Distribution of Kinks in an Ising Ferromagnet After Annealing and the Generalized Kibble-Zurek Mechanism

TL;DR

This paper analytically and numerically studies the distribution of kinks in a one-dimensional Ising ferromagnet after finite-time annealing, revealing universal scaling laws and deviations from thermal equilibrium.

Contribution

It provides a detailed analysis of kink distribution and universal scaling in Ising ferromagnets under various cooling schedules, extending the Kibble-Zurek mechanism.

Findings

Kink number distribution is Poissonian in the slow cooling limit.

Universal power-law scaling of cumulants with quench time.

Exponential cooling offers the most efficient shortcuts to cooling.

Abstract

We consider the annealing dynamics of a one-dimensional Ising ferromagnet induced by a temperature quench in finite time. In the limit of slow cooling, the asymptotic two-point correlator is analytically found under Glauber dynamics, and the distribution of the number of kinks in the final state is shown to be consistent with a Poissonian distribution. The mean kink number, the variance, and the third centered moment take the same value and obey a universal power-law scaling with the quench time in which the temperature is varied. The universal power-law scaling of cumulants is corroborated by numerical simulations based on Glauber dynamics for moderate cooling times away from the asymptotic limit, when the kink-number distribution takes a binomial form. We analyze the relation of these results to physics beyond the Kibble-Zurek mechanism for critical dynamics, using the kink number distribution to assess adiabaticity and its breakdown. We consider linear, nonlinear, and exponential cooling schedules, among which the latter provides the most efficient shortcuts to cooling in a given quench time. The non-thermal behavior of the final state is established by considering the trace norm distance to a canonical Gibbs state.