Zurek's scaling-law in the Ising model out of thermal equilibrium

TL;DR

This study tests Zurek's scaling-law in the Ising model during rapid cooling, finding consistent scaling behavior across dimensions and algorithms, suggesting broader applicability beyond condensed matter systems.

Contribution

The paper demonstrates that Zurek's scaling-law applies to the Ising model's domain dynamics during rapid cooling, extending its validity beyond systems with topological defects.

Findings

Scaling-laws observed in 1, 2, and 3 dimensions

Consistent exponents across different algorithms

Supports broader physical relevance of Zurek's conjecture

Abstract

The Kibble mechanism plays a prominent role in the theory of the early Universe, as an explanation of the possible formation of cosmic strings. Zurek suggested the analogous effect in liquid helium under rapid cooling, and he conjectured - together with del Campo and Kibble - a scaling-law for the relation between the density of remnant topological defects and the cooling rate, when a system passes through its critical temperature. Such scaling-laws were indeed observed in condensed matter experiments. Here we test the validity of Zurek's scaling-law in a different framework. We numerically study the behavior of the Ising model (with classical spins) in 1, 2 and 3 dimensions under rapid cooling. This model does not have topological defects, so we consider the dynamics of domains instead, i.e. we measure their evolution during a cooling process down to zero temperature. For several Markov chain Monte Carlo algorithms, we consistently observe scaling-laws along the lines of Zurek's conjecture, in all dimensions under consideration, which shows that this feature holds more generally than expected. It is highly remarkable that even the exponents of these scaling-laws are consistent for three different algorithms, which hints at a physical meaning.