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Physical review. B, Condensed matter

Ising spin glass: Replica-symmetric cluster expansion in finite dimensions

Anil Khurana

Published Sep 1, 1996• 2 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.54.7295

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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.
The n to 0 replica method is used to study the Ising spin glass below its transition temperature. The order parameter correlation functions are shown to contain massless modes ('replicon' modes) within mean-field theory. These become unstable at first order in perturbation theory about the mean-field solution. An alternative formulation of perturbation theory, however, yields a family of new solutions which break, in general, the symmetry between the replicas. Stability requirements single out a unique member of this family which exhibits (i) massless moded, and (ii) a restoration of replica symmetry. Within the framework of the new solution, perturbation theory for the order parameter (Q alpha beta ) breaks down for dimensionalities d<or=4 due to an infrared divergence. Four dimensions is interpreted as the 'lower critical dimensionality' of the theory, below which there can be no order of the Edwards-Anderson kind (1975).
Within the replica approach to mean-field spin-glasses the transition from ergodic high-temperature behaviour to the glassy low-temperature phase is marked by the instability of the replica-symmetric saddle-point. For general spin-glass models with non-Gaussian field distributions the corresponding Hessian is a 2n×2n2^n\times 2^n2n×2n matrix with the number nnn of replicas tending to zero eventually. We block-diagonalize this Hessian matrix using representation theory of the permutation group and identify the blocks related to the spin-glass susceptibility. Performing the limit n→0n\to 0n→0 within these blocks we derive expressions for the de~Almeida-Thouless line of general spin-glass models. Specifying these expressions to the cases of the Sherrington-Kirkpatrick, Viana-Bray, and the Lévy spin glass respectively we obtain results in agreement with previous findings using the cavity approach.
A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and XY\mathrm{XY}XY spins are evaluated using a many-replica procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for TTT greater than Tc{T}_{c}Tc​, the spin-glass transition temperature. At and below Tc{T}_{c}Tc​, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.
A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and XY\mathrm{XY}XY spins are evaluated using a "many-replica" procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for TTT greater than Tc{T}_{c}Tc​, the spin-glass transition temperature. At and below Tc{T}_{c}Tc​, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.
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.
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.