Symmetry and ergodicity breaking, mean-field study of the
Ising model

Rafael Mottl, ETH Zurich Advisor: Dr. Ingo Kirsch

Topics

Introduction Formalism The model system: Ising model

Solutions in one and more dimensions Symmetry and ergodicity breaking Conclusion

Introduction

• phase A
• phase B

T_c

temperature T phase transition

order parameter
ꢀ Critical exponents

T

T_c

Formalism

Statistical mechanical basics

Ω

sample region

d

i) certain dimension ii) volume

V (Ω) N(Ω)

iii) number of particles/sites iv) boundary conditions
Hamiltonian defined on the sample region

ꢀ

H_Ω

−

=

K_nΘ_n

k_BT

n

Depending on the degrees of freedom

Coupling constants

Partition function

1

β =

Z_Ω[{K_n}] = Tr exp^{−βH ({K },{Θ })}

Ω

• n
• n

k_BT

F_Ω[{K_n}] = F_Ω[K] = −k_BTlogZ_Ω[{K_n}]

Free energy Free energy per site

F_Ω[K] f_b[K] = lim
N(Ω)→∞ N(Ω)

i) Non-trivial existence of limit

Ω

ii) Independent of iii)

lim
N(Ω)

= const

N(Ω)→∞ V (Ω)

Phases and phase boundaries

f_b[K]

• Supp.: -
• exists

{K₁, . . . , K_D}

- there exist D coulping constants:

• -
• is analytic almost everywhere

f_b[K]

f [{K_n}] _{are points, lines, planes, hyperplanes in the}

- non-analyticities of b phase diagram

D_s = 0, 1, 2, . . .

ꢀ Dimension of these singular loci:
Codimension C for each type of singular loci:

C = D − D_s

f_b[K]

Phase: region of analyticity of

Phase boundaries: loci of codimension C = 1

Types of phase transitions

f_b[K]

is everywhere continuous.
Two types of phase transitions:

∂f_b[K]

a) Discontinuity across the phase boundary of

∂K_i

first-order phase transition

b) All derivatives of the free energy per site are continuous across the phase boundaries

Continuous phase transition

Overview

Z_Ω[K] = Tr exp^{−βH (K,{Θ })}

Ω

n

F_Ω[K] = −k_BTlogZ_Ω[K]

F_Ω[K] f_b[K] = lim
N(Ω)→∞ N(Ω)

∂
∂f_b[K]

ǫ_in[K] = (βf_b[K])

∂β

M[K] = −

∂H

• internal energy
• magnetization

∂M[K]
∂ǫ_in[K]

χ_T [K] =
C[K] =

∂H
∂T

magnetic susceptibility heat capacity

Critical exponents

How do the measurable quantities change in the neighbourhood of a critical point?

T − T_C t =

T_C

Critical temperature

T_C

C ∼|t|^−α

heat capacity

β

for H ≡ 0

M ∼|t|

Order parameter (t < 0) susceptibility

χ ∼|t|^−γ M ∼ H^1/δ

Equation of state

Correlation length

- Length scale over which the fluctuations of the microscopic degrees of freedom are significantly correlated with each other

(Cardy: Scaling and Renormalization in Statistical Physics)

- Strong dependence on temperature near a phase transition diverging to infinity at the transition itself

Critical exponents for the correlation function

t → 0

Γ(r) −→ r^−pe^−r/ξ

ξ ∼|t|^−ν

Correlation length

p = d − 2 + η

Power-law decay (t=0)

The model system: Ising model

- attempt to simulate a domain in a ferromagnetic material - Spontaneous magnetization in absence of an external magnetic field - Critical temperature: Curie temperature

Importance of Ising model

- Equivalent models (lattice gas, binary alloy) - The only non-trivial example of a phase transition that can be worked out be mathematical rigor in statistical mechanics

- Compare computer simulations of the model with exact solution

Characterization of the Ising model

i) Periodic lattice ii) Lattice contains in d dimensions

Ω

fixed points called lattice sited

N(Ω)

iii) For each site: classical spin variable S_i = +/- 1 (i = 1, …, N) in a definite direction (degrees of freedom)

iv) Most general Hamiltonian

• ꢀ
• ꢀ
• ꢀ

−H_Ω =

• H_iS_i +
• J_ijS_iS_j +

K_ijkS_iS_jS_k + . . .

i,j

i∈Ω

i,j,k

2^N(Ω)

v) Number of possible configurations:

• ꢀ ꢀ
• ꢀ
• ꢀ

Tr ≡

• · · ·
• ≡

S₁=±1 S₂=±1

S

_N(Ω)=±1

{S_i=±1}

Assumptions

K_ijk = 0, . . .

i) two-spin coupling only

H_i ≡ H

ii) External magnetic field spatially constant iii) Nearest-neighbour interactions only

• ꢀ
• ꢀ

→

• i,j
• <ij>

iv) Isotropic interactions v) Hypercubic lattice

J_ij ≡ J z = 2d J > 0

vi) Ferromagnetic material

• ꢀ
• ꢀ

−H_Ω = H S_i + J

S_iS_j

<i,j>

i∈Ω

ꢀ

• ꢀ
• ꢀ

_i∈Ω S_i+J

_<i,j> S_iS_j)

• Z_Ω[H, T, J] =
• exp^β(H

{S_i=±1}

Arguments for phase transition in d = 1,2 dimensions with H = 0

ꢀ

F_Ω[K] = E_in[K] − TS_Ω[K] = −J

S_iS_j − Tk_Blog(♯(states))

<i,j>

A
B

d = 1

N → ∞

F_N,B − F_N,A = 2J − k_BTlog(N − 1) → −∞

d = 2

A
B

n bonds

F_N,B − F_N,A = [2J − log(z − 1)k_BT]n
2J

T_C =

k_Blog(z − 1)

Long range order ꢀ short range order

- nearest neighbour interaction: short range interaction - What about:

J

J_ij =

σ

|r_i − r_j|

Answer:

σ < 1

thermodynamic limit does not exist

1 ꢀ σ ꢀ 2

long range order persist for 0 < T < T_C short-range interaction: no ferromagnetic state for T > 0

2 ꢀ σ

Solutions in one and more dimensions

- ad hoc methods - recursion method

H = 0 d = 1

- transfer matrix method

H = 0

(Kramers, Wannier 1941)
- low temperature expansion

• H = 0
• d = 2

- Onsager solution (1944) - mean-field method (Weiss)

d = 1, 2, 3

H = 0

• d = 1
• H = 0

Transfer matrix method

Reduce the problem of calculating the partition function to the problem of finding the eigenvalues of a matrix

S_N+1 = S₁

S_N+1 = S₁

Assumption:

S_N

S₂

Z_Ω[H, J] = Tr exp^{−βH (H,J,{S })}

S₃

Ω

i

• ꢀ
• ꢀ

• H_Ω(H, J, {S_i}) = −H
• S_i − J

S_iS_j

<i,j>

i∈Ω

• h := βH
• K := βJ

ꢀ ꢀ

• ꢀ
• ꢀ

• _i S_i+K
• _i S_iS_i+1

Z_Ω[h, K] = . . . e^h

S₁

S_N

• ꢁ
• ꢂ

e^h+K − λ

e^−K

e^−h+K − λ det

≡ 0

Calculation:

e^−K

ꢃ

• ⇒ λ_1,2 = e^K(cosh(h) ± sinh(h)² + e^−4K
• )

ꢃ

⇒ f_b(h, K, T) = −Jk_BTlog(cosh(h) + sinh(h)² + e^−4K)

Spatial correlation in one dimension (T > 0)

• H = 0
• d = 1

Definition: two point correlation function

G(i, j) := ꢁ(S_i − ꢁS_iꢂ)(S_j − ꢁS_jꢂ)ꢂ = ꢁS_iS_jꢂ − ꢁS_iꢂꢁS_jꢂ

G(i, i + j) = ꢁS_iS_i+jꢂ = (tanh(K))^j

G(i, i + j) = e^{−jlog(coth(K))} ≡ e^−j/ξ

1

J/(k_BT)

∼
⇒ ξ =

1/2 e
=

log(coth)K))

• H = 0
• d = 2

Onsager solution

Columns →

3

Notation and boundary conditions:

...

n

2

1

n + 1 ≡ 1

1

2

µ_α = {s₁, s₂, . . . , s_n}_{αth row}

s_n+1 = s₁ , µ_n+1 = µ₁

3

N = n²

n

Elements of the Hamiltonian:

n + 1 ≡ 1

n

ꢀ

E(µ, µ^′) = −ǫ

s_ks^′_k

k=1

• n
• n

• ꢀ
• ꢀ

E(µ) = −ǫ

s_ks_k+1 − H

s_k

$$2$$

• k=1
• k=1

Partition function:

ꢀ ꢀ

ꢀ

n

_α=1[E(µ_α,µ_α+1)+E(µ_α)]

Z_Ω[H, T] = . . . e^−β

µ₁

µ_n

−β[E(µ,µ^′)+E(µ)]

ꢁµ|P|µ^′ꢂ := e

• Properties of the matrix P:
• -

--

P : 2ⁿ × 2ⁿmatrix Z_Ω[H, T] = TrPⁿ

• 1
• 1

⇒ lim log(Z_Ω[H = 0, T]) = lim log(λ_max)

• ^N→∞ N
• ^n→∞ n

largest eigenvalue of P

• F_Ω[0, T]
• −k_BTlogZ_Ω[0, T ]

N(Ω)
1
= −k_BT lim log(λ_max

^n→∞ n

• f_b[H = 0, T] = lim
• = lim
• )

N(Ω)→∞ N(Ω)

N(Ω)→∞

ꢄ

π

ꢅ

• 1
• 1

• ⇒ βf_b[0, T] = − log (2 cosh 2βǫ) −
• dφ log 1 + 1 − κ²sin²φ

2π

2

0

κ = 2[cosh 2φ coth2φ]⁻¹