Replica-symmetry breaking in long-range glass models without quenched disorder

Abstract

We discuss mean field theory of glasses without quenched disorder focusing on the justification of the replica approach to thermodynamics. We emphasize the assumptions implicit in this method and discuss how they can be verified. The formalism is applied to the long range Ising model with orthogonal coupling matrix. We find the one step replica-symmetry breaking solution and show that it is stable in the intermediate temperature range that includes the glass state but excludes very low temperatures. At very low temperatures this solution becomes unstable and this approach fails.The thermodynamics of glasses without quenched disorder is a long standing problem in statistical physics. The interest to this problem was renewed recently when it was understood that powerful methods developed for the glasses with quenched disorder can be often applied to this problem. [1][2][3][4][5][6]. In both systems the local magnetization in the ground state varies from site to site and different sites are typically non-equivalent. The qualitative reason why glasses without quenched disorder are more difficult to describe theoretically than spin glasses is the following. The mean field theory has to operate with the average magnetization (or its moments), not with quantities which depend on a realization and a particular state. The average quantities appear naturally in spin glasses after averaging over quenched disorder which makes all sites equivalent.A few methods were suggested to overcome this difficulty for the glasses without quenched disorder. First, a mapping of some glass models to the quenched disordered problems was suggested [1], this method has an obvious disadvantage that such mapping is difficult to guess. Second, it was noted that a typical dynamics in a glassy system leads not to a ground state but to one of many metastable states providing an effective averaging mechanism [7] which makes all sites equivalent even for glasses without quenched disorder. This method has a disadvantage that dynamical equations are much more difficult to solve than the statical ones. Very recently the cloning method was proposed that is based on the idea that even at low T a system of m clones might be distributed in its phase space over many low lying metastable states if m is chosen correctly and the properties of all these states are essentially equivalent to those of the ground state. [3][4][5][6] Generally, the partition sum of m weakly coupled clones is F e −N (mβF −S conf (F )) , where sum goes over free energies (per site) of metastable states, F , and S conf (F ) is their configurational entropy (S conf = 1 N ln(N states )). Assuming that dS conf (F )/dF is finite at the lowest F associated with the ground state one needs to chose m ∝ T at low T in order to avoid a complete dominance by a single (ground) state and the problems with site nonequivalence mentioned above. Distributing the system in the phase space provides the effective averaging mechanism in this approach. The main assumptions implicit in this approach are that...

Mean-field glassy phase of the random-field Ising model

“…Our theory is closely related to other recent approaches 20,14,8,7 that address the emergence of glassy phases on a mean-field level. These theories taken as a whole appear to shed light on a number of experimentally relevant systems, and present a fairly complete and consistent picture of glassy behavior.…”

Structural glass on a lattice in the limit of infinite dimensions

“…This makes even a mean field theory difficult to construct because the thermodynamic ground state of the system (or any other particular state) has a unique configuration of the order parameter field, thus making it difficult to describe system in terms of average (and site independent) quantities. Few ways to resolve this difficulty were proposed recently [1][2][3][4][5][6][7]; they all share a common idea that states of the system with the same energy per site (but perhaps different total energies) are essentially equivalent and instead of studying one particular state one can average over all states with the same energy density.There are a few ways to implement averaging over states, the most convenient ones seem to be the clone method [5,6] and the introduction of a small random field conjugated to the order parameter (magnetization in case of spin glasses or density in structural glasses) [7]. The physical idea of the latter is that an infinitesimally small random field applied to a system with many metastable states rearranges the energies of low-lying states making the problem similar to one with quenched disorder.…”

“…If S conf (F ) is concave and dS conf (F )/dF is finite at the lower bound F = F 0 (corresponding to the ground state) the appropriate choice of m (m ∝ T at low T ) leads to a partition function dominated by a small vicinity of any given F > F 0 that still contains thermodynamic number of states providing the averaging mechanism in this approach. In this paper we shall apply the mean field formalism developed in our paper [7] that combines random field approach and the locator expansion developed for the glass physics in [9] to the simplest model of a realistic structural glass. The mean field theory for the structural glasses can be justified formally only in the limit of infinite dimensions; thus, in our derivation of the mean field theory below we shall consider the space of arbitrary dimension, d, and keep only the leading terms in 1/d.…”

“…There are a few ways to implement averaging over states, the most convenient ones seem to be the clone method [5,6] and the introduction of a small random field conjugated to the order parameter (magnetization in case of spin glasses or density in structural glasses) [7]. The physical idea of the latter is that an infinitesimally small random field applied to a system with many metastable states rearranges the energies of low-lying states making the problem similar to one with quenched disorder.…”

“…In this paper we shall apply the mean field formalism developed in our paper [7] that combines random field approach and the locator expansion developed for the glass physics in [9] to the simplest model of a realistic structural glass. The mean field theory for the structural glasses can be justified formally only in the limit of infinite dimensions; thus, in our derivation of the mean field theory below we shall consider the space of arbitrary dimension, d, and keep only the leading terms in 1/d.…”