Metastable states in the solvable spin glass model

Abstract

The solutions of the equations of Thouless, Anderson and Palmer represent metastable states of the Sherrington-Kirkpatrick model of an king spin glass. A replica symmetry breaking scheme of the Parisi kind is employed to enumerate their distribution with free energy. The characteristic non-ergodic behaviour of spin glasses arises from their large number of metastable states. For the Sherrington-Kirkpatrick (1975) (SK) infinite-range model of an king spin glass the metastable states are described by the solutions of the Thouless-Anderson-Palmer equations (1 977, referred to as TAP). We have previously shown that there are a large number of these solutions (of order exp aN), where N is the number of spins in the system, and that these solutions fall into two classes (Bray and Moore 1980a, referred to as I). The first class contains those whose free energies kBTf exceed a critical value kBTfc(T). Their properties and distribution with free energy are readily evaluated (see I, also de Dominicis et a1 1980, Tanaka and Edwards 1980). The average number (Ns(f)) of these metastable states of free energy kBTf is equal to the extremum with respect to n of exp(nfN)(Z) where (2) is the bond-averaged nth power of the partition function calculated within the two-group replica symmetry breaking scheme (Bray and Moore 1980b). These metastable states are 'uncorrelated'. The second class of metastable states have free energies f<fc(T) and are correlated, with the Edwards-Anderson order parameter a measure of the correlation between them. It is the purpose of this note to show how such states can be handled by the means of an extension of the Parisi symmetry breaking scheme (Parisi 1979). The N TAP equations for the magnetisation mi of the ith spin are