Ising model with random boundary condition A. C. D. van Enter # , K. - - PDF document

Equilibrium and Dynamics of Spin Glasses Monte Verita, April 18-23, 2004

Ising model with random boundary condition

• A. C. D. van Enter#, K. Netoˇ

cn´ y&, and H. G. Schaap#

#Institute for Theoretical Physics

R.U.G. Groningen, The Netherlands

&Eurandom

Eindhoven, The Netherlands

1

Model

We consider the d ≥ 2 dimensional Ising model Hη

Λ(σΛ) = −β ∑ x,y⊂Λ

σxσy − λβ

∑

x,y x∈Λ, y∈Λc

σxηy λ > 0 in cubic volumes Λ(N) = {x ∈ Zd; x ≤ N}

x = max{|x1|, |x2|}

under boundary conditions η ∈ Ω = {−1, 1}Zd sampled from the i.i.d. symmetric random field P{ηx = 1} = P{ηx = −1} = 1 2 and study the limit behavior of the sequence of finite-volume Gibbs measures (µη

Λ(N))N∈N

µη

Λ(σ) =

1 Zη

Λ

exp[−Hη

Λ(σΛ)] 1

l{σΛc=ηΛc} in the low-temperature regime

2

To be discussed:

• Existence vers. non-existence of the limit limN↑∞ µη

Λ(N)

• P-a.s. patterns in the set of the limit measures
• Cluster expansions as a tool to (dis)prove the existence of

thermodynamic limit

3

Thermodynamics of the Ising model

The Ising model is a simple example of a model exhibiting a non-trivial structure of the set of Gibbs measures, defined equivalently as 1) solutions of the DLR-equations µ(· | ηΛc) = µη

Λ(·)

for all finite volumes Λ ⊂ Zd and µ-almost every η 2) via the set of all (weak) limits limΛ↑Zd µη

Λ

A physical interpretation: The extremal translation-invariant Gibbs measures (or their symmetry-equivalent classes, in general) are interpreted as thermodynamic phases The mixtures are interpreted in terms of a lack of the knowledge about the thermodynamic state of the system The translationally non-invariant measures, including interfaces, correspond to ‘less stable’ physical states

4

d ≥ 2: For β > βc, exactly two extremal and translation-invariant Gibbs measures µ+, µ− exist. They are related by the spin-flip symmetry d ≥ 3: For β > ˜ βc > βc, translationally non-invariant Gibbs measures exist (e.g. Dobrushin interfaces) The extremal Gibbs measures µ+ (resp. µ−) are the limits µ± = lim

Λ µη≡±1 Λ

i.e. they can be constructed via the boundary conditions coinciding with the (local) ground states = a special instance of the Pirogov-Sinai theory Coherent vers. symmetric boundary conditions: Both periodic and free boundary conditions give rise to the statistical mixture lim

Λ µper Λ

= lim

Λ µfree Λ

= 1

2(µ+ + µ−)

5

Similarly as the periodic or the free boundary conditions, the symmetric random b.c. gives no preference to any of the extremal

• phases. However, a different picture is expected:

Conjecture (Newman and Stein ’92): At any temperature β > βc, the finite-volume Gibbs measures oscillate randomly between the ‘+’ and ‘-’ phases

− →

chaotic size-dependence with exactly two limit points coinciding with the pure thermodynamic phases CSD in general: expected for spin-glass models under a class of symmetric b.c. (Newman and Stein ’92) some rigorous results obtained for a class of mean-field models (e.g. K¨ ulske ’97; Bovier and Gayrard ’98; Enter, Bovier, Niederhauser ’00)

6

Results

Theorem 1 (Enter, Medved’, N., Markov Proc. Rel. Fields ’02). Assuming i) dimension d ≥ 2 ii) weak boundary coupling, λ ≤ λ∗ ≪ 1 Then at low enough temperatures, β > β0(λ∗) ≫ βc, we have: d ≥ 4: With P-probability 1, the set of limit points of {µη

Λ(N)}N∈I

N is {µ+, µ−} d = 2, 3: With P-probability 1, the set of limit points of any “sparse” sequence {µη

Λ(kN)}N∈I

N, kN ≥ N4−d+ω, ω > 0, is {µ+, µ−} Note in particular: An arbitrarily small boundary coupling λ > 0 causes chaotic size-dependence The set of limit Gibbs measures is P-a.s. constant and coincides with the set of pure thermodynamic phases

− → neither interfaces, nor mixtures occur

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The assumption of a weak enough boundary coupling is not essential and can be removed: Theorem 2 (Enter, N., Schaap, in preparation). Assuming d = 2, λ = 1, and β > β0 ≫ βc, the set of limit points of any “sparse” sequence {µη

Λ(kN)}N∈I

N, kN ≥ N2+ω, ω > 0, is

{µ+, µ−}, with P-probability 1

Interfaces are excluded in a strong sense: For the joint measure

(P × µ)Λ(σΛηΛc) = P(ηΛc)µη

Λ(σΛ)

we prove lim

N↑∞(P × µ)Λ(N){σΛ contains no interface} = 1

8

Outline of the proofs

• 1. Geometrical representation in terms of contours separating ‘+’

and ‘-’ regions

• 2. Decomposition of the configuration space into the ‘+’ and ‘-’

restricted ensembles, ΩΛ = Ω+

Λ ∪ Ω− Λ

• 3. Perturbative construction of the restricted Gibbs measures

ν±,η

Λ (σ) =

1

Z±η

Λ

exp[−Hη

Λ(σ)] 1{σΛ∈Ω±

Λ;σΛc=ηΛc}

(using a multi-scale generalization of the cluster expansions in the case λ = 1)

• 4. Proving the P-a.s. asymptotic triviality of the restricted Gibbs

measures: lim

Λ ν±,η Λ

= µ+

• 5. Proving the P-a.s. absence of finite limit points for the random

free energy difference: lim

Λ

• log Z+η

Λ

Z−η

Λ

• = ∞

(using a local central limit upper-bound for sums of weakly dependent random variables)

• 6. Representation of the Gibbs measure µη

Λ in terms of the

restricted Gibbs measures µη

Λ(σ) =

• 1 + Z−η

Λ

Z+η

Λ

−1 ν+,η

Λ (σ) +

• 1 + Z+η

Λ

Z−η

Λ

−1 ν−,η

Λ (σ)

9