Lecture 11: Ising model Outline: equilibrium theory d = 1 mean field theory, phase transition...

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 = − JijSi

ij

∑ S j − hi

i

∑ Si

= − 12 JijSi

ij

∑ S j − hi

i

∑ Si

“spins”


€

E = − JijSi

ij

∑ S j − hi

i

∑ Si

= − 12 JijSi

ij

∑ S j − hi

i

∑ Si (low energy is “favorable”)

“spins”


€

E = − JijSi

ij

∑ S j − hi

i

∑ Si

= − 12 JijSi

ij

∑ S j − hi

i


role of geometry: i,j can label points on a lattice of dimensionality d

“spins”


€

E = − JijSi

ij

∑ S j − hi

i

∑ Si

= − 12 JijSi

ij

∑ S j − hi

i


role of geometry: i,j can label points on a lattice of dimensionality dWe will consider especially connections between neighbors in d=1 (chain)

“spins”


€

E = − JijSi

ij

∑ S j − hi

i

∑ Si

= − 12 JijSi

ij

∑ S j − hi

i


role of geometry: i,j can label points on a lattice of dimensionality dWe will consider especially connections between neighbors in d=1 (chain) and d=∞ ( all-to-all connectivity)

“spins”


€

E = − JijSi

ij

∑ S j − hi

i

∑ Si

= − 12 JijSi

ij

∑ S j − hi

i


role of geometry: i,j can label points on a lattice of dimensionality dWe will consider especially connections between neighbors in d=1 (chain) and d=∞ ( all-to-all connectivity)Jij > 0: favour Si = Sj:

“spins”


€

E = − JijSi

ij

∑ S j − hi

i

∑ Si

= − 12 JijSi

ij

∑ S j − hi

i


role of geometry: i,j can label points on a lattice of dimensionality dWe will consider especially connections between neighbors in d=1 (chain) and d=∞ ( all-to-all connectivity)Jij > 0: favour Si = Sj: ferromagnetism

“spins”


€

E = − JijSi

ij

∑ S j − hi

i

∑ Si

= − 12 JijSi

ij

∑ S j − hi

i


role of geometry: i,j can label points on a lattice of dimensionality dWe will consider especially connections between neighbors in d=1 (chain) and d=∞ ( all-to-all connectivity)Jij > 0: favour Si = Sj: ferromagnetism hi > 0: favour Si = +1.

equilibrium stat mech

€

P[S] =1

Zexp(−βE)Gibbs distribution


€

P[S] =1

Zexp(−βE) =

1

Zexp 1

2 β JijSi

ij

∑ S j + β hi

i

∑ Si

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟Gibbs distribution


€

P[S] =1

Zexp(−βE) =

1

Zexp 1

2 β JijSi

ij

∑ S j + β hi

i

∑ Si

⎛

⎝ ⎜ ⎜

⎞


€

Z = exp 12 β JijSi

ij

∑ S j + β hi

i

∑ Si

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

{S}

∑


€

P[S] =1

Zexp(−βE) =

1

Zexp 1

2 β JijSi

ij

∑ S j + β hi

i

∑ Si

⎛

⎝ ⎜ ⎜

⎞


partition function

€

Z = exp 12 β JijSi

ij

∑ S j + β hi

i

∑ Si

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

{S}

∑


€

P[S] =1

Zexp(−βE) =

1

Zexp 1

2 β JijSi

ij

∑ S j + β hi

i

∑ Si

⎛

⎝ ⎜ ⎜

⎞


partition function

€

Z = exp 12 β JijSi

ij

∑ S j + β hi

i

∑ Si

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

{S}

∑

€

F = −T log Zfree energy:


€

P[S] =1

Zexp(−βE) =

1

Zexp 1

2 β JijSi

ij

∑ S j + β hi

i

∑ Si

⎛

⎝ ⎜ ⎜

⎞


partition function

the original Ising model: nearest-neighbor interactions, J > 0, d = 1, hi = 0€

Z = exp 12 β JijSi

ij

∑ S j + β hi

i

∑ Si

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

{S}

∑

€



€

P[S] =1

Zexp(−βE) =

1

Zexp 1

2 β JijSi

ij

∑ S j + β hi

i

∑ Si

⎛

⎝ ⎜ ⎜

⎞


partition function

the original Ising model: nearest-neighbor interactions, J > 0, d = 1, hi = 0

€

Z = exp βJ Si

i

∑ Si+1

⎛

⎝ ⎜

⎞

⎠ ⎟

{S}

∑€

Z = exp 12 β JijSi

ij

∑ S j + β hi

i

∑ Si

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

{S}

∑

€



€

P[S] =1

Zexp(−βE) =

1

Zexp 1

2 β JijSi

ij

∑ S j + β hi

i

∑ Si

⎛

⎝ ⎜ ⎜

⎞


partition function

the original Ising model: nearest-neighbor interactions, J > 0, d = 1, hi = 0

€

Z = exp βJ Si

i

∑ Si+1

⎛

⎝ ⎜

⎞

⎠ ⎟

{S}

∑€

Z = exp 12 β JijSi

ij

∑ S j + β hi

i

∑ Si

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

{S}

∑

can also have “3-spin interactions”, etc:

€

E = − 16 K ijkSiS jSk

ijk

∑ − 12 JijSi

ij

∑ S j − hi

i

∑ Si

€


solving 1-d Ising model by decimation

€

Z = exp βJ Si

i

∑ Si+1

⎛

⎝ ⎜

⎞

⎠ ⎟

{S}

∑ = eβJS1S2 eβJS2S3 eβJS3S4

S4

∑S3

∑ eβJS4 S5 LS5

∑S1S2

∑

€


€

Z = exp βJ Si

i

∑ Si+1

⎛

⎝ ⎜

⎞

⎠ ⎟

{S}


S4

∑S3

∑ eβJS4 S5 LS5

∑S1S2

∑

€

eβJS i Si+1 =1+ βJSiSi+1 + 12 (βJ)2(SiSi+1)

2 + 13! (βJ)3(SiSi+1)

3

= cosh(βJ) + SiSi+1 sinh(βJ)

= cosh(βJ) 1+ SiSi+1 tanh(βJ)[ ]

€


€

Z = exp βJ Si

i

∑ Si+1

⎛

⎝ ⎜

⎞

⎠ ⎟

{S}


S4

∑S3

∑ eβJS4 S5 LS5

∑S1S2

∑

€

Z = coshN βJ 1+ S1S2 tanh(βJ)( ) 1+ S2S3 tanh(βJ)( ) 1+ S3S4 tanh(βJ)( )S4

∑S3

∑ LS1S2

∑€


2 + 13! (βJ)3(SiSi+1)

3



€


€

Z = exp βJ Si

i

∑ Si+1

⎛

⎝ ⎜

⎞

⎠ ⎟

{S}


S4

∑S3

∑ eβJS4 S5 LS5

∑S1S2

∑

sum on every other spin:

€


∑S3

∑ LS1S2

∑€


2 + 13! (βJ)3(SiSi+1)

3



€ €

(1+ Si−1Si tanhβJ)(1+ SiSi+1 tanhβJ)Si

∑ = 2(1+ Si−1Si+1 tanh2 βJ)


€

Z = exp βJ Si

i

∑ Si+1

⎛

⎝ ⎜

⎞

⎠ ⎟

{S}


S4

∑S3

∑ eβJS4 S5 LS5

∑S1S2

∑

sum on every other spin:

€


∑S3

∑ LS1S2

∑€


2 + 13! (βJ)3(SiSi+1)

3



€

But this is an interaction J’ between Si-1 and Si+1 with

€

tanhβ ′ J = tanh2 βJ

€

(1+ Si−1Si tanhβJ)(1+ SiSi+1 tanhβJ)Si

∑ = 2(1+ Si−1Si+1 tanh2 βJ)

correlation function

Repeat (“renormalization group”):


Repeat (“renormalization group”):after m steps,

€

tanhβJm = (tanhβJ)2m


Repeat (“renormalization group”):after m steps,Suppose we started with N = 2M spins.

€



Repeat (“renormalization group”):after m steps,Suppose we started with N = 2M spins. After M decimation steps there are just 2 spins left:

€




€


€

Z ∝ exp βJM S1SN( )



€


€


€

S1SN =2exp(βJM ) − 2exp(−βJM )

2exp(βJM ) + 2exp(−βJM )



€


€


€


2exp(βJM ) + 2exp(−βJM )= tanhβJM



€


€


€


2exp(βJM ) + 2exp(−βJM )= tanhβJM = tanhβJ( )

N



€


€


€



N

€

SiSi+n = exp −n /ξ( )



€


€


€



N

=> correlation length

€


€

ξ =−1

logtanhβJ



€


€


€



N


€


€

tanhβJ →1− 2exp(−2βJ)low T:

€

ξ =−1

logtanhβJ



€


€


€



N


€


€

tanhβJ →1− 2exp(−2βJ) ⇒ ξ → 12 exp(2βJ)low T:

€

ξ =−1

logtanhβJ



€


€


€



N


€


€

tanhβJ →1− 2exp(−2βJ) ⇒ ξ → 12 exp(2βJ)low T:

€

ξ =−1

logtanhβJ

Correlation length grows toward ∞ at low T, but no ordering

infinite-range modelThe opposite limit: every Jij = J/Nalso soluble:


d = ∞


€

Z = expβJ

2NSi

ij

∑ S j + βh Si

i

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

{S}

∑

d = ∞


€

Z = expβJ

2NSi

ij

∑ S j + βh Si

i

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

{S}

∑ = expβJ

2NSi

i

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

2

+ βh Si

i

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

{S}

∑

d = ∞


€

Z = expβJ

2NSi

ij

∑ S j + βh Si

i

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

{S}

∑ = expβJ

2NSi

i

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

2

+ βh Si

i

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

{S}

∑

€

Z = dmδ Nm − Si

i

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟∫ exp 1

2 NβJm2 + Nβhm( ){S}

∑

d = ∞


€

Z = expβJ

2NSi

ij

∑ S j + βh Si

i

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

{S}

∑ = expβJ

2NSi

i

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

2

+ βh Si

i

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

{S}

∑

€

Z = dmδ Nm − Si

i

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟∫ exp 1


∑

€

Z = dmdy

2πi∫ exp −y Nm − Si

i

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥∫ exp 1


∑

d = ∞


€

Z = expβJ

2NSi

ij

∑ S j + βh Si

i

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

{S}

∑ = expβJ

2NSi

i

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

2

+ βh Si

i

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

{S}

∑

€

Z = dmδ Nm − Si

i

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟∫ exp 1


∑

€

Z = dmdy


i

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥∫ exp 1


∑

€

Z = dmdy

2πi∫∫ exp N −ym + log(2cosh y) + 1

2 βJm2 + βhm( )[ ]

d = ∞


€

Z = expβJ

2NSi

ij

∑ S j + βh Si

i

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

{S}

∑ = expβJ

2NSi

i

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

2

+ βh Si

i

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

{S}

∑

€

Z = dmδ Nm − Si

i

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟∫ exp 1


∑

€

Z = dmdy


i

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥∫ exp 1


∑

€

Z = dmdy


2 βJm2 + βhm( )[ ]

exponent prop to N => can use saddle point to evaluate Z

d = ∞

saddle point equations

€

Z = dmdy


2 βJm2 + βhm( )[ ]


€

Z = dmdy


2 βJm2 + βhm( )[ ]

€

∂∂y

−ym + log(2cosh y) + 12 βJm2 + βhm( ) = 0 ⇒ m = tanh y


€

Z = dmdy


2 βJm2 + βhm( )[ ]

€

∂∂y


∂

∂m−ym + log(2cosh y) + 1

2 βJm2 + βhm( ) = 0 ⇒ y = β (Jm + h)


€

Z = dmdy


2 βJm2 + βhm( )[ ]

€

∂∂y


∂


2 βJm2 + βhm( ) = 0 ⇒ y = β (Jm + h)

put them together:

€

m = tanh β (Jm + h)[ ]


€

Z = dmdy


2 βJm2 + βhm( )[ ]

€

∂∂y


∂


2 βJm2 + βhm( ) = 0 ⇒ y = β (Jm + h)

put them together:

€


h = 0:

€

m = tanhβJm


€

Z = dmdy


2 βJm2 + βhm( )[ ]

€

∂∂y


∂


2 βJm2 + βhm( ) = 0 ⇒ y = β (Jm + h)

put them together:

€


h = 0:

€

m = tanhβJm has 2 solutions (i.e 2 saddle points) for βJ > 1, i.e., T < J


€

Z = dmdy


2 βJm2 + βhm( )[ ]

€

∂∂y


∂


2 βJm2 + βhm( ) = 0 ⇒ y = β (Jm + h)

put them together:

€


h = 0:

€

m = tanhβJm has 2 solutions (i.e 2 saddle points) for βJ > 1, i.e., T < J

solution m: spontaneous magnetization

intuition: heuristic approach

€

P[S] =1

Zexp 1

2 β JijSi

ij

∑ S j + βh Si

i

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟from


€

P[S] =1

Zexp 1

2 β JijSi

ij

∑ S j + βh Si

i

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟from

total field on Si is

€

H i = h + JijS j

j

∑


€

P[S] =1

Zexp 1

2 β JijSi

ij

∑ S j + βh Si

i

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟from


replace it by its mean:

€

H i = h + JijS j

j

∑

€

H = h + Jij S j

j

∑ = h + Jm


€

P[S] =1

Zexp 1

2 β JijSi

ij

∑ S j + βh Si

i

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟from



and calculate m as the average S of a single spin in field H: €

H i = h + JijS j

j

∑

€

H = h + Jij S j

j

∑ = h + Jm


€

P[S] =1

Zexp 1

2 β JijSi

ij

∑ S j + βh Si

i

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟from



and calculate m as the average S of a single spin in field H: €

H i = h + JijS j

j

∑

€

H = h + Jij S j

j

∑ = h + Jm

€

m = S =eβH − eβH

eβH + eβH= tanhβ (h + Jm)

broken symmetry

T > TcT = TcT < Tc

Near Tc

expand

€

m = tanh(βJm + βh) = βJm + βh − 13 (βJm)3

Near Tc

expand

€


€

(βJ −1)m = 13 (βJm)3h = 0:

Near Tc

expand

€


€

(βJ −1)m = 13 (βJm)3 ⇒ m =

3(Tc − T)

Tc

, Tc = Jh = 0:

Near Tc

expand

€


€

(βJ −1)m = 13 (βJm)3 ⇒ m =

3(Tc − T)

Tc

, Tc = Jh = 0:

1st order in h:

€

m = βJm + βh

Near Tc

expand

€


€

(βJ −1)m = 13 (βJm)3 ⇒ m =

3(Tc − T)

Tc

, Tc = Jh = 0:

1st order in h:

€

m = βJm + βh ⇒ m =h

T − Tc

Near Tc

expand

€


€

(βJ −1)m = 13 (βJm)3 ⇒ m =

3(Tc − T)

Tc

, Tc = Jh = 0:

1st order in h:

€


T − Tc

, i.e., χ =1

T − Tc

Near Tc

expand

€


€

(βJ −1)m = 13 (βJm)3 ⇒ m =

3(Tc − T)

Tc

, Tc = Jh = 0:

T = Tc:

€

βh = 13 (βJm)3 ⇒ m∝ h

13

1st order in h:

€


T − Tc

, i.e., χ =1

T − Tc

Near Tc

expand

€


€

(βJ −1)m = 13 (βJm)3 ⇒ m =

3(Tc − T)

Tc

, Tc = Jh = 0:

T = Tc:

€

βh = 13 (βJm)3 ⇒ m∝ h

13

1st order in h:

€


T − Tc

, i.e., χ =1

T − Tc

mean field critical behaviour

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 exponentsare different.


€

m∝ (T − Tc )β , χ ∝ (T − Tc )−γ , m∝ h1δ

β < 12 , γ >1, δ > 3


€

m∝ (T − Tc )β , χ ∝ (T − Tc )−γ , m∝ h1δ

β < 12 , γ >1, δ > 3

d = 2: Onsager exact solution:


€

m∝ (T − Tc )β , χ ∝ (T − Tc )−γ , m∝ h1δ

β < 12 , γ >1, δ > 3

d = 2: Onsager exact solution:

€

Tc = 2.269J, β = 18 , γ = 7

4 , δ =15

Dynamics: Glauber model

74

Ising model: Binary “spins” Si(t) = ±1


75

Ising model: Binary “spins” Si(t) = ±1Dynamics: at every time step,


76


(1) choose a spin (i) at random


77


(1) choose a spin (i) at random(2) compute “field” of neighbors hi(t) = ΣjJijSj(t)


78


(1) choose a spin (i) at random(2) compute “field” of neighbors hi(t) = ΣjJijSj(t) Jij = Jji


79


(1) choose a spin (i) at random(2) compute “field” of neighbors hi(t) = ΣjJijSj(t) Jij = Jji (3) Si(t + Δt) = ±1 with probability

€

P(hi) =exp ±βhi( )

exp βhi( ) + exp −βhi( )=

1

1+ exp(m2βhi)


80



€



1

1+ exp(m2βhi)= 1

2 1± tanh(βhi)( )


81



(equilibration of Si, given current values of other S’s)

€



1

1+ exp(m2βhi)= 1

2 1± tanh(βhi)( )


82



(equilibration of Si, given current values of other S’s)

€



1

1+ exp(m2βhi)= 1

2 1± tanh(βhi)( )

master equation for Glauber model

€

τ 0

dP({S},t)

dt= 1

2 1+ Si tanh βhi(t)( )[ ]i

∑ P(S1L − SiL SN )

− 12 1− Si tanh βhi(t)( )[ ]

i

∑ P(S1L SiL SN )


€

τ 0

dP({S},t)

dt= 1



− 12 1− Si tanh βhi(t)( )[ ]

i

∑ P(S1L SiL SN )

magnetization

€

Si(t) = Si

{S}

∑ P({S}, t)


€

τ 0

dP({S},t)

dt= 1



− 12 1− Si tanh βhi(t)( )[ ]

i

∑ P(S1L SiL SN )

magnetization

€

Si(t) = Si

{S}

∑ P({S}, t)

€

τ 0

d Si

dt= τ 0 Si

{S}

∑ dP({S},t)

dttime evolution:


€

τ 0

dP({S},t)

dt= 1



− 12 1− Si tanh βhi(t)( )[ ]

i

∑ P(S1L SiL SN )

magnetization

€

Si(t) = Si

{S}

∑ P({S}, t)

€

τ 0

d Si

dt= τ 0 Si

{S}

∑ dP({S},t)

dt

= 12 Si + tanh βhi(t)( )[ ]

i


− 12 Si − tanh βhi(t)( )[ ]

i

∑ P(S1L SiL SN )

time evolution:


€

τ 0

dP({S},t)

dt= 1



− 12 1− Si tanh βhi(t)( )[ ]

i

∑ P(S1L SiL SN )

magnetization

€

Si(t) = Si

{S}

∑ P({S}, t)

€

τ 0

d Si

dt= τ 0 Si

{S}

∑ dP({S},t)

dt

= 12 Si + tanh βhi(t)( )[ ]

i


− 12 Si − tanh βhi(t)( )[ ]

i

∑ P(S1L SiL SN )

= − Si + tanh βhi(t)( )

time evolution:

mean field dynamics

€

τ 0

d Si

dt= − Si(t) + tanh β JijS j (t) + βh

j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

mean field dynamics

€

τ 0

d Si


j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

mean field approximation:

mean field dynamics

€

τ 0

d Si


j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟


€

S j (t) → S j (t) = m(t)

mean field dynamics

€

τ 0

d Si


j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟


€

S j (t) → S j (t) = m(t)

€

⇒ τ 0

dm

dt= −m + tanhβ (Jm + h)

mean field dynamics

€

τ 0

d Si


j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟


€

S j (t) → S j (t) = m(t)

€

⇒ τ 0

dm


t -> ∞: recover equilibrium result

mean field dynamics

€

τ 0

d Si


j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟


€

S j (t) → S j (t) = m(t)

€

⇒ τ 0

dm



T > Tc, h = 0:

€

τ 0

dm

dt= −m(1− βJ)

mean field dynamics

€

τ 0

d Si


j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟


€

S j (t) → S j (t) = m(t)

€

⇒ τ 0

dm



T > Tc, h = 0:

€

τ 0

dm

dt= −m(1− βJ) ⇒ m = m(0)exp −

T − Tc

Tτ 0

⎛

⎝ ⎜

⎞

⎠ ⎟t

⎡

⎣ ⎢

⎤

⎦ ⎥

mean field dynamics

€

τ 0

d Si


j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟


€

S j (t) → S j (t) = m(t)

€

⇒ τ 0

dm



T > Tc, h = 0:

€

τ 0

dm

dt= −m(1− βJ) ⇒ m = m(0)exp −

T − Tc

Tτ 0

⎛

⎝ ⎜

⎞

⎠ ⎟t

⎡

⎣ ⎢

⎤

⎦ ⎥

“critical slowing down”

€

τ =τ 0

T

T − Tc

⎛

⎝ ⎜

⎞

⎠ ⎟

below Tc

€

τ 0

dm


below Tc

€

τ 0

dm


€

m0 = tanh(βJm0)expand around

below Tc

€

τ 0

dm


€

m0 = tanh(βJm0)

τ 0

dδm

dt= −δm + sech2(βJm0) ⋅βJδm

expand around

below Tc

€

τ 0

dm


€

m0 = tanh(βJm0)

τ 0

dδm


= −δm + (1− m02)βJδm

expand around

below Tc

€

τ 0

dm


€

m0 = tanh(βJm0)

τ 0

dδm


= −δm + (1− m02)βJδm

expand around

€

τ 0

dδm

dt= − 1− (βJ)(1− m0

2)( )δmjust below Tc:

below Tc

€

τ 0

dm


€

m0 = tanh(βJm0)

τ 0

dδm


= −δm + (1− m02)βJδm

expand around

€

τ 0

dδm

dt= − 1− (βJ)(1− m0

2)( )δm

= − 1− (βJ)(1− 3(Tc − T) /Tc )( )δm

just below Tc:

below Tc

€

τ 0

dm


€

m0 = tanh(βJm0)

τ 0

dδm


= −δm + (1− m02)βJδm

expand around

€

τ 0

dδm

dt= − 1− (βJ)(1− m0

2)( )δm

= − 1− (βJ)(1− 3(Tc − T) /Tc )( )δm

= −2(Tc − T)δm

just below Tc:

below Tc

€

τ 0

dm


€

m0 = tanh(βJm0)

τ 0

dδm


= −δm + (1− m02)βJδm

expand around

€

τ 0

dδm

dt= − 1− (βJ)(1− m0

2)( )δm

= − 1− (βJ)(1− 3(Tc − T) /Tc )( )δm

= −2(Tc − T)δm

€

τ =12 τ 0

Tc

Tc − T

⎛

⎝ ⎜

⎞

⎠ ⎟

just below Tc:

correlation time

beyond MFT

still have critical slowing down, different critical exponents

Landau-Ginzburg model

continuous-valued spins in continuous space:



€

Si → φ(x)



€

Si → φ(x)

€

Z = exp −βE({S})( ){S}

∑



€

Si → φ(x)

€

Z = exp −βE({S})( ){S}

∑

→ Z = Dφexp∫ −βE[φ]( )



€

Si → φ(x)

€

Z = exp −βE({S})( ){S}

∑

→ Z = Dφexp∫ −βE[φ]( ) functional integral:



€

Si → φ(x)

€

Z = exp −βE({S})( ){S}

∑


€

Dφ ≡ limxi+1 −xi →0

∫ dφi

i

∏



€

Si → φ(x)

€

Z = exp −βE({S})( ){S}

∑


€


∫ dφi

i

∏

€

E[φ] = 12 dd x∫ r0φ

2(x) + 12 u0φ

4 (x) + (∇φ(x))2[ ]

(free) energyfunctional:



€

Si → φ(x)

€

Z = exp −βE({S})( ){S}

∑


€


∫ dφi

i

∏

€

E[φ] = 12 dd x∫ r0φ

2(x) + 12 u0φ

4 (x) + (∇φ(x))2[ ]


“potential”

€

V (φ) = 12 r0φ

2 + 14 u0φ

4



€

Si → φ(x)

€

Z = exp −βE({S})( ){S}

∑


€


∫ dφi

i

∏

€

E[φ] = 12 dd x∫ r0φ

2(x) + 12 u0φ

4 (x) + (∇φ(x))2[ ]


“potential” “bending energy”

€

∇φ(x)( )2

€

V (φ) = 12 r0φ

2 + 14 u0φ

4



€

Si → φ(x)

€

Z = exp −βE({S})( ){S}

∑


€


∫ dφi

i

∏

€

E[φ] = 12 dd x∫ r0φ

2(x) + 12 u0φ

4 (x) + (∇φ(x))2[ ]


“potential” “bending energy”

€

∇φ(x)( )2

€

V (φ) = 12 r0φ

2 + 14 u0φ

4

€

− Si+1Si −1( )i

∑ = 12 Si+1 − Si( )

i

∑2

continuum limit of

mean field theory

no fluctuations: ϕ uniform

mean field theory


€

dV

dφ= 0

mean field theory


€

dV

dφ= 0 ⇒ r0φ0 + u0φ0

3 = 0

mean field theory


€

dV

dφ= 0 ⇒ r0φ0 + u0φ0

3 = 0

€

φ0 = 0

φ0 = 0, ± −r0 u0

solutions:

mean field theory


€

dV

dφ= 0 ⇒ r0φ0 + u0φ0

3 = 0

€

φ0 = 0

φ0 = 0, ± −r0 u0

solutions:

take r0 prop to T - Tc

fluctuations around MFT

r0 > 0: ignore ϕ4 term:



€

E[φ] = 12 dd x∫ r0φ

2(x) + (∇φ(x))2[ ]



€

E[φ] = 12 dd x∫ r0φ

2(x) + (∇φ(x))2[ ]

Fourier transform:

€

E[φ] = 12

dd p

(2π )d∫ r0 φ( p)2

+ p2 φ( p)2

[ ]



€

E[φ] = 12 dd x∫ r0φ

2(x) + (∇φ(x))2[ ]

Fourier transform:

€

E[φ] = 12

dd p

(2π )d∫ r0 φ( p)2

+ p2 φ( p)2

[ ] ≡ 12 r0 φ( p)

2+ p2 φ( p)

2

[ ]p

∫



€

E[φ] = 12 dd x∫ r0φ

2(x) + (∇φ(x))2[ ]

Fourier transform:

€

E[φ] = 12

dd p

(2π )d∫ r0 φ( p)2

+ p2 φ( p)2

[ ] ≡ 12 r0 φ( p)

2+ p2 φ( p)

2

[ ]p

∫

€

⇒ P[φ] =1

Zexp −βE[φ]( )



€

E[φ] = 12 dd x∫ r0φ

2(x) + (∇φ(x))2[ ]

Fourier transform:

€

E[φ] = 12

dd p

(2π )d∫ r0 φ( p)2

+ p2 φ( p)2

[ ] ≡ 12 r0 φ( p)

2+ p2 φ( p)

2

[ ]p

∫

€

⇒ P[φ] =1

Zexp −βE[φ]( ) =

1

Zexp − 1

2 (r0 + p2)φ(p)2

[ ]p

∏



€

E[φ] = 12 dd x∫ r0φ

2(x) + (∇φ(x))2[ ]

Fourier transform:

€

E[φ] = 12

dd p

(2π )d∫ r0 φ( p)2

+ p2 φ( p)2

[ ] ≡ 12 r0 φ( p)

2+ p2 φ( p)

2

[ ]p

∫

€

⇒ P[φ] =1


1

Zexp − 1

2 (r0 + p2)φ(p)2

[ ]p

∏ (T = 1)



€

E[φ] = 12 dd x∫ r0φ

2(x) + (∇φ(x))2[ ]

Fourier transform:

€

E[φ] = 12

dd p

(2π )d∫ r0 φ( p)2

+ p2 φ( p)2

[ ] ≡ 12 r0 φ( p)

2+ p2 φ( p)

2

[ ]p

∫

€

⇒ P[φ] =1


1

Zexp − 1

2 (r0 + p2)φ(p)2

[ ]p

∏

product of independent Gaussians, one for each pair (p,-p)

(T = 1)



€

E[φ] = 12 dd x∫ r0φ

2(x) + (∇φ(x))2[ ]

Fourier transform:

€

E[φ] = 12

dd p

(2π )d∫ r0 φ( p)2

+ p2 φ( p)2

[ ] ≡ 12 r0 φ( p)

2+ p2 φ( p)

2

[ ]p

∫

€

⇒ P[φ] =1


1

Zexp − 1

2 (r0 + p2)φ(p)2

[ ]p

∏


€

φ( p)2

=1

r0 + p2

(T = 1)



€

E[φ] = 12 dd x∫ r0φ

2(x) + (∇φ(x))2[ ]

Fourier transform:

€

E[φ] = 12

dd p

(2π )d∫ r0 φ( p)2

+ p2 φ( p)2

[ ] ≡ 12 r0 φ( p)

2+ p2 φ( p)

2

[ ]p

∫

€

⇒ P[φ] =1


1

Zexp − 1

2 (r0 + p2)φ(p)2

[ ]p

∏


€

φ( p)2

=1

r0 + p2

€

⇒ φ(x)φ(y) =1

4π x − yexp − r0 x − y( )

(T = 1)



€

E[φ] = 12 dd x∫ r0φ

2(x) + (∇φ(x))2[ ]

Fourier transform:

€

E[φ] = 12

dd p

(2π )d∫ r0 φ( p)2

+ p2 φ( p)2

[ ] ≡ 12 r0 φ( p)

2+ p2 φ( p)

2

[ ]p

∫

€

⇒ P[φ] =1


1

Zexp − 1

2 (r0 + p2)φ(p)2

[ ]p

∏


€

φ( p)2

=1

r0 + p2

€

⇒ φ(x)φ(y) =1

4π x − yexp − r0 x − y( )

correlation length

€

ξ =1

r0

∝1

T − Tc

(T = 1)



€

E[φ] = 12 dd x∫ r0φ

2(x) + (∇φ(x))2[ ]

Fourier transform:

€

E[φ] = 12

dd p

(2π )d∫ r0 φ( p)2

+ p2 φ( p)2

[ ] ≡ 12 r0 φ( p)

2+ p2 φ( p)

2

[ ]p

∫

€

⇒ P[φ] =1


1

Zexp − 1

2 (r0 + p2)φ(p)2

[ ]p

∏


€

φ( p)2

=1

r0 + p2

€

⇒ φ(x)φ(y) =1

4π x − yexp − r0 x − y( )

correlation length

€

ξ =1

r0

∝1

T − Tc

diverges at Tc

(T = 1)

Langevin dynamics

€

∂φ(x)

∂t= −

δE

δφ(x)+ η (x, t), η (x, t)η ( ′ x , ′ t ) = 2Tδ d (x − ′ x )δ(t − ′ t )

Langevin dynamics

€

∂φ(x)

∂t= −

δE


€

∂φ( p)

∂t= −

∂E

∂φ*( p)+ η (p, t), η (p, t)η (−p, ′ t ) = 2Tδ(t − ′ t )

Fourier transform :

Langevin dynamics

€

∂φ(x)

∂t= −

δE


€

∂φ( p)

∂t= −

∂E


Fourier transform :

€

E[φ] = 12 r0 φ( p)

2+ p2 φ( p)

2

[ ]p

∫ + 14 u0 φ(p)φ( ′ p )φ( ′ ′ p )φ(−p − ′ p − ′ ′ p )

p, ′ p , ′ ′ p

∫

Langevin dynamics

€

∂φ(x)

∂t= −

δE


€

∂φ( p)

∂t= −

∂E


Fourier transform :

€

E[φ] = 12 r0 φ( p)

2+ p2 φ( p)

2

[ ]p

∫ + 14 u0 φ(p)φ( ′ p )φ( ′ ′ p )φ(−p − ′ p − ′ ′ p )

p, ′ p , ′ ′ p

∫

€

∂φ( p)

∂t= − r0 + p2

( )φ(p) − u0 φ( ′ p )φ( ′ ′ p )φ(p − ′ p − ′ ′ p )′ p , ′ ′ p

∫ + η ( p, t)

Langevin dynamics

€

∂φ(x)

∂t= −

δE


€

∂φ( p)

∂t= −

∂E


Fourier transform :

€

E[φ] = 12 r0 φ( p)

2+ p2 φ( p)

2

[ ]p

∫ + 14 u0 φ(p)φ( ′ p )φ( ′ ′ p )φ(−p − ′ p − ′ ′ p )

p, ′ p , ′ ′ p

∫

€

∂φ( p)

∂t= − r0 + p2

( )φ(p) − u0 φ( ′ p )φ( ′ ′ p )φ(p − ′ p − ′ ′ p )′ p , ′ ′ p

∫ + η ( p, t)

or, back in real space,

€

∂φ(x)

∂t= − r0 −∇ 2

( )φ(x) − u0φ3(x) + η (x, t)

small fluctuations above Tc

€

∂φ( p)

∂t= − r0 + p2

( )φ(p) + η ( p, t)ignoring u0:


€

∂φ( p)

∂t= − r0 + p2


We have solved this before:


€

∂φ( p)

∂t= − r0 + p2



€

φ( p, t) = exp −(r0 + p2)(t − ′ t )[ ]−∞

t

∫ η (p, ′ t )d ′ t


€

∂φ( p)

∂t= − r0 + p2



€

φ( p, t) = exp −(r0 + p2)(t − ′ t )[ ]−∞

t

∫ η (p, ′ t )d ′ t

average over noise (we did this before, too)


€

∂φ( p)

∂t= − r0 + p2



€

φ( p, t) = exp −(r0 + p2)(t − ′ t )[ ]−∞

t

∫ η (p, ′ t )d ′ t


€

φ( p, t)φ(p, ′ t ) =T

r0 + p2exp −(r0 + p2) t − ′ t [ ]


€

∂φ( p)

∂t= − r0 + p2



€

φ( p, t) = exp −(r0 + p2)(t − ′ t )[ ]−∞

t

∫ η (p, ′ t )d ′ t


€

φ( p, t)φ(p, ′ t ) =T

r0 + p2exp −(r0 + p2) t − ′ t [ ]

critical slowing down (long correlation time) of fluctuations near Tc (small r0) for small p (long wavelengths)

Lecture 11: Ising model Outline: equilibrium theory d = 1 mean field theory, phase transition...

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Transcript of Lecture 11: Ising model Outline: equilibrium theory d = 1 mean field theory, phase transition...