Free energy barriers in the Sherrington-Kirkpatrick model

TL;DR

This paper analyzes the free energy landscape of the SK spin glass model, revealing small barriers for certain states and larger barriers for marginally stable states, with implications for numerical studies and replica symmetry breaking.

Contribution

It provides a detailed calculation of free energy barriers in the SK model and discusses their implications for state stability and numerical observations.

Findings

Small barriers (~1/N^2) for states with free energy above f_c.

States with marginal stability have barriers at least of order 1.

Potentially large barriers scaling as N^{1/3} below f_c.

Abstract

The free energy landscape of the Sherrington-Kirkpatrik (SK) Ising spin glass is simple in the framework of the Thouless-Anderson-Palmer (TAP) equations as each solution (which are minima of the free energy) has associated with it a nearby index-one saddle point. The free energy barrier to escape the minimum is just the difference between the saddle point free energy and that at its associated minimum. This difference is calculated for the states with free energies $f > f_c$. It is very small for these states, decreasing as $1/N^2$, where $N$ is the number of spins in the system. These states are not marginally stable. We argue that such small barriers are why numerical studies never find these states when $N$ is large. Instead the states which are found are those which have marginal stability. For them the barriers are at least of $O(1)$. $f_c$ is the free energy per spin below which the states develop broken replica-symmetry like overlaps with each other. In the regime $f < f_c$ we can only offer some possibilities based around scaling arguments. One of these suggest that the barriers might become as large as $N^{1/3}$. That might be consistent with recent numerical studies on the Viana-Bray model, which were at variance with the expectations of Cugliandolo and Kurchan for the SK model.