Distribution of barrier heights in infinite-range spin glass models

Abstract

The spherical model of a spin glass with infinite-range interactions is studied to obtain the distribution of barrier heights between time-reversed states. The problem is shown to be equivalent at leading order to that of finding the distribution of the difference between the largest two eigenvalues of the exchange matrix. For the case where the exchange matrix is Gaussian orthogonal an approximate distribution is quoted and verified numerically. The barriers between time-reversed states scale with system size, N, as (1-T/Tc)N1/3. The barrier heights in the Sherrington and Kirkpatrick model of a spin glass are also studied. Using the mean-field equations of Thouless, Anderson and Palmer (1977) a perturbation analysis in terms of the eigenvectors of the Hessian is performed. In the scaling limit T to Tc, N to infinity with (1-T/Tc)N1/3 fixed, macroscopic condensation into the largest eigenvector occurs and the barrier between time-reversed states scales with system size as (1-T/Tc)N1/3.