Lecture 11: Ising model
Outline:
• equilibrium theory
•d=1
• mean field theory, phase transition
• critical phenomena
• kinetics (Glauber model)
• critical dynamics
• continuum description: Landau-Ginzburg model
•Langevin dynamics
“spins”
binary variables Si = ±1 (or 0/1)
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)
representing up/down, firing/not firing, opinions, decisions, atom present
/not present, …
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)
representing up/down, firing/not firing, opinions, decisions, atom present
/not present, …
(always an idealization)
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)
representing up/down, firing/not firing, opinions, decisions, atom present
/not present, …
(always an idealization)
Energy:
E   Jij Si S j   hi Si
ij
i
  12  Jij Si S j   hi Si
ij

i
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)
representing up/down, firing/not firing, opinions, decisions, atom present
/not present, …
(always an idealization)
Energy:
E   Jij Si S j   hi Si
ij
i
  12  Jij Si S j   hi Si
ij

i
(low energy is “favorable”)
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)
representing up/down, firing/not firing, opinions, decisions, atom present
/not present, …
(always an idealization)
Energy:
E   Jij Si S j   hi Si
ij
i
  12  Jij Si S j   hi Si
ij
(low energy is “favorable”)
i
role of geometry: i,j can label points on a lattice of dimensionality d

“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)
representing up/down, firing/not firing, opinions, decisions, atom present
/not present, …
(always an idealization)
Energy:
E   Jij Si S j   hi Si
ij
i
  12  Jij Si S j   hi Si
ij
(low energy is “favorable”)
i
role of geometry: i,j can label points on a lattice of dimensionality d
We will consider especially connections between neighbors in d=1

(chain)
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)
representing up/down, firing/not firing, opinions, decisions, atom present
/not present, …
(always an idealization)
Energy:
E   Jij Si S j   hi Si
ij
i
  12  Jij Si S j   hi Si
ij
(low energy is “favorable”)
i
role of geometry: i,j can label points on a lattice of dimensionality d
We will consider especially connections between neighbors in d=1
 and d=∞ ( all-to-all connectivity)
(chain)
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)
representing up/down, firing/not firing, opinions, decisions, atom present
/not present, …
(always an idealization)
Energy:
E   Jij Si S j   hi Si
ij
i
  12  Jij Si S j   hi Si
ij
(low energy is “favorable”)
i
role of geometry: i,j can label points on a lattice of dimensionality d
We will consider especially connections between neighbors in d=1
 and d=∞ ( all-to-all connectivity)
(chain)
Jij > 0: favour Si = Sj:
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)
representing up/down, firing/not firing, opinions, decisions, atom present
/not present, …
(always an idealization)
Energy:
E   Jij Si S j   hi Si
ij
i
  12  Jij Si S j   hi Si
ij
(low energy is “favorable”)
i
role of geometry: i,j can label points on a lattice of dimensionality d
We will consider especially connections between neighbors in d=1
 and d=∞ ( all-to-all connectivity)
(chain)
Jij > 0: favour Si = Sj: ferromagnetism
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)
representing up/down, firing/not firing, opinions, decisions, atom present
/not present, …
(always an idealization)
Energy:
E   Jij Si S j   hi Si
ij
i
  12  Jij Si S j   hi Si
ij
(low energy is “favorable”)
i
role of geometry: i,j can label points on a lattice of dimensionality d
We will consider especially connections between neighbors in d=1
 and d=∞ ( all-to-all connectivity)
(chain)
Jij > 0: favour Si = Sj: ferromagnetism
hi > 0: favour Si = +1.
equilibrium stat mech
Gibbs distribution P[S] 

1
exp(E)
Z
equilibrium stat mech


1
1
1
Gibbs distribution P[S]  exp(E)  exp
 J S S    hi Si 
2  ij i j


Z
Z
 ij
i



equilibrium stat mech


1
1
1
Gibbs distribution P[S]  exp(E)  exp
 J S S    hi Si 
2  ij i j


Z
Z
 ij
i



1
Z   exp
2   J ij Si S j    hi Si 

{S}
 ij
i


equilibrium stat mech


1
1
1
Gibbs distribution P[S]  exp(E)  exp
 J S S    hi Si 
2  ij i j


Z
Z
 ij
i


 partition
1
Z   exp
2   J ij Si S j    hi Si 
 function
{S}
 ij
i


equilibrium stat mech


1
1
1
Gibbs distribution P[S]  exp(E)  exp
 J S S    hi Si 
2  ij i j


Z
Z
 ij
i


 partition
1
Z   exp
2   J ij Si S j    hi Si 
 function
{S}
 ij
i


free energy:
F  T log Z

equilibrium stat mech


1
1
1
Gibbs distribution P[S]  exp(E)  exp
 J S S    hi Si 
2  ij i j


Z
Z
 ij
i


 partition
1
Z   exp
2   J ij Si S j    hi Si 
 function
{S}
 ij
i

the original Ising model:
nearest-neighbor interactions, J > 0, d = 1,
 hi = 0
free energy:
F  T log Z

equilibrium stat mech


1
1
1
Gibbs distribution P[S]  exp(E)  exp
 J S S    hi Si 
2  ij i j


Z
Z
 ij
i


 partition
1
Z   exp
2   J ij Si S j    hi Si 
 function
{S}
 ij
i

free energy:
F  T log Z
the original Ising model:


J  Si Si1 
nearest-neighbor interactions, J > 0, d = 1,
 hi = 0 Z   exp
 i

{S}


equilibrium stat mech


1
1
1
Gibbs distribution P[S]  exp(E)  exp
 J S S    hi Si 
2  ij i j


Z
Z
 ij
i


 partition
1
Z   exp
2   J ij Si S j    hi Si 
 function
{S}
 ij
i

free energy:
F  T log Z
the original Ising model:


J  Si Si1 
nearest-neighbor interactions, J > 0, d = 1,
 hi = 0 Z   exp
 i

{S}
can also have “3-spin interactions”, etc:

E   16 K ijkSi S j Sk  12  Jij Si S j   hi Si
ijk

ij
i
solving 1-d Ising model by decimation


Z   expJ  Si Si1   e JS 1 S2  e JS 2 S3  e JS 3 S4  e JS 4 S5 L
 i
 S1 S2
{S}
S3
S4
S5


solving 1-d Ising model by decimation


Z   expJ  Si Si1   e JS 1 S2  e JS 2 S3  e JS 3 S4  e JS 4 S5 L
 i
 S1 S2
{S}
S3
S4
S5
e JS i Si 1  1 JSi Si1  12 (J) 2 (Si Si1 ) 2  3!1 (J) 3 (Si Si1 ) 3

 cosh( J)  Si Si1 sinh( J)
 cosh( J)1 Si Si1 tanh( J)


solving 1-d Ising model by decimation


Z   expJ  Si Si1   e JS 1 S2  e JS 2 S3  e JS 3 S4  e JS 4 S5 L
 i
 S1 S2
{S}
S3
S4
S5
e JS i Si 1  1 JSi Si1  12 (J) 2 (Si Si1 ) 2  3!1 (J) 3 (Si Si1 ) 3

 cosh( J)  Si Si1 sinh( J)
 cosh( J)1 Si Si1 tanh( J)
Z  cosh N J 1 S1S2 tanh( J)1 S2 S3 tanh( J)1 S3S4 tanh( J)L

S1 S2
S3

S4
solving 1-d Ising model by decimation


Z   expJ  Si Si1   e JS 1 S2  e JS 2 S3  e JS 3 S4  e JS 4 S5 L
 i
 S1 S2
{S}
S3
S4
S5
e JS i Si 1  1 JSi Si1  12 (J) 2 (Si Si1 ) 2  3!1 (J) 3 (Si Si1 ) 3
 cosh( J)  Si Si1 sinh( J)

 cosh( J)1 Si Si1 tanh( J)
Z  cosh N J 1 S1S2 tanh( J)1 S2 S3 tanh( J)1 S3S4 tanh( J)L

S1 S2
S3
S4
sum on every other spin:
(1 S
2
S
tanh

J)(1
S
S
tanh

J)

2(1
S
S
tanh
J)
i1 i
i i1
i1 i1
Si


solving 1-d Ising model by decimation


Z   expJ  Si Si1   e JS 1 S2  e JS 2 S3  e JS 3 S4  e JS 4 S5 L
 i
 S1 S2
{S}
S3
S4
S5
e JS i Si 1  1 JSi Si1  12 (J) 2 (Si Si1 ) 2  3!1 (J) 3 (Si Si1 ) 3
 cosh( J)  Si Si1 sinh( J)

 cosh( J)1 Si Si1 tanh( J)
Z  cosh N J 1 S1S2 tanh( J)1 S2 S3 tanh( J)1 S3S4 tanh( J)L

S1 S2
S3
S4
sum on every other spin:
(1 S
2
S
tanh

J)(1
S
S
tanh

J)

2(1
S
S
tanh
J)
i1 i
i i1
i1 i1
Si
But this is an interaction J’ between Si-1 and Si+1 with tanh J  tanh 2 J



correlation function
Repeat (“renormalization group”):
correlation function
Repeat (“renormalization group”):
m
after m steps, tanh Jm  (tanh J)2

correlation function
Repeat (“renormalization group”):
m
after m steps, tanh Jm  (tanh J)2
Suppose we started with N = 2M spins.

correlation function
Repeat (“renormalization group”):
m
after m steps, tanh Jm  (tanh J)2
Suppose we started with N = 2M spins.
After M decimation steps there are just 2 spins left:

correlation function
Repeat (“renormalization group”):
m
after m steps, tanh Jm  (tanh J)2
Suppose we started with N = 2M spins.
After M decimation steps there are just 2 spins left:
Z  exp J M S1SN 


correlation function
Repeat (“renormalization group”):
m
after m steps, tanh Jm  (tanh J)2
Suppose we started with N = 2M spins.
After M decimation steps there are just 2 spins left:
Z  exp J M S1SN 

2exp( JM )  2exp(JM )
S1SN 
2exp( JM )  2exp(JM )


correlation function
Repeat (“renormalization group”):
m
after m steps, tanh Jm  (tanh J)2
Suppose we started with N = 2M spins.
After M decimation steps there are just 2 spins left:
Z  exp J M S1SN 

2exp( JM )  2exp(JM )
S1SN 
 tanh JM
2exp( JM )  2exp(JM )


correlation function
Repeat (“renormalization group”):
m
after m steps, tanh Jm  (tanh J)2
Suppose we started with N = 2M spins.
After M decimation steps there are just 2 spins left:
Z  exp J M S1SN 

2exp( JM )  2exp(JM )
N
S1SN 
 tanh JM  tanh J 
2exp( JM )  2exp(JM )


correlation function
Repeat (“renormalization group”):
m
after m steps, tanh Jm  (tanh J)2
Suppose we started with N = 2M spins.
After M decimation steps there are just 2 spins left:
Z  exp J M S1SN 

2exp( JM )  2exp(JM )
N
S1SN 
 tanh JM  tanh J 
2exp( JM )  2exp(JM )

Si Sin  exp n / 


correlation function
Repeat (“renormalization group”):
m
after m steps, tanh Jm  (tanh J)2
Suppose we started with N = 2M spins.
After M decimation steps there are just 2 spins left:
Z  exp J M S1SN 

2exp( JM )  2exp(JM )
N
S1SN 
 tanh JM  tanh J 
2exp( JM )  2exp(JM )

1
Si Sin  exp n /  => correlation length   
log tanh J



correlation function
Repeat (“renormalization group”):
m
after m steps, tanh Jm  (tanh J)2
Suppose we started with N = 2M spins.
After M decimation steps there are just 2 spins left:
Z  exp J M S1SN 

2exp( JM )  2exp(JM )
N
S1SN 
 tanh JM  tanh J 
2exp( JM )  2exp(JM )

1
Si Sin  exp n /  => correlation length   
log tanh J

low T:

tanh J 1 2exp(2J)


correlation function
Repeat (“renormalization group”):
m
after m steps, tanh Jm  (tanh J)2
Suppose we started with N = 2M spins.
After M decimation steps there are just 2 spins left:
Z  exp J M S1SN 

2exp( JM )  2exp(JM )
N
S1SN 
 tanh JM  tanh J 
2exp( JM )  2exp(JM )

1
Si Sin  exp n /  => correlation length   
log tanh J

low T:

tanh J 1 2exp(2J)    12 exp(2J)


correlation function
Repeat (“renormalization group”):
m
after m steps, tanh Jm  (tanh J)2
Suppose we started with N = 2M spins.
After M decimation steps there are just 2 spins left:
Z  exp J M S1SN 

2exp( JM )  2exp(JM )
N
S1SN 
 tanh JM  tanh J 
2exp( JM )  2exp(JM )

1
Si Sin  exp n /  => correlation length   
log tanh J


low T:
tanh J 1 2exp(2J)    12 exp(2J)

Correlation length grows toward ∞ at low T, but no ordering

infinite-range model
The opposite limit: every Jij = J/N
also soluble:
infinite-range model
The opposite limit: every Jij = J/N
also soluble:
d=∞
infinite-range model
The opposite limit: every Jij = J/N
also soluble:
J

Z   exp
2N  Si S j  h  Si 

{S}

ij
i


d=∞
infinite-range model
The opposite limit: every Jij = J/N
also soluble:
d=∞
2
 

J



J


Z  exp
2N  Si S j  h Si 
 exp2N  Si   h Si 
{S}

ij
i
 {S}
i
  i 


infinite-range model
The opposite limit: every Jij = J/N
also soluble:
d=∞
2
 

J



J


Z  exp
2N  Si S j  h Si 
 exp2N  Si   h Si 
{S}

ij
i
 {S}
i
  i 



Z    dmNm   Si exp 12 NJm 2  Nhm


{S}
i


infinite-range model
The opposite limit: every Jij = J/N
also soluble:



d=∞
2
 

J



J


Z  exp
2N  Si S j  h Si 
 exp2N  Si   h Si 
{S}

ij
i
 {S}
i
  i 



Z    dmNm   Si exp 12 NJm 2  Nhm


{S}
i
 

dy
Z    dm 
expyNm   Si exp 12 NJm 2  Nhm
2i

 
{S}
i
infinite-range model
The opposite limit: every Jij = J/N
also soluble:




d=∞
2
 

J



J


Z  exp
2N  Si S j  h Si 
 exp2N  Si   h Si 
{S}

ij
i
 {S}
i
  i 



Z    dmNm   Si exp 12 NJm 2  Nhm


{S}
i
 

dy
Z    dm 
expyNm   Si exp 12 NJm 2  Nhm
2i

 
{S}
i
dy
Z   dm 
exp N ym  log( 2cosh y)  12 Jm 2  hm
2i


infinite-range model
The opposite limit: every Jij = J/N
also soluble:




d=∞
2
 

J



J


Z  exp
2N  Si S j  h Si 
 exp2N  Si   h Si 
{S}

ij
i
 {S}
i
  i 



Z    dmNm   Si exp 12 NJm 2  Nhm


{S}
i
 

dy
Z    dm 
expyNm   Si exp 12 NJm 2  Nhm
2i

 
{S}
i
dy
Z   dm 
exp N ym  log( 2cosh y)  12 Jm 2  hm
2i

exponent prop to N => can use saddle point to evaluate Z

saddle point equations
Z

 dm 


dy
exp N ym  log( 2cosh y)  12 Jm 2  hm
2i
saddle point equations
Z


 dm 


dy
exp N ym  log( 2cosh y)  12 Jm 2  hm
2i

ym  log( 2cosh y)  12 Jm 2  hm 0  m  tanh y

y
saddle point equations
Z


 dm 


dy
exp N ym  log( 2cosh y)  12 Jm 2  hm
2i

ym  log( 2cosh y)  12 Jm 2  hm 0  m  tanh y

y

ym  log( 2cosh y)  12 Jm 2  hm 0  y   (Jm  h)

m
saddle point equations
Z

 dm 


dy
exp N ym  log( 2cosh y)  12 Jm 2  hm
2i

ym  log( 2cosh y)  12 Jm 2  hm 0  m  tanh y

y

ym  log( 2cosh y)  12 Jm 2  hm 0  y   (Jm  h)

m
put them together:


m  tanh  (Jm  h)
saddle point equations
Z

 dm 


dy
exp N ym  log( 2cosh y)  12 Jm 2  hm
2i

ym  log( 2cosh y)  12 Jm 2  hm 0  m  tanh y

y

ym  log( 2cosh y)  12 Jm 2  hm 0  y   (Jm  h)

m
put them together:

h = 0: m  tanh Jm


m  tanh  (Jm  h)
saddle point equations
Z

 dm 


dy
exp N ym  log( 2cosh y)  12 Jm 2  hm
2i

ym  log( 2cosh y)  12 Jm 2  hm 0  m  tanh y

y

ym  log( 2cosh y)  12 Jm 2  hm 0  y   (Jm  h)

m
put them together:

m  tanh  (Jm  h)
h = 0: m  tanh Jm has 2 solutions (i.e 2 saddle points)
for βJ > 1, i.e., T < J


saddle point equations
Z

 dm 


dy
exp N ym  log( 2cosh y)  12 Jm 2  hm
2i

ym  log( 2cosh y)  12 Jm 2  hm 0  m  tanh y

y

ym  log( 2cosh y)  12 Jm 2  hm 0  y   (Jm  h)

m
put them together:

m  tanh  (Jm  h)
h = 0: m  tanh Jm has 2 solutions (i.e 2 saddle points)
for βJ > 1, i.e., T < J

solution m: spontaneous magnetization

intuition: heuristic approach
from



1
1
P[S]  exp
 J S S  h  Si 
2  ij i j


Z
 ij
i

intuition: heuristic approach
from


1
1
P[S]  exp
 J S S  h  Si 
2  ij i j


Z
 ij
i

total field on Si is


Hi  h   Jij S j
j
intuition: heuristic approach
from


1
1
P[S]  exp
 J S S  h  Si 
2  ij i j


Z
 ij
i

total field on Si is Hi  h   Jij S j

j
replace it by its mean: H  h   Jij S j  h  Jm
j


intuition: heuristic approach
from


1
1
P[S]  exp
 J S S  h  Si 
2  ij i j


Z
 ij
i

total field on Si is Hi  h   Jij S j

j
replace it by its mean: H  h   Jij S j  h  Jm
j
and calculate m as the average S of a single spin in field H:


intuition: heuristic approach
from


1
1
P[S]  exp
 J S S  h  Si 
2  ij i j


Z
 ij
i

total field on Si is Hi  h   Jij S j

j
replace it by its mean: H  h   Jij S j  h  Jm
j
and calculate m as the average S of a single spin in field H:

e H  e H
m  S  H
 tanh  (h  Jm)
H
e

e


broken symmetry
T > Tc
T = Tc
T < Tc
Near Tc
expand

m  tanh( Jm  h)  Jm  h  13 (Jm)3
Near Tc
expand
m  tanh( Jm  h)  Jm  h  13 (Jm)3
(J 1)m  13 (Jm)3
h = 0:


Near Tc
expand
h = 0:


m  tanh( Jm  h)  Jm  h  13 (Jm)3
(J 1)m  13 (Jm) 3  m 
3(Tc  T)
, Tc  J
Tc
Near Tc
m  tanh( Jm  h)  Jm  h  13 (Jm)3
expand
(J 1)m  13 (Jm) 3  m 
h = 0:

1st order in h:


m  Jm  h
3(Tc  T)
, Tc  J
Tc
Near Tc
m  tanh( Jm  h)  Jm  h  13 (Jm)3
expand
(J 1)m  13 (Jm) 3  m 
h = 0:

1st order in h:


m  Jm  h  m 
h
T  Tc
3(Tc  T)
, Tc  J
Tc
Near Tc
m  tanh( Jm  h)  Jm  h  13 (Jm)3
expand
(J 1)m  13 (Jm) 3  m 
h = 0:

1st order in h:


m  Jm  h  m 
3(Tc  T)
, Tc  J
Tc
h
1
, i.e.,  
T  Tc
T  Tc
Near Tc
m  tanh( Jm  h)  Jm  h  13 (Jm)3
expand
(J 1)m  13 (Jm) 3  m 
h = 0:

1st order in h:

T = Tc:
h  13 (Jm)3  m  h


m  Jm  h  m 
1
3
3(Tc  T)
, Tc  J
Tc
h
1
, i.e.,  
T  Tc
T  Tc
Near Tc
m  tanh( Jm  h)  Jm  h  13 (Jm)3
expand
(J 1)m  13 (Jm) 3  m 
h = 0:

1st order in h:

T = Tc:
m  Jm  h  m 
h  13 (Jm)3  m  h

mean field critical behaviour

1
3
3(Tc  T)
, Tc  J
Tc
h
1
, i.e.,  
T  Tc
T  Tc
critical behaviour:
MFT describes the phase transition qualitatively correctly
(right exponents) for d > 4.
critical behaviour:
MFT describes the phase transition qualitatively correctly
(right exponents) for d > 4.
For 1 < d < 4, there is a phase transition, but the critical exponents
are different.
critical behaviour:
MFT describes the phase transition qualitatively correctly
(right exponents) for d > 4.
For 1 < d < 4, there is a phase transition, but the critical exponents
are different.
m  (T  Tc )  ,   (T  Tc ) , m  h 
1
  12 ,   1,   3

critical behaviour:
MFT describes the phase transition qualitatively correctly
(right exponents) for d > 4.
For 1 < d < 4, there is a phase transition, but the critical exponents
are different.
m  (T  Tc )  ,   (T  Tc ) , m  h 
1
  12 ,   1,   3
d = 2: Onsager exact solution:

critical behaviour:
MFT describes the phase transition qualitatively correctly
(right exponents) for d > 4.
For 1 < d < 4, there is a phase transition, but the critical exponents
are different.
m  (T  Tc )  ,   (T  Tc ) , m  h 
1
  12 ,   1,   3
d = 2: Onsager exact solution:


Tc  2.269J,   18 ,   47 ,   15
Dynamics: Glauber model
Ising model: Binary “spins” Si(t) = ±1
74
Dynamics: Glauber model
Ising model: Binary “spins” Si(t) = ±1
Dynamics: at every time step,
75
Dynamics: Glauber model
Ising model: Binary “spins” Si(t) = ±1
Dynamics: at every time step,
(1) choose a spin (i) at random
76
Dynamics: Glauber model
Ising model: Binary “spins” Si(t) = ±1
Dynamics: at every time step,
(1) choose a spin (i) at random
(2) compute “field” of neighbors hi(t) = ΣjJijSj(t)
77
Dynamics: Glauber model
Ising model: Binary “spins” Si(t) = ±1
Dynamics: at every time step,
(1) choose a spin (i) at random
(2) compute “field” of neighbors hi(t) = ΣjJijSj(t)
Jij = Jji
78
Dynamics: Glauber model
Ising model: Binary “spins” Si(t) = ±1
Dynamics: at every time step,
(1) choose a spin (i) at random
(2) compute “field” of neighbors hi(t) = ΣjJijSj(t)
(3) Si(t + Δt) = ±1 with probability
P(hi ) 
Jij = Jji
exp hi 
1

exp hi   exp hi  1 exp(m2hi )

79
Dynamics: Glauber model
Ising model: Binary “spins” Si(t) = ±1
Dynamics: at every time step,
(1) choose a spin (i) at random
(2) compute “field” of neighbors hi(t) = ΣjJijSj(t)
(3) Si(t + Δt) = ±1 with probability
P(hi ) 
Jij = Jji
exp hi 
1

 12 1 tanh( hi )
exp hi   exp hi  1 exp(m2hi )

80
Dynamics: Glauber model
Ising model: Binary “spins” Si(t) = ±1
Dynamics: at every time step,
(1) choose a spin (i) at random
(2) compute “field” of neighbors hi(t) = ΣjJijSj(t)
(3) Si(t + Δt) = ±1 with probability
Jij = Jji
exp hi 
1

 12 1 tanh( hi )
exp hi   exp hi  1 exp(m2hi )
(equilibration of Si, given
current values of other S’s)
P(hi ) 

81
Dynamics: Glauber model
Ising model: Binary “spins” Si(t) = ±1
Dynamics: at every time step,
(1) choose a spin (i) at random
(2) compute “field” of neighbors hi(t) = ΣjJijSj(t)
(3) Si(t + Δt) = ±1 with probability
Jij = Jji
exp hi 
1

 12 1 tanh( hi )
exp hi   exp hi  1 exp(m2hi )
(equilibration of Si, given
current values of other S’s)
P(hi ) 

82
master equation for Glauber model
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

master equation for Glauber model
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
magnetization
Si (t)   Si P({S},t)
{S}


master equation for Glauber model
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
magnetization
Si (t)   Si P({S},t)
{S}

time evolution:


d Si
dP({S},t)
0
  0  Si
dt
dt
{S}
master equation for Glauber model
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
magnetization
Si (t)   Si P({S},t)
{S}

time evolution:

d Si
dP({S},t)
0
  0  Si
dt
dt
{S}
 12  Si  tanh hi (t)P(S1 L  Si L SN )
i
 12  Si  tanh hi (t)P(S1 L Si L SN )
i
master equation for Glauber model
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
magnetization
Si (t)   Si P({S},t)
{S}

time evolution:

d Si
dP({S},t)
0
  0  Si
dt
dt
{S}
 12  Si  tanh hi (t)P(S1 L  Si L SN )
i
 12  Si  tanh hi (t)P(S1 L Si L SN )
i
  Si  tanh hi (t)
mean field dynamics


d Si
0
  Si (t)  tanh 
  Jij S j (t)  h


dt
 j


mean field dynamics


d Si
0
  Si (t)  tanh 
  Jij S j (t)  h


dt
 j

mean field approximation:

mean field dynamics


d Si
0
  Si (t)  tanh 
  Jij S j (t)  h


dt
 j

mean field approximation:


S j (t)  S j (t)  m(t)
mean field dynamics


d Si
0
  Si (t)  tanh 
  Jij S j (t)  h


dt
 j


mean field approximation: S j (t)  S j (t)  m(t)
dm
 0
 m  tanh  (Jm  h)
dt


mean field dynamics


d Si
0
  Si (t)  tanh 
  Jij S j (t)  h


dt
 j


mean field approximation: S j (t)  S j (t)  m(t)
dm
 0
 m  tanh  (Jm  h)
dt
t -> ∞: recover equilibrium
result


mean field dynamics


d Si
0
  Si (t)  tanh 
  Jij S j (t)  h


dt
 j


mean field approximation: S j (t)  S j (t)  m(t)
dm
 0
 m  tanh  (Jm  h)
dt
t -> ∞: recover equilibrium
result

T > Tc, h = 0:


0
dm
 m(1 J)
dt
mean field dynamics


d Si
0
  Si (t)  tanh 
  Jij S j (t)  h


dt
 j


mean field approximation: S j (t)  S j (t)  m(t)
dm
 0
 m  tanh  (Jm  h)
dt
t -> ∞: recover equilibrium
result

T > Tc, h = 0:


 T  T  
dm
c
0
 m(1 J)  m  m(0)exp
t
dt
  T 0  
mean field dynamics


d Si
0
  Si (t)  tanh 
  Jij S j (t)  h


dt
 j


mean field approximation: S j (t)  S j (t)  m(t)
dm
 0
 m  tanh  (Jm  h)
dt
t -> ∞: recover equilibrium
result

 T  T  
dm
c
T > Tc, h = 0:  0
 m(1 J)  m  m(0)exp
t

dt
  T 0  


“critical slowing down”     T 
0
T  Tc 


below Tc
0

dm
 m  tanh  (Jm  h)
dt
below Tc
0
dm
 m  tanh  (Jm  h)
dt
expand around


m0  tanh( Jm0 )
below Tc
0
dm
 m  tanh  (Jm  h)
dt
m0  tanh( Jm0 )
expand around
0


dm
 m  sech 2 (Jm 0 )  Jm
dt
below Tc
0
dm
 m  tanh  (Jm  h)
dt
m0  tanh( Jm0 )
expand around
0


dm
 m  sech 2 (Jm 0 )  Jm
dt
 m  (1 m02 )Jm
below Tc
0
dm
 m  tanh  (Jm  h)
dt
m0  tanh( Jm0 )
expand around
0

just below Tc:  0


dm
 m  sech 2 (Jm 0 )  Jm
dt
 m  (1 m02 )Jm
dm
 1 (J)(1 m02 )m
dt
below Tc
0
dm
 m  tanh  (Jm  h)
dt
m0  tanh( Jm0 )
expand around
0

just below Tc:  0


dm
 m  sech 2 (Jm 0 )  Jm
dt
 m  (1 m02 )Jm
dm
 1 (J)(1 m02 )m
dt
 1 (J)(1 3(Tc  T) /Tc )m
below Tc
0
dm
 m  tanh  (Jm  h)
dt
m0  tanh( Jm0 )
expand around
0

just below Tc:  0

dm
 m  sech 2 (Jm 0 )  Jm
dt
 m  (1 m02 )Jm
dm
 1 (J)(1 m02 )m
dt
 1 (J)(1 3(Tc  T) /Tc )m
 2(Tc  T)m

below Tc
0
dm
 m  tanh  (Jm  h)
dt
m0  tanh( Jm0 )
expand around
0

just below Tc:  0

dm
 m  sech 2 (Jm 0 )  Jm
dt
 m  (1 m02 )Jm
dm
 1 (J)(1 m02 )m
dt
 1 (J)(1 3(Tc  T) /Tc )m
 2(Tc  T)m
correlation time

 Tc 
   0

Tc  T 
1
2
beyond MFT
still have critical slowing down, different critical exponents
Landau-Ginzburg model
continuous-valued spins in continuous space:
Landau-Ginzburg model
continuous-valued spins in continuous space: Si  (x)

Landau-Ginzburg model
continuous-valued spins in continuous space: Si  (x)
Z  exp E({S})
{S}


Landau-Ginzburg model
continuous-valued spins in continuous space: Si  (x)
Z   exp E({S})
{S}
 Z

 D exp E[ ]

Landau-Ginzburg model
continuous-valued spins in continuous space: Si  (x)
Z   exp E({S})
{S}
 Z

 D exp E[ ]
functional

integral:
Landau-Ginzburg model
continuous-valued spins in continuous space: Si  (x)
Z   exp E({S})
{S}
 Z
 D exp E[ ]
functional

integral:


 D 
lim
 d
x i 1 x i 0
i
i
Landau-Ginzburg model
continuous-valued spins in continuous space: Si  (x)
Z   exp E({S})
{S}
 Z

 D exp E[ ]
(free) energy
functional:
E[]  12
functional

integral:
 d xr  (x) 
d
2
0


 D 
1
2
lim
 d
x i 1 x i 0
i
i
u0 4 (x)  ((x)) 2 
Landau-Ginzburg model
continuous-valued spins in continuous space: Si  (x)
Z   exp E({S})
{S}
 Z

 D exp E[ ]
(free) energy
functional:
E[]  12
functional

integral:
 d xr  (x) 
d

“potential” V()  12 r0 2  14 u0 4


2
0
 D 
1
2
lim
 d
x i 1 x i 0
i
i
u0 4 (x)  ((x)) 2 
Landau-Ginzburg model
continuous-valued spins in continuous space: Si  (x)
Z   exp E({S})
{S}
 Z

 D exp E[ ]
(free) energy
functional:
E[]  12
functional

integral:
 d xr  (x) 
d

“potential” V()  12 r0 2  14 u0 4


2
0
 D 
1
2
lim
 d
x i 1 x i 0
i
i
u0 4 (x)  ((x)) 2 
Landau-Ginzburg model
continuous-valued spins in continuous space: Si  (x)
Z   exp E({S})
{S}
 Z

 D exp E[ ]
(free) energy
functional:
E[]  12
functional

integral:
 d xr  (x) 
d
2
0
 D 
1
2
lim
 d
x i 1 x i 0
i
i
u0 4 (x)  ((x)) 2 
2

“potential” V()  12 r0 2  14 u0 4 “bending energy”  (x)



Landau-Ginzburg model
continuous-valued spins in continuous space: Si  (x)
Z   exp E({S})
{S}
 Z

 D exp E[ ]
(free) energy
functional:
E[]  12
functional

integral:
 d xr  (x) 
d
2
0
 D 
1
2
 d
lim
x i 1 x i 0
i
i
u0 4 (x)  ((x)) 2 
2

“potential” V()  12 r0 2  14 u0 4 “bending energy”  (x)
continuum limit of

Si1Si 1 

i


1
2
S
i
 Si 
2
i1
mean field theory
no fluctuations: ϕ uniform
mean field theory
no fluctuations: ϕ uniform
dV
0
d

mean field theory
no fluctuations: ϕ uniform
dV
 0  r0 0  u0 03  0
d

mean field theory
no fluctuations: ϕ uniform
dV
 0  r0 0  u0 03  0
d
solutions:


0  0
0  0,  r0 u0
mean field theory
no fluctuations: ϕ uniform
dV
 0  r0 0  u0 03  0
d
solutions:

0  0
0  0,  r0 u0
take r0 prop to T - Tc

fluctuations around MFT
r0 > 0: ignore ϕ4 term:
fluctuations around MFT
r0 > 0: ignore ϕ4 term:

E[]  12
 d xr  (x)  ((x)) 
d
2
0
2
fluctuations around MFT
E[]  12
r0 > 0: ignore ϕ4 term:
 d xr  (x)  ((x)) 
d
Fourier transform:
dd p
2
2
2
1
E[ ]  2 
r

(
p)

p

(
p)
(2
 )d 0


2
0

2
fluctuations around MFT
E[]  12
r0 > 0: ignore ϕ4 term:
 d xr  (x)  ((x)) 
d
2
Fourier transform:
dd p
2
2
2
1
1
E[ ]  2 
r

(
p)

p

(
p)

0
2
(2
 )d


2
0


r0  ( p)  p2  ( p)
p
2
2


fluctuations around MFT
E[]  12
r0 > 0: ignore ϕ4 term:
 d xr  (x)  ((x)) 
d
2
Fourier transform:
dd p
2
2
2
1
1
E[ ]  2 
r

(
p)

p

(
p)

0
2
(2
 )d
1
 P[ ]  exp E[ ]
Z


2
0


r0  ( p)  p2  ( p)
p
2
2


fluctuations around MFT
E[]  12
r0 > 0: ignore ϕ4 term:
 d xr  (x)  ((x)) 
d
2
2
0
Fourier transform:
dd p
2
2
2
2
2
2
1
1
E[ ]  2 
r

(
p)

p

(
p)

r

(
p)

p

(
p)
0
2 
(2
 )d 0
p
1
1
2
 P[ ]  exp E[ ]   exp  12 (r0  p 2 )  ( p)
Z
Z p








fluctuations around MFT
E[]  12
r0 > 0: ignore ϕ4 term:
 d xr  (x)  ((x)) 
d
2
2
0
Fourier transform:
dd p
2
2
2
2
2
2
1
1
E[ ]  2 
r

(
p)

p

(
p)

r

(
p)

p

(
p)
0
2 
(2
 )d 0
p
1
1
2
 P[ ]  exp E[ ]   exp  12 (r0  p 2 )  ( p)
(T = 1)
Z
Z p








fluctuations around MFT
E[]  12
r0 > 0: ignore ϕ4 term:
 d xr  (x)  ((x)) 
d
2
2
0
Fourier transform:
dd p
2
2
2
2
2
2
1
1
E[ ]  2 
r

(
p)

p

(
p)

r

(
p)

p

(
p)
0
2 
(2
 )d 0
p
1
1
2
 P[ ]  exp E[ ]   exp  12 (r0  p 2 )  ( p)
(T = 1)
Z
Z p







product of independent Gaussians, one for each pair (p,-p)
fluctuations around MFT
E[]  12
r0 > 0: ignore ϕ4 term:
 d xr  (x)  ((x)) 
d
2
2
0
Fourier transform:
dd p
2
2
2
2
2
2
1
1
E[ ]  2 
r

(
p)

p

(
p)

r

(
p)

p

(
p)
0
2 
(2
 )d 0
p
1
1
2
 P[ ]  exp E[ ]   exp  12 (r0  p 2 )  ( p)
(T = 1)
Z
Z p









product of independent Gaussians, one for each pair (p,-p)
1
2
( p) 
r0  p2
fluctuations around MFT
E[]  12
r0 > 0: ignore ϕ4 term:
 d xr  (x)  ((x)) 
d
2
2
0
Fourier transform:
dd p
2
2
2
2
2
2
1
1
E[ ]  2 
r

(
p)

p

(
p)

r

(
p)

p

(
p)
0
2 
(2
 )d 0
p
1
1
2
 P[ ]  exp E[ ]   exp  12 (r0  p 2 )  ( p)
(T = 1)
Z
Z p







product of independent Gaussians, one for each pair (p,-p)
1
1
2


(x)

(y)

exp  r0 x  y
( p) 
2
4 x  y
r0  p





fluctuations around MFT
E[]  12
r0 > 0: ignore ϕ4 term:
 d xr  (x)  ((x)) 
d
2
2
0
Fourier transform:
dd p
2
2
2
2
2
2
1
1
E[ ]  2 
r

(
p)

p

(
p)

r

(
p)

p

(
p)
0
2 
(2
 )d 0
p
1
1
2
 P[ ]  exp E[ ]   exp  12 (r0  p 2 )  ( p)
(T = 1)
Z
Z p







product of independent Gaussians, one for each pair (p,-p)
1
1
2


(x)

(y)

exp  r0 x  y
( p) 
2
4 x  y
r0  p
1
1
correlation length


r0
T  Tc






fluctuations around MFT
E[]  12
r0 > 0: ignore ϕ4 term:
 d xr  (x)  ((x)) 
d
2
2
0
Fourier transform:
dd p
2
2
2
2
2
2
1
1
E[ ]  2 
r

(
p)

p

(
p)

r

(
p)

p

(
p)
0
2 
(2
 )d 0
p
1
1
2
 P[ ]  exp E[ ]   exp  12 (r0  p 2 )  ( p)
(T = 1)
Z
Z p







product of independent Gaussians, one for each pair (p,-p)
1
1
2


(x)

(y)

exp  r0 x  y
( p) 
2
4 x  y
r0  p
1
1
correlation length


diverges at Tc
r0
T  Tc







Langevin dynamics
 (x)
E

 (x,t),
t
 (x)
(x,t)( x , t )  2T d (x  x ) (t  t )


Langevin dynamics
 (x)
E

 (x,t),
t
 (x)
(x,t)( x , t )  2T d (x  x ) (t  t )
Fourier transform :
 ( p)
E
 *
 ( p,t),
t
 ( p)
( p,t)( p, t )  2T (t  t )



Langevin dynamics
 (x)
E

 (x,t),
t
 (x)
(x,t)( x , t )  2T d (x  x ) (t  t )
Fourier transform :
 ( p)
E
 *
 ( p,t),
t
 ( p)
E[ ]  12

r0  ( p)  p2  ( p)
p
2
( p,t)( p, t )  2T (t  t )
2

1
4
u0
  ( p) ( p) ( p) ( p  p p)
p, p , p 




Langevin dynamics
 (x)
E

 (x,t),
t
 (x)
(x,t)( x , t )  2T d (x  x ) (t  t )
Fourier transform :
 ( p)
E
 *
 ( p,t),
t
 ( p)
E[ ]  12

r0  ( p)  p2  ( p)
p
2
( p,t)( p, t )  2T (t  t )
2

1
4
u0
  ( p) ( p) ( p) ( p  p p)
p, p , p 
 ( p)
 r0  p 2  ( p)  u0   ( p) ( p) ( p  p p)  ( p,t)
t
p , p 



Langevin dynamics
 (x)
E

 (x,t),
t
 (x)
(x,t)( x , t )  2T d (x  x ) (t  t )
Fourier transform :
 ( p)
E
 *
 ( p,t),
t
 ( p)
E[ ]  12

r0  ( p)  p2  ( p)
2
p
( p,t)( p, t )  2T (t  t )
2

1
4
u0
  ( p) ( p) ( p) ( p  p p)
p, p , p 
 ( p)
 r0  p 2  ( p)  u0   ( p) ( p) ( p  p p)  ( p,t)
t
p , p 
or, back in real space,


 (x)
 r0   2  (x)  u0 3 (x)  (x,t)
t
small fluctuations above Tc
ignoring u0:

 ( p)
 r0  p2  ( p)  ( p,t)
t
small fluctuations above Tc
 ( p)
 r0  p2  ( p)  ( p,t)
t
We have solved this before:
ignoring u0:

small fluctuations above Tc
 ( p)
 r0  p2  ( p)  ( p,t)
t
We have solved this before:
ignoring u0:
( p,t) 



t

exp (r0  p2 )(t  t )(p, t )dt 
small fluctuations above Tc
 ( p)
 r0  p2  ( p)  ( p,t)
t
We have solved this before:
ignoring u0:
( p,t) 

t

exp (r0  p2 )(t  t )(p, t )dt 

average over noise (we did this before, too)

small fluctuations above Tc
 ( p)
 r0  p2  ( p)  ( p,t)
t
We have solved this before:
ignoring u0:
( p,t) 

t

exp (r0  p2 )(t  t )(p, t )dt 

average over noise (we did this before, too)


 ( p,t) ( p, t ) 
T
2
exp
(r

p
) t  t 

0
2
r0  p
small fluctuations above Tc
 ( p)
 r0  p2  ( p)  ( p,t)
t
We have solved this before:
ignoring u0:
( p,t) 

t

exp (r0  p2 )(t  t )(p, t )dt 

average over noise (we did this before, too)

 ( p,t) ( p, t ) 
T
2
exp
(r

p
) t  t 

0
2
r0  p
critical slowing down (long correlation time) of fluctuations
near Tc (small r0) for small p (long wavelengths)
