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Journal de Physique IV (Proceedings)

Cluster approximation for Ising spin glasses

Roberto Baviera, Michele Pasquini, Maurizio Serva

Published Oct 1, 1998• 0 citations
Publisher

Abstract

We report about a new variational method [9] which approximates in a hierarchical way the random Ising spin glass on lattice in d dimensions. At the lowest level our approximation coincides with the Sherrington-Kirkpatrick model, while at the highest level it coincides with the true d-dimensional system. The attention is focused on finite size clusters of spins where the action of the rest of the system is taken into account by a coupling field, which is the variational parameter of the problem.

DOI
10.1051/jp4:1998609

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We report about a new variational method [9] which approximates in a hierarchical way the random Ising spin glass on lattice in d dimensions. At the lowest level our approximation coincides with the Sherrington-Kirkpatrick model, while at the highest level it coincides with the true d-dimensional system. The attention is focused on finite size clusters of spins where the action of the rest of the system is taken into account by a coupling field, which is the variational parameter of the problem.
We introduce a method for computing corrections to the Bethe approximation for spin models on arbitrary lattices. Unlike cluster variational methods, the new approach takes into account fluctuations on all length scales. The derivation of the leading correction is explained and applied to two simple examples: the ferromagnetic Ising model on d -dimensional lattices, and the spin glass on random graphs (both in their high-temperature phases). In the first case we rederive the well known Ginzburg criterion and the upper critical dimension. In the second, we compute finite-size corrections to the free energy.
We study the d-dimensional random Ising model using a suitable type of Bethe-Peierls approximation in the framework of the replica method. We take into account the correct interaction only inside replicated clusters of spins. Our ansatz is that the interaction of the borders of the clusters with the external world can be described via an effective interaction among replicas. The Bethe-Peierls model can be mapped into a single Ising model with a random Gaussian field, whose strength (related to the effective coupling between two replicas) is determined via a self-consistency equation. This allows us to obtain analytic estimates of the internal energy and of the critical temperature in d dimensions. \textcopyright{} 1996 The American Physical Society.
We introduce a method for computing corrections to Bethe approximation for spin models on arbitrary lattices. Unlike cluster variational methods, the new approach takes into account fluctuations on all length scales. The derivation of the leading correction is explained and applied to two simple examples: the ferromagnetic Ising model on d-dimensional lattices, and the spin glass on random graphs (both in their high-temperature phases). In the first case we rederive the well-known Ginzburg criterion and the upper critical dimension. In the second, we compute finite-size corrections to the free energy. UMR 8549, Unité Mixte de Recherche du Centre National de la Recherche Scientifique et de l ’ Ecole
We present and solve the replica symmetric equations in the context of the replica cluster variational method for the 2D random bond Ising model (including the 2D Edwards–Anderson spin-glass model). First, we solve a linearized version of these equations to obtain the phase diagrams of the model on the square and triangular lattices. In both cases, the spin-glass transition temperatures and the multicritical point estimations improve largely over the Bethe predictions. Moreover, we show that this phase diagram is consistent with the behavior of inference algorithms on single instances of the problem. Finally, we present a method to consistently find approximate solutions to the equations in the glassy phase. The method is applied to the triangular lattice down to T = 0, also in the presence of an external field.
We study in Ising spin glasses the finite-size effects near the spin-glass transition in zero field and at the de Almeida-Thouless transition in a field by Monte Carlo methods and by analytical approximations. In zero field, the finite-size scaling function associated with the spin-glass susceptibility of the Sherrington-Kirkpatrick mean-field spin-glass model is of the same form as that of one-dimensional spin-glass models with power-law long-range interactions in the regime where they can be a proxy for the Edwards-Anderson short-range spin-glass model above the upper critical dimension. We also calculate a simple analytical approximation for the spin-glass susceptibility crossover function. The behavior of the spin-glass susceptibility near the de Almeida-Thouless transition line has also been studied, but here we have only been able to obtain analytically its behavior in the asymptotic limit above and below the transition. We have also simulated the one-dimensional system in a field in the non-mean-field regime to illustrate that when the Imry-Ma droplet length scale exceeds the system size one can then be erroneously lead to conclude that there is a de Almeida-Thouless transition even though it is absent.