Counting metastable states of Ising spin glasses on arbitrary graphs

TL;DR

This paper introduces a field-theoretical method to compute the number of local energy minima in Ising spin glasses on arbitrary graphs, revealing dependence on local graph properties.

Contribution

We develop a systematic field-theoretical approach to count metastable states in Ising spin glasses on arbitrary graphs, extending previous methods to more general graph structures.

Findings

Accurate computation of local minima for various graph types

Number of minima depends mainly on local graph properties

Method applicable to diverse graph structures

Abstract

Using a field-theoretical representation of the Tanaka-Edwards integral we develop a method to systematically compute the number N_s of 1-spin-stable states (local energy minima) of a glassy Ising system with nearest-neighbor interactions and random Gaussian couplings on an arbitrary graph. In particular, we use this method to determine N_s for K-regular random graphs and d-dimensional regular lattices for d=2,3. The method works also for other graphs. Excellent accuracy of the results allows us to observe that the number of local energy minima depends mainly on local properties of the graph on which the spin glass is defined.