An investigation of the hidden structure of states in a mean field spin glass model

TL;DR

This paper explores the complex geometrical structure of states in a mean field spin glass model at low temperatures, revealing a continuous overlap distribution and insights into the role of threshold states in system dynamics.

Contribution

It introduces a novel entropic method based on TAP equations to analyze the geometrical structure of states in the p-spin spherical model, which was previously inaccessible.

Findings

Overlap distribution is continuous and non-random.

The shape of the threshold landscape is characterized.

Threshold states influence the dynamical behavior of the system.

Abstract

We study the geometrical structure of the states in the low temperature phase of a mean field model for generalized spin glasses, the p-spin spherical model. This structure cannot be revealed by the standard methods, mainly due to the presence of an exponentially high number of states, each one having a vanishing weight in the thermodynamic limit. Performing a purely entropic computation, based on the TAP equations for this model, we define a constrained complexity which gives the overlap distribution of the states. We find that this distribution is continuous, non-random and highly dependent on the energy range of the considered states. Furthermore, we show which is the geometrical shape of the threshold landscape, giving some insight into the role played by threshold states in the dynamical behaviour of the system.