Dynamics of an Ising Spin Glass on the Bethe Lattice

TL;DR

This paper investigates the low-temperature dynamics of the Ising spin glass on the Bethe lattice using a cavity-like approach, revealing solutions for binary couplings and highlighting complexities for general distributions.

Contribution

It introduces a cavity-like Ansatz for analyzing slow dynamics in spin glasses and derives a perturbative framework near the critical temperature.

Findings

Binary couplings yield a spin glass solution similar to the SK model.

No stable solution found for general coupling distributions.

The nature of the low-temperature phase remains unclear for non-binary couplings.

Abstract

We study the dynamical low temperature behaviour of the Ising spin glass on the Bethe lattice. Starting from Glauber dynamics we propose a cavity like Ansatz that allows for the treatment of the slow (low temperature) part of dynamics. Assuming a continuous phase transitions and ultrametricity with respect to long time scales we approach the problem perturbatively near the critical temperature. The theory is formulated in terms of correlation-response-functions of arbitrary order. They can, however, be broken down completely to products of pair functions depending on two time arguments only. For binary couplings $J=\pm I$ a spin glass solution is found which approaches the corresponding solution for the SK-model in the limit of high connectivity. For more general distributions $P(J)$ no stable or marginal solution of this type appears to exist. The nature of the low temperature phase in this more general case is unclear.