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Progress of Theoretical Physics Supplement

Thermodynamics of 2dIsing Spin Glasses with Binary Couplings on Large Lattices Using Exact Computations of Partition Functions

Jovanka Lukic, Enzo Marinari, Olivier Martin

Published Jan 1, 2005• 0 citations
Publisher

Abstract

We study 2d Ising spin glasses with binary couplings, using exact computations of the partition functions on large lattices with periodic boundary conditions. We describe the algorithm used to compute the partition functions and then give some of our main results. A Migdal-Kadanoff approximation illustrates further the anomalous scaling of these systems.

DOI
10.1143/ptps.157.17

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We study 2d Ising spin glasses with binary couplings, using exact computations of the partition functions on large lattices with periodic boundary conditions. We describe the algorithm used to compute the partition functions and then give some of our main results. A Migdal-Kadanoff approximation illustrates further the anomalous scaling of these systems.
We compute the exact partition function of 2d Ising spin glasses with binary couplings. In these systems, the ground state is highly degenerate and is separated from the first excited state by a gap of size 4J. Nevertheless, we find that the low temperature specific heat density scales as exp(-2J/T), corresponding to an "effective" gap of size 2J; in addition, an associated crossover length scale grows as exp(J/T). We justify these scalings via the degeneracy of the low lying excitations and by the way low energy domain walls proliferate in this model.
We analyze the exact partition function of the anisotropic Ising model on finite M × N rectangular lattices under four different boundary conditions (periodic–periodic (pp), periodic–antiperiodic (pa), antiperiodic–periodic (ap) and antiperiodic–antiperiodic (aa)) obtained by Kaufman (1949 Phys. Rev. 76 1232), Wu and Hu (2002 J. Phys. A: Math. Gen. 35 5189) and Kastening (2002 Phys. Rev. E 66 057103)). We express the partition functions in terms of the partition functions Zα, β(J, k) with (α, β) = (0, 0), (1/2, 0), (0, 1/2) and (1/2, 1/2), J is an interaction coupling and k is an anisotropy parameter. Based on such expressions, we then extend the algorithm of Ivashkevich et al (2002 J. Phys. A: Math. Gen. 35 5543) to derive the exact asymptotic expansion of the logarithm of the partition function for all boundary conditions mentioned above. Our result is f = fbulk + ∑∞p = 0fp(ρ, k)S−p − 1, where f is the free energy of the system, fbulk is the free energy of the bulk, S = MN is the area of the lattice and ρ = M/N is the aspect ratio. All coefficients in this expansion are expressed through analytical functions. We have introduced the effective aspect ratio ρeff = ρ/sinh 2Jc and show that for pp and aa boundary conditions all finite size correction terms are invariant under the transformation ρeff → 1/ρeff.This article is part of ‘Lattice models and integrability’, a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday.
The exact partition functions of the Ising model on L × L square lattices with free boundary conditions are evaluated up to L = 22 for the first time by using a microcanonical transfer matrix and counting all possible spin states. The total number of states is 2 22×22 = 2 484 ≈ 5.0 × 10 145 for L = 22. Using the partition function zeros in the complex temperature plane of the Ising model on L × L square lattices with free boundary conditions, we analyze the critical behavior of the square-lattice Ising antiferromagnet and compare them with the results for periodic boundary conditions (the most popular boundary conditions). Our results show clearly that free boundary conditions are much more efficient than periodic boundary conditions in the study of an antiferromagnet.
We approximate a two-dimensional spin glass by tiling an infinite lattice with large identical unit cells. The interactions within the unit cell are chosen at random, just as when one studies finite-size systems with periodic boundary conditions. But here the unit cells are instead connected to form an infinite lattice, so one may examine correlations on all length scales, and the system can have true phase transitions. For such approximations to the Ising spin glass on the square lattice, we apply the free-fermion method of Onsager and the anticommuting operator approach of Kaufman to obtain numerically exact results for each realization of the quenched disorder. Each such sample shows one or more critical points, with the distribution of critical temperatures scaling with the unit cell size, consistent with what is expected from the scaling theory of low-dimensional spin glasses. Due to ``chaos,'' the correlations between unit cells can change sign with changing temperature. We examine the scaling of this chaos with unit cell size. Many samples have multiple critical points due to the interactions between unit cells changing sign at temperatures within the ordered phases.