Relaxation time for competing short- and long-range interactions in the model A dynamic universality class

TL;DR

This paper analyzes the critical relaxation dynamics of a one-dimensional spin model with competing interactions, revealing universal decay behaviors and classifying its dynamical universality class.

Contribution

It provides an analytical derivation of the relaxation decay exponents and confirms the model's classification within the Model A universality class.

Findings

Magnetization decays as t^{-1/2} along the critical line.

At the tricritical point, decay follows t^{-1/4}.

First passage times follow an Arrhenius law.

Abstract

We study the relaxation dynamics at criticality in the one-dimensional spin-$1/2$ Nagle-Kardar model, where short- and long-range interactions can compete. The phase diagram of this model shows lines of first and second-order phase transitions, separated by a tricritical point. We consider Glauber dynamics, focusing on the slowing-down of the magnetization $m$ both along the critical line and at the tricritical point. Starting from the master equation and performing a coarse-graining procedure, we obtain a Fokker-Planck equation for $m$ and the fraction of defects. Using central manifold theory, we analytically show that $m$ decays asymptotically as $t^{-1/2}$ along the critical line, and as $t^{-1/4}$ at the tricritical point. This result implies that the dynamical critical exponent is $z=2$, proving that the macroscopic critical dynamics of the Nagle-Kardar model falls within the dynamic universality class of purely relaxational dynamics with a non-conserved order parameter (model A). Large deviation techniques enable us to show that the average first passage time between local equilibrium states follows an Arrhenius law.