Slow Cooling of an Ising Ferromagnet

TL;DR

This paper derives exact and asymptotic solutions for the domain wall density in a ferromagnetic Ising chain undergoing slow cooling, providing precise insights beyond traditional Kibble-Zurek approximations.

Contribution

It presents an exact integral equation for domain wall density under arbitrary annealing and its asymptotic solution for very slow cooling, advancing understanding of phase transition dynamics.

Findings

Exact integral equation for domain wall density derived

Asymptotic behavior of domain walls at zero temperature obtained

Kibble-Zurek predictions validated by exact asymptotics

Abstract

A ferromagnetic Ising chain which is endowed with a single-spin-flip Glauber dynamics is investigated. For an arbitrary annealing protocol, we derive an exact integral equation for the domain wall density. This integral equation admits an asymptotic solution in the limit of extremely slow cooling. For instance, we extract an asymptotic of the density of domain walls at the end of the cooling procedure when the temperature vanishes. Slow annealing is usually studied using a Kibble-Zurek argument; in our setting, this argument leads to approximate predictions which are in good agreement with exact asymptotics.