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 y & , and H. G. Schaap # cn´ # Institute for Theoretical Physics R.U.G. Groningen, The Netherlands & Eurandom Eindhoven, The Netherlands 1

Model We consider the d ≥ 2 dimensional Ising model Λ ( σ Λ ) = − β ∑ ∑ H η σ x σ y − λβ λ > 0 σ x η y � x , y �⊂ Λ � x , y � x ∈ Λ , y ∈ Λ c in cubic volumes Λ ( N ) = { x ∈ Z d ; � x � ≤ N } � x � = max {| x 1 | , | x 2 |} under boundary conditions η ∈ Ω = {− 1, 1 } Z d sampled from the i.i.d. symmetric random field P { η x = 1 } = P { η x = − 1 } = 1 2 and study the limit behavior of the sequence of finite-volume Gibbs measures ( µ η Λ ( N ) ) N ∈ N 1 µ η exp [ − H η Λ ( σ ) = Λ ( σ Λ )] 1 l { σ Λ c = η Λ c } Z η Λ in the low-temperature regime 2

To be discussed: • Existence vers. non-existence of the limit lim N ↑ ∞ µ η 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 ( · | η Λ c ) = µ η Λ ( · ) for all finite volumes Λ ⊂ Z d and µ -almost every η 2) via the set of all (weak) limits lim Λ ↑ Z d µ η 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 > β c , exactly two extremal and translation-invariant Gibbs measures µ + , µ − exist. They are related by the spin-flip symmetry d ≥ 3 : For β > ˜ β c > β c , translationally non-invariant Gibbs measures exist (e.g. Dobrushin interfaces) The extremal Gibbs measures µ + (resp. µ − ) are the limits µ ± = lim Λ µ η ≡± 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 = 1 2 ( µ + + µ − ) Λ µ per Λ µ free lim = lim Λ Λ 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 > β c , 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 > β 0 ( λ ∗ ) ≫ β c , we have: d ≥ 4 : With P -probability 1, the set of limit points of { µ η Λ ( N ) } N ∈ I N is { µ + , µ − } d = 2, 3 : With P -probability 1, the set of limit points of any “sparse” sequence { µ η N, k N ≥ N 4 − d + ω , ω > 0, is { µ + , µ − } Λ ( k N ) } N ∈ I 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 7

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 > β 0 ≫ β c , the set of limit points of any “sparse” sequence { µ η N, k N ≥ N 2 + ω , ω > 0, is Λ ( k N ) } N ∈ I { µ + , µ − } , with P -probability 1 Interfaces are excluded in a strong sense: For the joint measure ( P × µ ) Λ ( σ Λ η Λ c ) = P ( η Λ c ) µ η Λ ( σ Λ ) we prove N ↑ ∞ ( P × µ ) Λ ( N ) { σ Λ contains no interface } = 1 lim 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, Ω Λ = Ω + Λ ∪ Ω − Λ 3. Perturbative construction of the restricted Gibbs measures 1 ν ± , η exp [ − H η Λ ( σ ) = Λ ( σ )] 1 { σ Λ ∈ Ω ± Z ± η Λ ; σ Λ c = η Λ c } Λ (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: Λ ν ± , η = µ + lim Λ 5. Proving the P -a.s. absence of finite limit points for the random free energy difference: � log Z + η � � Λ � = ∞ lim � � Z − η Λ Λ (using a local central limit upper-bound for sums of weakly dependent random variables) 6. Representation of the Gibbs measure µ η Λ in terms of the restricted Gibbs measures 1 + Z − η 1 + Z + η � − 1 � − 1 � � ν + , η ν − , η µ η Λ Λ Λ ( σ ) = Λ ( σ ) + Λ ( σ ) Z + η Z − η Λ Λ 9