Instantons in the Langevin dynamics: an application to spin glasses

TL;DR

This paper introduces a new method using instanton techniques within Langevin dynamics to compute transition probabilities in spin glasses, simplifying complex calculations and applying them to the spherical Sherrington-Kirkpatrick model.

Contribution

It develops a novel instanton-based approach for analyzing barrier transitions in spin glasses within dynamical theory, including a simplification via time-reversal mapping.

Findings

Derived equations for instantons in spin-glass dynamics

Mapped instanton processes to time-reversed processes for simplification

Calculated transition probabilities consistent with physical expectations

Abstract

We develop a general technique to calculate the probability of transitions over the barriers in spin-glasses in the framework of the dynamical theory. We use Lagrangian formulation of the instanton dynamics in which the transitions are represented by instantons. We derive the full set of the equations that determine the instantons but instead of solving them directly we prove that an instanton process can be mapped into a usual process going back in time which simplifies the problem significantly. We apply this general considerations to a simple example of the spherical Sherrington-Kirkpatrick model and we find the probability of the transition between the metastable states which is in agreement with physical expectations.