The Effect of Pure State Structure on Nonequilibrium Dynamics

TL;DR

This paper reviews rigorous results on how the structure of pure states in spin glasses influences their nonequilibrium dynamics, highlighting the challenges in achieving equilibrium and the impact of initial conditions.

Contribution

It provides a detailed analysis of the relationship between pure state structure and nonequilibrium behavior in spin glasses, with new insights into the conditions affecting equilibration.

Findings

Almost all initial configurations are on the boundary of multiple pure states.

No pure state equilibration occurs unless there are uncountably many pure states with zero overlap.

The results shed light on the difficulty of spin glass equilibration.

Abstract

Motivated by short-range Ising spin glasses, we review some rigorous results and their consequences for the relation between the number/nature of equilibrium pure states and nonequilibrium dynamics. Two of the consequences for spin glass dynamics following a deep quench to a temperature with broken spin flip symmetry are: (1) Almost all initial configurations lie on the boundary between the basins of attraction of multiple pure states. (2) Unless there are uncountably many pure states with almost all pairs having zero overlap, there can be no equilibration to a pure state as time goes to infinity. We discuss the relevance of these results to the difficulty of equilibration of spin glasses. We also review some results concerning the ``nature vs. nurture'' problem of whether the large-time behavior of both ferromagnets and spin glasses following a deep quench is determined more by the initial configuration or by the dynamics realization.