 and d=2 integral diverges at the lower bound k=0

Unphysical result indicates absence of orderedlow temperature phase in d=1 and d=2