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arXiv:cond-mat/9605158 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Beyond the mean field approximation for spin glasses

Maurizio Serva, Giovanni Paladin

Published Nov 1, 1996• 7 citations
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Abstract

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.

DOI
10.1103/physreve.54.4637

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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.
The partition function of the 2D Ising model with random nearest-neighbour coupling is expressed in the dual lattice made of square plaquettes. The dual model is solved in the mean field and in different types of Bethe-Peierls approximations, using the replica method. The application of methods of mean-field type to Ising models allows one to obtain very accurate approximations of the thermodynamic quantities. However, in the presence of quenched disorder this approach is difficult to implement. In this paper, we show that rather good results can be obtained after performing a duality transformation of the Ising model with random nearest-neighbour coupling that assumes the values Jij D1 with equal probability. The model is thus defined on a dual lattice where the spin variables are attached to the square plaquettes. The advantage is that the quadratic term of the dual Hamiltonian has constant coefficients instead of random ones. It is therefore possible to use the standard methods of the mean field to estimate the quenched free energy. Our results can be generalized to higher dimensions, although the approximations become rougher, because the number of plaquette spins over the number of interaction links increases with the dimensionality (1). In particular, we obtain an extremely good estimate of the ground- state energy of the random Ising model, by applying the Bethe-Peierls approximation where part of the short-range order is taken into account. In section 1, we introduce the dual lattice made of elementary square plaquettes. On this lattice the partition function can be expressed as a function of the inverse temperature e D 1 ln tanh.1=T/ whereT D 1 is the temperature of the original lattice. In section 2, we apply the mean-field approximation to the dual model, using the replica method. In section 3, we introduce the Bethe-Peierls approximation. This allows us to obtain a very precise estimate of the ground-state energy of the two-dimensional Ising model with random coupling. In section 4, we show that it is possible to improve the Bethe-Peierls approximation by introducing an interaction between different replicas. In section 5, the reader will find some remarks and conclusions.
We construct a cluster-mean-field theory for short-range spin glasses in the replica space, in which only the effective fields which are conjugate to the order parameters are taken into account. In the one-step replica-symmetry breaking (RSB) scheme just below the critical temperature, the solutions of the self-consistent equations for general clusters are given in terms of the thermal and quenched random averages of spin products. From the comparison of the free energies we have introduced, we find the (one-step) RSB solution is more stable than the replica-symmetric (RS) solution for arbitrary clusters so long as the former exists. Analytic solutions of the Bethe cluster approximation suggest that the RSB effects are rather enhanced by short-range fluctuations.
The replica-symmetric cluster expansion for the Ising spin glass is reviewed and explained. The expansion allows computation of various quantities and correlations directly in finite dimensions, without recourse to the mean-field theory or the saddle point. The expansion for the free energy is recovered with the use of external fields that couple to the replica-symmetric Edwards-Anderson order parameter. The expansion scheme is applied to the Edwards-Anderson susceptibility. The expansion diverges in the mean-field [infinite dimensionality] limit. The divergent expansion, when naively summed, yields the negative replicon mass of the mean-field theory. This so-called Almeida-Thouless instability of the replica-symmetric mean-field theory is argued to be absent in finite dimensions. The argument, similar to the one already reported for the entropy, depends on the low-temperature behavior of terms in the expansion at finite dimensions. \textcopyright{} 1996 The American Physical Society.
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 consider a system of linear chains of Ising spins with short-range nearest-neighbor ferromagnetic interactions of strength K along the chains, and with infinite-range random interchain interactions of a Sherrington-Kirkpatrick type. Also included in the model is a local Gaussian random field with variance \ensuremath{\Delta}. In the replica symmetric case, the system is mapped exactly on the one-dimensional Ising model in a random field and a recursion method for calculating the spin-correlation functions is developed. For \ensuremath{\Delta}=0, an analytic relation for freezing temperature Tf{\mathit{T}}_{\mathit{f}}Tf​(K) is obtained. The replica-symmetric spin-glass phase is shown to be stable above the freezing temperature Tf{\mathit{T}}_{\mathit{f}}Tf​(K,\ensuremath{\Delta}), which is determined numerically.
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.