On the number of metastable states in spin glasses

TL;DR

This paper demonstrates that Bray and Moore's formulas for counting metastable states in spin glasses can be derived using a replica approach with broken symmetry, providing new solutions for the SK model.

Contribution

It introduces a replica-based derivation of Bray and Moore's formulas and finds solutions where the free energy matches the static free energy, applicable to mean-field spin glasses.

Findings

Derived Bray and Moore's formulas via replica method

Found solutions with lower band-edge free energy equal to static free energy

Results applicable to all mean-field spin glasses

Abstract

In this letter, we show that the formulae of Bray and Moore for the average logarithm of the number of metastable states in spin glasses can be obtained by calculating the partition function with $m$ coupled replicas with the symmetry among these explicitly broken according to a generalization of the `two-group' ansatz. This equivalence allows us to find solutions of the BM equations where the lower `band-edge' free energy equals the standard static free energy. We present these results for the Sherrington-Kirkpatrick model, but we expect them to apply to all mean-field spin glasses.