Lecture 14: Spin glasses
Outline:
• the EA and SK models
• heuristic theory
• dynamics I: using random matrices
• dynamics II: using MSR
Random Ising model
So far we dealt with “uniform systems” Jij was the same for all
pairs of neighbours.
Random Ising model
So far we dealt with “uniform systems” Jij was the same for all
pairs of neighbours.
What if every Jij is picked (independently) from some distribution?
Random Ising model
So far we dealt with “uniform systems” Jij was the same for all
pairs of neighbours.
What if every Jij is picked (independently) from some distribution?
We want to know the average of physical quantities (thermodynamic
functions, correlation functions, etc) over the distribution of Jij’s.
Random Ising model
So far we dealt with “uniform systems” Jij was the same for all
pairs of neighbours.
What if every Jij is picked (independently) from some distribution?
We want to know the average of physical quantities (thermodynamic
functions, correlation functions, etc) over the distribution of Jij’s.
Today: a simple model with <Jij> = 0
Random Ising model
So far we dealt with “uniform systems” Jij was the same for all
pairs of neighbours.
What if every Jij is picked (independently) from some distribution?
We want to know the average of physical quantities (thermodynamic
functions, correlation functions, etc) over the distribution of Jij’s.
Today: a simple model with <Jij> = 0: spin glass
Simple model (Edwards-Anderson)
Nearest-neighbour model with z neighbours
J2
2
Jij av  0, Jij av  z (Jij  J ji )

Simple model (Edwards-Anderson)
Nearest-neighbour model with z neighbours
J2
2
Jij av  0, Jij av  z (Jij  J ji )
note averages over different “samples” (1 sample = 1 realization of
choices of Jij’s for all pairs (ij) indicated by [ … ]av

Simple model (Edwards-Anderson)
Nearest-neighbour model with z neighbours
J2
2
Jij av  0, Jij av  z (Jij  J ji )
note averages over different “samples” (1 sample = 1 realization of
choices of Jij’s for all pairs (ij) indicated by [ … ]av

E   Jij Si S j   hi Si
ij
i
  12  Jij Si S j   hi Si
ij

i
Simple model (Edwards-Anderson)
Nearest-neighbour model with z neighbours
J2
2
Jij av  0, Jij av  z (Jij  J ji )
note averages over different “samples” (1 sample = 1 realization of
choices of Jij’s for all pairs (ij) indicated by [ … ]av

E   Jij Si S j   hi Si
ij
i
  12  Jij Si S j   hi Si
ij
i
non-uniform J: anticipate nonuniform magnetization mi  Si


Sherrington-Kirkpatrick model
Every spin is a neighbour of every other one: z = (N – 1)
Sherrington-Kirkpatrick model
Every spin is a neighbour of every other one: z = (N – 1)
J 
2
ij av

J2
J2


N 1 N
Sherrington-Kirkpatrick model
Every spin is a neighbour of every other one: z = (N – 1)
J 
2
ij av
J2
J2


N 1 N
Mean field theory is exact for this model

Sherrington-Kirkpatrick model
Every spin is a neighbour of every other one: z = (N – 1)
J 
2
ij av
J2
J2


N 1 N
Mean field theory is exact for this model
(but 
it is not simple)
Heuristic mean field theory
replace total field on Si,
Heuristic mean field theory
replace total field on Si, Hi   Jij S j
j

Heuristic mean field theory
replace total field on Si, Hi   Jij S j
j

(take hi = 0)
Heuristic mean field theory
replace total field on Si, Hi   Jij S j
j
by its mean

(take hi = 0)
Heuristic mean field theory
replace total field on Si, Hi   Jij S j
by its mean
Jij m j
j


j
(take hi = 0)
Heuristic mean field theory
replace total field on Si, Hi   Jij S j
(take hi = 0)
Jij m j
and calculate mj i as the average S of a single spin in field H:
j
by its mean


Heuristic mean field theory
replace total field on Si, Hi   Jij S j
(take hi = 0)
Jij m j
and calculate mj i as the average S of a single spin in field H:


 mi  tanh  J ij m j 


 j


by its mean

j
Heuristic mean field theory
replace total field on Si, Hi   Jij S j
(take hi = 0)
Jij m j
and calculate mj i as the average S of a single spin in field H:


 mi  tanh  J ij m j 


 j


no preference for mi > 0 or <0:
by its mean

j
Heuristic mean field theory
replace total field on Si, Hi   Jij S j
(take hi = 0)
Jij m j
and calculate mj i as the average S of a single spin in field H:


 mi  tanh  J ij m j 


 j


no preference for mi > 0 or <0: [mij]av = 0
by its mean

j
Heuristic mean field theory
replace total field on Si, Hi   Jij S j
(take hi = 0)
Jij m j
and calculate mj i as the average S of a single spin in field H:


 mi  tanh  J ij m j 


 j


no preference for mi > 0 or <0: [mij]av = 0
by its mean
j
if there are local spontaneous magnetizations mi ≠ 0,
measure
 them by the order parameter (Edwards-Anderson)
Heuristic mean field theory
replace total field on Si, Hi   Jij S j
(take hi = 0)
Jij m j
and calculate mj i as the average S of a single spin in field H:


 mi  tanh  J ij m j 


 j


no preference for mi > 0 or <0: [mij]av = 0
j
by its mean
if there are local spontaneous magnetizations mi ≠ 0,
measure
 them by the order parameter (Edwards-Anderson)
q  mi2 av

self-consistent calculation of q:
To compute q: Hi is a sum of many (seemingly) independent terms
self-consistent calculation of q:
To compute q: Hi is a sum of many (seemingly) independent terms
=> Hi is Gaussian
self-consistent calculation of q:
To compute q: Hi is a sum of many (seemingly) independent terms
=> Hi is Gaussian with variance
2

  

 J m     J J m m   [J 2 ] [m 2 ]
ij
j 
ij ik
j k
ij av
j




av
   jk
j
 j
av

self-consistent calculation of q:
To compute q: Hi is a sum of many (seemingly) independent terms
=> Hi is Gaussian with variance

2 

 J m     J J m m   [J 2 ] [m 2 ]
ij
j 
ij ik
j k
ij av
j




av
   jk
j
 j
av
 [Jij2 ]av q  J 2q
j

self-consistent calculation of q:
To compute q: Hi is a sum of many (seemingly) independent terms
=> Hi is Gaussian with variance

2 

 J m     J J m m   [J 2 ] [m 2 ]
ij
j 
ij ik
j k
ij av
j




av
   jk
j
 j
av
 [Jij2 ]av q  J 2q
j
q
so



 H 2 
tanh H exp 2 
2
 2J q 
2J q
dH
2
self-consistent calculation of q:
To compute q: Hi is a sum of many (seemingly) independent terms
=> Hi is Gaussian with variance

2 

 J m     J J m m   [J 2 ] [m 2 ]
ij
j 
ij ik
j k
ij av
j




av
   jk
j
 j
av
 [Jij2 ]av q  J 2q
j
q
so



 H 2 
tanh H exp 2 
2
 2J q 
2J q
dH
2
(solve for q)
spin glass transition:
q


 H 2 
tanh H exp 2 
2
 2J q 
2J q
dH
2
spin glass transition:
q
expand in β:


 H 2 
tanh H exp 2 
2
 2J q 
2J q
dH
2
spin glass transition:
q

expand in β: q 


 H 2 
tanh H exp 2 
2
 2J q 
2J q
dH

2
 H 2 
H  (H) L  exp 2 

2
 2J q 
2J q
dH
1
3
3
2
spin glass transition:
q
 H 2 
tanh H exp 2 
2
 2J q 
2J q

dH
expand in β: q 





2
dH
2J q
2
H 
1
3
(H)  L
3

2
 H 2 
exp 2 
 2J q 
2 

H
2
(

H)


3 (H)  L exp
2 
2
2J
q 

2J q
dH
2
4
spin glass transition:
q
 H 2 
tanh H exp 2 
2
 2J q 
2J q

dH
expand in β: q 


2

 H 2 
H  (H)  L  exp 2 

2
 2J q 
2J q

2 

H
2
(

H)


3 (H)  L exp
2 
2
2J
q 

2J q
dH
dH
1
3
2
  2 J 2q  23  3 4 J 4 q 2  L

2
3
4
spin glass transition:
q
 H 2 
tanh H exp 2 
2
 2J q 
2J q

dH
expand in β: q 


2

 H 2 
H  (H)  L  exp 2 

2
 2J q 
2J q

2 

H
2
(

H)


3 (H)  L exp
2 
2
2J
q 

2J q
dH
dH
1
3
2
  2 J 2q  23  3 4 J 4 q 2  L
critical temperature: Tc = J

2
3
4
spin glass transition:
q
 H 2 
tanh H exp 2 
2
 2J q 
2J q

dH
expand in β: q 


2

 H 2 
H  (H)  L  exp 2 

2
 2J q 
2J q

2 

H
2
(

H)


3 (H)  L exp
2 
2
2J
q 

2J q
dH
dH
1
3
2
2
3
4
  2 J 2q  23  3 4 J 4 q 2  L
critical temperature: Tc = J
below
 Tc:

 J
2 2
1q  2q2  q  Tc  T
spin glass transition:
q
 H 2 
tanh H exp 2 
2
 2J q 
2J q

dH
expand in β: q 


2

 H 2 
H  (H)  L  exp 2 

2
 2J q 
2J q

2 

H
2
(

H)


3 (H)  L exp
2 
2
2J
q 

2J q
dH
dH
1
3
2
2
3
4
  2 J 2q  23  3 4 J 4 q 2  L
critical temperature: Tc = J
below
 Tc:
 J
2 2
1q  2q2  q  Tc  T
This heuristic theory is right up to this point, but wrong below Tc.

the trouble below Tc
In the ferromagnet, it was safe to approximate
Hi   Jij S j   Jij m j
j

j
the trouble below Tc
In the ferromagnet, it was safe to approximate
Hi   Jij S j   Jij m j
j
j
because the next term in a systematic expansion in β,

the trouble below Tc
In the ferromagnet, it was safe to approximate
Hi   Jij S j   Jij m j
j
j
because the next term in a systematic expansion in β,


Hi  Jij S j  Jij m j  mi Jij2 (1 m2j ) L
j
j
j
the trouble below Tc
In the ferromagnet, it was safe to approximate
Hi   Jij S j   Jij m j
j
j
because the next term in a systematic expansion in β,

was O(1/z).

Hi  Jij S j  Jij m j  mi Jij2 (1 m2j ) L
j
j
j
the trouble below Tc
In the ferromagnet, it was safe to approximate
Hi   Jij S j   Jij m j
j
j
because the next term in a systematic expansion in β,
Hi  Jij S j  Jij m j  mi Jij2 (1 m2j ) L
j
j
j

was O(1/z). But here, the average of the 1st term is zero and you
have to keep the second order term, the mean of which is of the
order of the rms value of the first term.

the trouble below Tc
In the ferromagnet, it was safe to approximate
Hi   Jij S j   Jij m j
j
j
because the next term in a systematic expansion in β,
Hi  Jij S j  Jij m j  mi Jij2 (1 m2j ) L
j
j
j

was O(1/z). But here, the average of the 1st term is zero and you
have to keep the second order term, the mean of which is of the
order of the rms value of the first term.

 Thouless-Anderson-Palmer (TAP) equations):


2
2
2
mi  tanh  J ij m j   mi  J ij (1 m j )  hi 


 j

j

the trouble below Tc
In the ferromagnet, it was safe to approximate
Hi   Jij S j   Jij m j
j
j
because the next term in a systematic expansion in β,
Hi  Jij S j  Jij m j  mi Jij2 (1 m2j ) L
j
j
j

was O(1/z). But here, the average of the 1st term is zero and you
have to keep the second order term, the mean of which is of the
order of the rms value of the first term.

 Thouless-Anderson-Palmer (TAP) equations):


2
2
2
mi  tanh  J ij m j   mi  J ij (1 m j )  hi 


______________
 j

j
Onsager correction
to mean field

Dynamics (I: simple way)
Glauber dynamics:
Dynamics (I: simple way)
Glauber dynamics:
0
dP({S},t) 1
 2 1 Si tanhhi (t)P(S1 L  Si L SN )
dt
i
 12 1 Si tanhhi (t)P(S1 L Si L SN )
i

Dynamics (I: simple way)
Glauber dynamics:
0
dP({S},t) 1
 2 1 Si tanhhi (t)P(S1 L  Si L SN )
dt
i
 12 1 Si tanhhi (t)P(S1 L Si L SN )
i
recall we derived from this


d Si (t)
0
  Si (t)  tanh
  Jij S j (t)


dt
 j



Dynamics (I: simple way)
Glauber dynamics:
0
dP({S},t) 1
 2 1 Si tanhhi (t)P(S1 L  Si L SN )
dt
i
 12 1 Si tanhhi (t)P(S1 L Si L SN )
i
recall we derived from this


d Si (t)
0
  Si (t)  tanh
  Jij S j (t)


dt
 j


mean field:

Dynamics (I: simple way)
Glauber dynamics:
0
dP({S},t) 1
 2 1 Si tanhhi (t)P(S1 L  Si L SN )
dt
i
 12 1 Si tanhhi (t)P(S1 L Si L SN )
i
recall we derived from this


d Si (t)
0
  Si (t)  tanh
  Jij S j (t)


dt
 j


mean field:


0
dmi
 mi  tanhHi 
dt
Dynamics I (continued)
0

dmi
 mi  tanhHi 
dt
Dynamics I (continued)
dmi
 mi  tanhH i 
dt
linearize (above Tc):  mi  H i
0

Dynamics I (continued)
dmi
 mi  tanhH i 
dt
linearize (above Tc):  mi  H i
0
use TAP:
 mi    Jij m j   2 mi  Jij2 (1 q)  hi
j

j
Dynamics I (continued)
dmi
 mi  tanhH i 
dt
linearize (above Tc):  mi  H i
0
use TAP:
 mi    J ij m j   2 mi  J ij2 (1 q)  hi
j
j
 mi    J ij m j  mi 2 J 2  hi (q  0, T  Tc )
j

Dynamics I (continued)
dmi
 mi  tanhH i 
dt
linearize (above Tc):  mi  H i
0
use TAP:
 mi    J ij m j   2 mi  J ij2 (1 q)  hi
j
j
 mi    J ij m j  mi 2 J 2  hi (q  0, T  Tc )
In basis where J is diagonal:

j
Dynamics I (continued)
dmi
 mi  tanhH i 
dt
linearize (above Tc):  mi  H i
0
use TAP:
 mi    J ij m j   2 mi  J ij2 (1 q)  hi
j
j
 mi    J ij m j  mi 2 J 2  hi (q  0, T  Tc )
j
In basis where J is diagonal:
dm
 0   m 1  2J 2  J  h
dt


Dynamics I (continued)
dmi
 mi  tanhH i 
dt
linearize (above Tc):  mi  H i
0
use TAP:
 mi    J ij m j   2 mi  J ij2 (1 q)  hi
j
j
 mi    J ij m j  mi 2 J 2  hi (q  0, T  Tc )
j
In basis where J is diagonal:
dm
 0   m 1  2J 2  J  h
dt

m ( )


susceptibility:    
h ( ) 1 i 0   2 J 2  J


Dynamics I (continued)
dmi
 mi  tanhH i 
dt
linearize (above Tc):  mi  H i
0
use TAP:
 mi    J ij m j   2 mi  J ij2 (1 q)  hi
j
j
 mi    J ij m j  mi 2 J 2  hi (q  0, T  Tc )
j
In basis where J is diagonal:
dm
 0   m 1  2J 2  J  h
dt

m ( )


susceptibility:    
h ( ) 1 i 0   2 J 2  J
instability
 (transition) reached when maximum eigenvalue

Dynamics I (continued)
dmi
 mi  tanhH i 
dt
linearize (above Tc):  mi  H i
0
use TAP:
 mi    J ij m j   2 mi  J ij2 (1 q)  hi
j
j
 mi    J ij m j  mi 2 J 2  hi (q  0, T  Tc )
j
In basis where J is diagonal:
dm
 0   m 1  2J 2  J  h
dt

m ( )


susceptibility:    
h ( ) 1 i 0   2 J 2  J
instability
 (transition) reached when maximum eigenvalue
1  2 J 2
(J  ) max 


eigenvalue spectrum of a random matrix
For a dense random matrix with mean square element value J2/N,
the eigenvalue density is “semicircular”:
eigenvalue spectrum of a random matrix
For a dense random matrix with mean square element value J2/N,
the eigenvalue density is “semicircular”:
1
(J ) 
4J 2  J2
2J

eigenvalue spectrum of a random matrix
For a dense random matrix with mean square element value J2/N,
the eigenvalue density is “semicircular”:
1
(J ) 
4J 2  J2
2J
so


(J  ) max  2   c  1
eigenvalue spectrum of a random matrix
For a dense random matrix with mean square element value J2/N,
the eigenvalue density is “semicircular”:
1
(J ) 
4J 2  J2
2J
so
(J  ) max  2   c  1
local
 susceptibility

eigenvalue spectrum of a random matrix
For a dense random matrix with mean square element value J2/N,
the eigenvalue density is “semicircular”:
1
(J ) 
4J 2  J2
2J
so
(J  ) max  2   c  1
1
1

(

)


(

)

  ( ) 
local
susceptibility


ii

N i
N 


 d()  ( )
eigenvalue spectrum of a random matrix
For a dense random matrix with mean square element value J2/N,
the eigenvalue density is “semicircular”:
1
(J ) 
4J 2  J2
2J
so
(J  ) max  2   c  1
1
1

(

)


(

)

  ( )   d()  ( )
local
susceptibility


ii

N i
N 

 2J
4J 2  2

d

2J
2J
1 i 0   2J 2  


eigenvalue spectrum of a random matrix
For a dense random matrix with mean square element value J2/N,
the eigenvalue density is “semicircular”:
1
(J ) 
4J 2  J2
2J
so
(J  ) max  2   c  1
1
1

(

)


(

)

  ( )   d()  ( )
local
susceptibility


ii

N i
N 

 2J
4J 2  2

d

2J
2J
1 i 0   2J 2  

1 2J
4J 2  x 2
2
2
use
dx

y

y

4J

 2J
yx




eigenvalue spectrum of a random matrix
For a dense random matrix with mean square element value J2/N,
the eigenvalue density is “semicircular”:
1
(J ) 
4J 2  J2
2J
so
(J  ) max  2   c  1
1
1

(

)


(

)

  ( )   d()  ( )
local
susceptibility


ii

N i
N 

 2J
4J 2  2

d

2J
2J
1 i 0   2J 2  

1 2J
4J 2  x 2
2
2
use
dx

y

y

4J

 2J
yx

with y 1 i 0   2J 2, x  J




critical slowing down

  ( )  12  T 2 (1 i 0 )  1

2
2
2 
T
(1
i

)

1

4T



0


critical slowing down

  ( )  12  T 2 (1 i 0 )  1

2
2
2 
T
(1
i

)

1

4T



0

(J = 1)

critical slowing down

  ( )  12  T 2 (1 i 0 )  1

small ω:
2
2
2 
T
(1
i

)

1

4T



0

(J = 1)
critical slowing down

  ( )  12  T 2 (1 i 0 )  1

small ω:


 1( )  T(1 i )
2
2
2 
T
(1
i

)

1

4T



0

(J = 1)
critical slowing down

  ( )  12  T 2 (1 i 0 )  1

small ω:


 1 ( )  T(1 i ),
2
2
2 
T
(1
i

)

1

4T



0


1
T  Tc
(J = 1)
critical slowing down

  ( )  12  T 2 (1 i 0 )  1

small ω:


 1 ( )  T(1 i ),
2
2
2 
T
(1
i

)

1

4T



0


1
T  Tc
(J = 1)
critical slowing down
critical slowing down

  ( )  12  T 2 (1 i 0 )  1

small ω:

 1 ( )  T(1 i ),
2
2
2 
T
(1
i

)

1

4T



0


1
T  Tc
critical slowing down
but note: for the softest mode (with eigenvalue 2J)

(J = 1)
critical slowing down

  ( )  12  T 2 (1 i 0 )  1

small ω:
 1 ( )  T(1 i ),

2
2
2 
T
(1
i

)

1

4T



0


1
T  Tc
critical slowing down
but note: for the softest mode (with eigenvalue 2J)


 
1 i 0   2 J 2  2J

(J = 1)
critical slowing down

  ( )  12  T 2 (1 i 0 )  1

small ω:
 1 ( )  T(1 i ),

2
2
2 
T
(1
i

)

1

4T



0


1
T  Tc
critical slowing down
but note: for the softest mode (with eigenvalue 2J)


 
1 i 0   2 J 2  2J


(1 J) 2  i 0

(J = 1)
critical slowing down

  ( )  12  T 2 (1 i 0 )  1

small ω:
 1 ( )  T(1 i ),

2
2
2 
T
(1
i

)

1

4T



0


1
T  Tc
critical slowing down
but note: for the softest mode (with eigenvalue 2J)


 
1 i 0   2 J 2  2J


(1 J) 2  i 0
so its relaxation time diverges twice as strongly:

(J = 1)
critical slowing down

  ( )  12  T 2 (1 i 0 )  1

small ω:
 1 ( )  T(1 i ),

2
2
2 
T
(1
i

)

1

4T



0


1
T  Tc
critical slowing down
but note: for the softest mode (with eigenvalue 2J)


 
1 i 0   2 J 2  2J


(1 J) 2  i 0
so its relaxation time diverges twice as strongly:


(J = 1)

1
(T  Tc ) 2
Dynamics II: using MSR
Use a “soft-spin” SK model:
Dynamics II: using MSR
Use a “soft-spin” SK model:
E[]  12 r0i2  14 u0i4  12 Jiji j  hii
i

ij
i
Dynamics II: using MSR
Use a “soft-spin” SK model:
E[]   r   14 u   12 Jiji j  hii
2
1
2 0 i
i

4
0 i
ij
i

J 
2
ij av
J2

N
Dynamics II: using MSR
Use a “soft-spin” SK model:
E[]   r   14 u   12 Jiji j  hii
2
1
2 0 i
i
4
0 i
ij
J 
2
ij av
i
Langevin dynamics:


E[]
i

 i (t)  r0i  u0i3   Jij
 j  hi  i (t)
t
i
j
J2

N
Dynamics II: using MSR
Use a “soft-spin” SK model:
E[]   r   14 u   12 Jiji j  hii
2
1
2 0 i
4
0 i
i
ij
J 
2
ij av
i
Langevin dynamics:

E[]
i

 i (t)  r0i  u0i3   Jij
 j  hi  i (t)
t
i
j
Generating functional:

J2

N
Dynamics II: using MSR
Use a “soft-spin” SK model:
E[]   r   14 u   12 Jiji j  hii
2
1
2 0 i
i
4
0 i
ij
J 
2
ij av
i
J2

N
Langevin dynamics:

E[]
i

 i (t)  r0i  u0i3   Jij
 j  hi  i (t)
t
i
j
Generating functional: Z[J,h, ] 


 DDˆ exp S[,ˆ, J,h, ]
Dynamics II: using MSR
Use a “soft-spin” SK model:
E[]   r   14 u   12 Jiji j  hii
2
1
2 0 i
i
4
0 i
ij
J 
2
ij av
i
J2

N
Langevin dynamics:

E[]
i

 i (t)  r0i  u0i3   Jij
 j  hi  i (t)
t
i
j
Generating functional: Z[J,h, ] 


 DDˆ exp S[,ˆ, J,h, ]




2
3
i
ˆ, J,h, ]   dt T
ˆ  i
ˆ   r   u    J   h  i  
S[, 
i
i 
0 i
0 i
ij j
i 
i i

t


i 

j



averaging over the Jij
 expi  dt(ˆ 
i
j
ˆ  )J
 
j i
ij

ij
 J 2

ˆ (t)
ˆ (t') (t) (t')  
ˆ (t) ( t ) (t)
ˆ ( t ) 
 exp
dt dt  


i
i
j
j
i
i
j
j
2N




ij



averaging over the Jij
 expi  dt(ˆ 
i
j
ˆ  )J
 
j i
ij

ij
 J 2

ˆ (t)
ˆ (t') (t) (t')  
ˆ (t) ( t ) (t)
ˆ ( t ) 
 exp
dt dt  


i
i
j
j
i
i
j
j
2N




ij

The exponent contains
1
i
ˆ (t'),
C(t  t )    i (t) i (t'), R(t  t )    i (t)
i
N i
N i



averaging over the Jij
 expi  dt(ˆ 
i
j
ˆ  )J
 
j i
ij

ij
 J 2

ˆ (t)
ˆ (t') (t) (t')  
ˆ (t) ( t ) (t)
ˆ ( t ) 
 exp
dt dt  


i
i
j
j
i
i
j
j
2N




ij


The exponent contains
1
i
ˆ (t'),
C(t  t )    i (t) i (t'), R(t  t )    i (t)
i
N i
N i

so replace them in the exponent

 expi  dt(ˆ 
i
j
ˆ  )J
 
j i
ij

ij
 2

1
ˆ
ˆ
ˆ
 exp 2 J   dt dt  i (t)i (t')C(t  t )  ii (t) i (t')R(t  t ) 


i


decoupling sites
and introduce delta functions



ˆ
ˆ
1   DCDC exp  dtdt C(t  t )NC(t  t )    i (t) i (t')



i



ˆ
ˆ
1   DRDCRexp  dtdt R(t  t )NR(t  t )  i  i (t)i (t')



i

decoupling sites
and introduce delta functions



ˆ
ˆ
1   DCDC exp  dtdt C(t  t )NC(t  t )    i (t) i (t')



i



ˆ
ˆ
1   DRDCRexp  dtdt R(t  t )NR(t  t )  i  i (t)i (t')



i
We are left with

W  Z[0,0,J]av
 DC DCˆ DRDRˆ expN  dt dt Cˆ (t  t )C(t  t )  Rˆ (t  t )R(t  t )
ˆ expS [, 
ˆ, C, Cˆ ,R, Rˆ ]
expN log  DD


loc

(almost there)
where
ˆ, C, Cˆ ,R, Rˆ ] 
Sloc [, 

 dt Tˆ  iˆ Ý r   u  
2
3
0
0

ˆ (t)
ˆ (t )  iR(t  t )
ˆ (t) (t )
 12 J 2  dt dt  C(t  t )


ˆ (t )
  dt dt  Cˆ (t  t ) (t) (t )  iRˆ (t  t ) (t)

(almost there)
where
ˆ, C, Cˆ ,R, Rˆ ] 
Sloc [, 

 dt Tˆ  iˆ Ý r   u  
2
3
0
0

ˆ (t)
ˆ (t )  iR(t  t )
ˆ (t) (t )
 12 J 2  dt dt  C(t  t )


ˆ (t )
  dt dt  Cˆ (t  t ) (t) (t )  iRˆ (t  t ) (t)
saddle-point equations:

(almost there)
where
ˆ, C, Cˆ ,R, Rˆ ] 
Sloc [, 

 dt Tˆ  iˆ Ý r   u  
2
3
0
0

ˆ (t)
ˆ (t )  iR(t  t )
ˆ (t) (t )
 12 J 2  dt dt  C(t  t )


ˆ (t )
  dt dt  Cˆ (t  t ) (t) (t )  iRˆ (t  t ) (t)
saddle-point equations:
wrt Cˆ : C(t  t')  (t)(t )


loc
(almost there)
where
ˆ, C, Cˆ ,R, Rˆ ] 
Sloc [, 

 dt Tˆ  iˆ Ý r   u  
2
3
0
0

ˆ (t)
ˆ (t )  iR(t  t )
ˆ (t) (t )
 12 J 2  dt dt  C(t  t )


ˆ (t )
  dt dt  Cˆ (t  t ) (t) (t )  iRˆ (t  t ) (t)
saddle-point equations:
wrt Cˆ : C(t  t')   (t) (t ) loc

ˆ ( t)
ˆ ( t )
wrt C : Cˆ (t  t')   12 J 2 

loc
0
(almost there)
where
ˆ, C, Cˆ ,R, Rˆ ] 
Sloc [, 

 dt Tˆ  iˆ Ý r   u  
2
3
0
0

ˆ (t)
ˆ (t )  iR(t  t )
ˆ (t) (t )
 12 J 2  dt dt  C(t  t )


ˆ (t )
  dt dt  Cˆ (t  t ) (t) (t )  iRˆ (t  t ) (t)
saddle-point equations:
wrt Cˆ : C(t  t')   (t) ( t ) loc

ˆ ( t)
ˆ ( t )
wrt C : Cˆ (t  t')   12 J 2 
ˆ ( t )
wrt Rˆ : R(t  t')  i  (t)
loc
loc
0
(almost there)
where
ˆ, C, Cˆ ,R, Rˆ ] 
Sloc [, 

 dt Tˆ  iˆ Ý r   u  
2
3
0
0

ˆ (t)
ˆ (t )  iR(t  t )
ˆ (t) (t )
 12 J 2  dt dt  C(t  t )


ˆ (t )
  dt dt  Cˆ (t  t ) (t) (t )  iRˆ (t  t ) (t)
saddle-point equations:
wrt Cˆ : C(t  t')   (t) (t ) loc

ˆ ( t)
ˆ ( t )
wrt C : Cˆ (t  t')   12 J 2 
ˆ ( t )
wrt Rˆ : R(t  t')  i  (t)
loc
0
loc
ˆ (t) (t )
wrt R : Rˆ (t  t )  12 iJ 2 
loc
 12 J 2 R( t  t)
effective 1-spin problem:
The average correlation and response functions are equal to
those of a self-consistent single-spin problem with action
effective 1-spin problem:
The average correlation and response functions are equal to
those of a self-consistent single-spin problem with action
ˆ, C, Cˆ ,R, Rˆ ]   dt T
ˆ 2  i
ˆ 
Ý r   u  3 
Sloc [, 
0
0


ˆ (t)C(t  t )
ˆ (t )
 12 J 2  dt dt 


ˆ ( t)R(t  t ) ( t )
J 2  dt dt  i

effective 1-spin problem:
The average correlation and response functions are equal to
those of a self-consistent single-spin problem with action
ˆ, C, Cˆ ,R, Rˆ ]   dt T
ˆ 2  i
ˆ 
Ý r   u  3 
Sloc [, 
0
0


ˆ (t)C(t  t )
ˆ (t )
 12 J 2  dt dt 


ˆ ( t)R(t  t ) ( t )
J 2  dt dt  i
describing a single spin

effective 1-spin problem:
The average correlation and response functions are equal to
those of a self-consistent single-spin problem with action
ˆ, C, Cˆ ,R, Rˆ ]   dt T
ˆ 2  i
ˆ 
Ý r   u  3 
Sloc [, 
0
0


ˆ (t)C(t  t )
ˆ (t )
 12 J 2  dt dt 


ˆ ( t)R(t  t ) ( t )
J 2  dt dt  i
describing a single spin subject to
noise with correlation function 2Tδ(t – t’) +J2C(t - t’)

effective 1-spin problem:
The average correlation and response functions are equal to
those of a self-consistent single-spin problem with action
ˆ, C, Cˆ ,R, Rˆ ]   dt T
ˆ 2  i
ˆ 
Ý r   u  3 
Sloc [, 
0
0


ˆ (t)C(t  t )
ˆ (t )
 12 J 2  dt dt 


ˆ ( t)R(t  t ) ( t )
J 2  dt dt  i

describing a single spin subject to
noise with correlation function 2Tδ(t – t’) +J2C(t - t’)
and retarded self-interaction J2R(t - t’)
local response function
single effective spin obeys
local response function
single effective spin obeys
t
dS
3
2
 r0S  u0S  J dt R(t  t )S(t )  h(t)   (t)
dt

local response function
single effective spin obeys
t
dS
3
2
 r0 S  u0 S  J  dt R(t  t )S( t )  h(t)   (t)

dt
 (t) ( t )  2T (t  t )  J 2C(t  t )

local response function
single effective spin obeys
t
dS
3
2
 r0 S  u0 S  J  dt R(t  t )S( t )  h(t)   (t)

dt
 (t) ( t )  2T (t  t )  J 2C(t  t )
Fourier transform (u0 = 0)

local response function
single effective spin obeys
t
dS
3
2
 r0 S  u0 S  J  dt R(t  t )S( t )  h(t)   (t)

dt
 (t) ( t )  2T (t  t )  J 2C(t  t )


Fourier transform (u0 = 0)
iS()  r0S()  J 2R0 ()S()  h()   ()
local response function
single effective spin obeys
t
dS
3
2
 r0 S  u0 S  J  dt R(t  t )S( t )  h(t)   (t)

dt
 (t) ( t )  2T (t  t )  J 2C(t  t )



Fourier transform (u0 = 0)
iS()  r0S()  J 2R0 ()S()  h()   ()
response function (susceptibility)
 S( )
1
R0 ( ) 

h( ) r0  J 2 R0 ( )
local response function
single effective spin obeys
t
dS
3
2
 r0 S  u0 S  J  dt R(t  t )S( t )  h(t)   (t)

dt
 (t) ( t )  2T (t  t )  J 2C(t  t )


Fourier transform (u0 = 0)
iS()  r0S()  J 2R0 ()S()  h()   ()
response function (susceptibility)
 S( )
1
R0 ( ) 

h( ) r0  J 2 R0 ( )
(Can solve quadratic equation for R0 to find it explicitly)

critical slowing down
at small ω, R0-1(ω) ~ 1 - iωτ
critical slowing down
at small ω, R0-1(ω) ~ 1 - iωτ
from

R01 ( )  r0  J 2 R0 ()
critical slowing down
at small ω, R0-1(ω) ~ 1 - iωτ
from
R01 ( )  r0  J 2 R0 ()
compute


 R1( ) 
R01 ( ) 
2 2
  lim
 1 J R0 (0)lim 0


 0 (i ) 
 0 (i ) 


critical slowing down
at small ω, R0-1(ω) ~ 1 - iωτ
from
R01 ( )  r0  J 2 R0 ()
compute

 R1 ( ) 
R01 ( ) 
2 2
  lim
 1 J R0 (0)lim 0


 0 (i ) 
 0  (i ) 


  1 J 2 R02 (0)

critical slowing down
at small ω, R0-1(ω) ~ 1 - iωτ
from
R01 ( )  r0  J 2 R0 ()
compute

 R1 ( ) 
R01 ( ) 
2 2
  lim
 1 J R0 (0)lim 0


 0 (i ) 
 0  (i ) 


  1 J 2 R02 (0)


1
 
1 J 2 R02 (0)

1 
R(0)  

T 
critical slowing down
at small ω, R0-1(ω) ~ 1 - iωτ
from
R01 ( )  r0  J 2 R0 ()
compute

 R1 ( ) 
R01 ( ) 
2 2
  lim
 1 J R0 (0)lim 0


 0 (i ) 
 0  (i ) 


  1 J 2 R02 (0)


1
 
1 J 2 R02 (0)

1 
R(0)  

T 
critical slowing down
at Tc = J
critical slowing down
at small ω, R0-1(ω) ~ 1 - iωτ
from
R01 ( )  r0  J 2 R0 ()
compute

 R1 ( ) 
R01 ( ) 
2 2
  lim
 1 J R0 (0)lim 0


 0 (i ) 
 0  (i ) 


  1 J 2 R02 (0)

1
 
1 J 2 R02 (0)

1 
R(0)  

T 
critical slowing down
at Tc = J
(u0 > 0: perturbation theory does not change this qualitatively)
