Marginal States in Mean Field Glasses

TL;DR

This paper explores the nature of marginally stable states in mean field glassy systems, developing and connecting various theoretical techniques to better understand their physical properties and the role of soft modes.

Contribution

It provides a unified interpretation of replica symmetry breaking, cavity method, and solution counting in the context of marginal states in mean field glasses.

Findings

Clarifies the physical meaning of the two-group replica symmetry breaking scheme.

Establishes the relation between different methods for analyzing marginal states.

Re-examines the properties of marginal states in the Sherrington-Kirkpatrick model.

Abstract

We study mean field systems whose free energy landscape is dominated by marginally stable states. We review and develop various techniques to describe such states, elucidating their physical meaning and the interrelation between them. In particular, we give a physical interpretation of the two-group replica symmetry breaking scheme and confirm it by establishing the relation to the cavity method and to the counting of solutions of the Thouless-Anderson-Palmer equations. We show how these methods all incorporate the presence of a soft mode in the free energy landscape and interpret the occurring order parameter functions in terms of correlations between the soft mode and the local magnetizations. The general formalism is applied to the prototypical case of the Sherrington-Kirkpatrick-model where we re-examine the physical properties of marginal states under a new perspective.