### Spin waves and magnons

```Spin waves and magnons
Consider an almost perfectly ordered ferromagnet at low temperatures T<<Tc
At
z
rn
y
x
J
rn
magnetic moment characterized by spin:
S n   S nx , S ny , S nz 
Interaction between spins and magnetic field given by Hamiltonian
H   J  S n S m  g B 0 H  S n
( n,m)
Spin waves:
n
Thermal properties of ferromagnet determined at T<<Tc by low
energy excitations, quantized spin waves
Strategy similar to photons and phonons
Phonons: 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
Derivation of spin waves in the classical limit
For simplicity let’s consider classical Heisenberg spin chain
S n1 S n S n1
J
J
Classical spin vectors S of length
S S
Ground state : all spins parallel with energy
h : g B 0 H
E0   NJS 2  NhS
Deviations from ground state are spin wave excitations which can be pictured as
S
Deriving the spin wave dispersion relation
Spin is an angular momentum
Classical mechanics
Here:
dL
T
dt
Torque changes angular momentum
dSn
 T  S n  H A  S n  J  S n1  S n1 
dt
H A  J  S n1  S n1 
S n1 S n S n1
J
J
Exchange field, exchange interaction with neighbors can effectively be considered
as a magnetic field acting on spin at position n
dSn
 J  S n  S n 1  S n  S n 1 
dt
 dSnx 


dt
 ex e y ez
ex e y ez



 x

 dSny 
y
z
x
y
z
 Sn Sn Sn

 dt   J  Sn Sn Sn
 x
y
z
x
y
z 
 z
S
S
S
S
S
S
n 1
n 1
n 1
n 1
n 1 
 n 1

 dSn 
 dt 


 dSnx

 dt
 dSny
 dt
 z
 dSn
 dt



 ex e y ez
ex e y ez


 x


y
z
x
y
z

J
S
S
S

S
S
S
 n

n
n
n
n
n


x
y
z
x
y
z 

S
S
S
S
S
S
n

1
n

1
n

1
n

1
n

1
n
1 






Let’s write down the x-component
the rest follows from cyclic permutation
(be careful with the signs though!)
dSnx
 J  Sny Snz1  Snz Sny1  Sny Snz1  Snz Sny1   J  Sny  Snz1  Snz1   Snz  Sny1  Sny1  
dt
We consider excitations with small amplitude
Snz  S , Snx, y  S
dSnx
 J  Sny 2S  S  Sny1  Sny1    JS  2Sny  Sny1  Sny1 
dt
dSny
  JS  2Snx  Snx1  Snx1 
dt
dSnz
0
Takes care of the fact that spins are at discrete lattice positions x =n a
dt
i nka t 
S nx  uSe 
n
Solution with plane wave ansatz:
S ny  vSe 
i nka t 
With
S nx  uSe
i  nka t 
S ny  vSe 
i nka t 
i nka t 
x
n
dS
i nka t 
 iuSe 
dt
into
and
dSny
i nka t 
 i vSe 
dt
dSnx
 JS  2Sny  Sny1  Sny1 
dt
dSny
  JS  2Snx  Snx1  Snx1 
dt
iuSe 
i nka t 
i nka t   ika
i nka t   ika
 JS  2vSe 
 vSe 
e  vSe 
e 
ivSe 
i nka t 
i nka t   ika
i nka t   ika
  JS  2uSe 
 uSe 
e  uSe 
e 
i nka t 
iu  2vJS 1  cos ka 
iv  2uJS 1  cos ka 
i
2 JS 1  cos ka    u 


   0
i
 2 JS 1  cos ka 
 v 
Non-trivial solution meaning other than u=v=0 for:
i
2 JS 1  cos ka 
0
2 JS 1  cos ka 
i
  2JS 1 cos ka 
Magnon dispersion relation
Thermodynamics of magnons
Calculation of the internal energy:
E
k
1

k  nk  
2

in complete analogy to the photons
and phonons
U  E 
k
k
1  E 

k  nk  

0
 k
e
1
2
k

We consider the limit T->0:
Only low energy magnons near k=0 excited
  2JS 1 cos ka   JSa2k 2
With
 ... 
k
V
3
d
k ...
3 
(2 )
V
JSa 2 k 2
2
U  E0 
4 k  JSa2k 2
dk
3 
(2 )
e
1
With
xk
JSa 2
D
k
k BT
k BT
and hence
dx  dk
D
k BT
2
V
D
k BT
4  k BT 
U  E0 
4

x
dx

 x2
3 
(2 )
 D  e 1 D
 E0 
V
dx
5/ 2
3/ 2
4
D
k
T
x
 B   x2
2
2
e 1
Just a number which becomes with integration to infinity

4
x

0
dx
e 1
x2

3 
3 
 (5 / 2) 
1.3419
8
8

1
s
k 1 k
 ( s)  
Exponent different than for phonons due to difference in dispersion
 U 
 k BT 
CV  




 T V  D 
3/ 2
Magnetization and the celebrated
T3/2
Bloch law:
Z. Physik 61, 206 (1930):
The internal energy
U  E0    (k ) nk
k
can alternatively be expressed as
U  E0    (k )
k
where S n 
2
1
Skx  Sky
2S
ik r n
S
e
 k
Intuitive/hand-waving interpretation:
# of excitations in a mode <-> average of classical amplitude squared
k
2
1
Skx  S ky
2S
2
2
1
1
m x2  m  2 x2
2 1
2
1
 E  m  x 2   m  2  x 2  m  2  x 2     n 
2
2
E
 nk
 x2    n 
Magnetization and its deviation from full alignment in z-direction is determined as
g B
M (T ) 
V

n
S
z
n

g B
V

n

S 2   Snx    Sny 
2
2


S 2   Snx    Sny 
Let’s closer inspect

S 2   Snx    Sny 
M (T ) 
with
2
g B
V

n
Sn  Sk e
2
2
 Snx    Sny 
2
1

and remember
S2
2
 Snx    Sny   S 2 for T->0
2
using
2
  S x 2   S y 2 
n
n

 S 1 
2


2S



  S x 2   S y 2 
n
n
  g  B  NS  
S 1 


2S 2
V 
n



ik r n
k
S
2
 Snx    Sny 
2
2S
i ( k k ) r n
e
  k ,k  and S  k  S k

2





n
g B 
1
x 2
M (T ) 
Sk  Sky
 NS  
V 
k 2S

g B 
M (T ) 
 NS   nk 
V 
k

2



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

g B 
M (T ) 
 NS   nk  with
V 
k

M (T ) 
Again with
 ... 
k
V
3
d
k ... and
3 
(2 )
  JSa 2 k 2

g B 
V
dk
2
NS

4

k


 JSa2 k 2
V 
(2 )3 
e
1 
xk
g B
M (T ) 
V
JSa 2
D
k
k BT
k BT
and hence
dx  dk
D
k BT
3/ 2

V  k BT 
dx 
2
 NS  2 

  x x2
2

D


e 1 

3/ 2

V  k BT 
  (3 / 2) 
M (T )  M (T  0) 1  2

 2 NS  D 
4


g  B 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
H  J  S n S m
( n,m)
Heisenberg Hamiltonian example for continuous rotational symmetry
which can be spontaneously broken depending on the dimension, d
When a continuous symmetry is broken there must exist
a Goldstone mode (boson) with 0 for k0
Let’s have a look to spin wave approach for
d=1
d=2
M  M (T  0)  M (T )
in various spatial dimensions d
From

g B 
M (T ) 
 NS   nk 
V 

k
dk k d 1
M  
k2
e
JSa 2 k 2
and
Ld
d
...

d
k (2 )d  k ...
In low dimensions d=1 and d=2 integral diverges at the
lower bound k=0
 1   JSa 2 k 2
k 0
Unphysical result indicates absence of ordered
low temperature phase in d=1 and d=2
```