The Stability of the Replica Symmetric State in Finite Dimensional Spin Glasses

TL;DR

This paper investigates the stability of the replica symmetric state in finite-dimensional spin glasses, showing it is unstable in high dimensions but can be exact, and discusses implications for lower dimensions.

Contribution

It demonstrates the instability of the replica symmetric solution in high dimensions and suggests a possible stable replica symmetry broken phase in lower dimensions.

Findings

Replica symmetric solution unstable in dimensions >6

Exact solutions can be replica-symmetric

Potential for replica symmetry breaking in lower dimensions

Abstract

According to the droplet picture of spin glasses, the low-temperature phase of spin glasses should be replica symmetric. However, analysis of the stability of this state suggested that it was unstable and this instability lends support to the Parisi replica symmetry breaking picture of spin glasses. The finite-size scaling functions in the critical region of spin glasses below T_c in dimensions greater than 6 can be determined and for them the replica symmetric solution is unstable order by order in perturbation theory. Nevertheless the exact solution can be shown to be replica-symmetric. It is suggested that a similar mechanism might apply in the low-temperature phase of spin glasses in less than six dimensions, but that a replica symmetry broken state might exist in more than six dimensions.