Critical relaxational dynamics at the continuous transitions of three-dimensional spin models with ${\mathbb Z}_2$ gauge symmetry

TL;DR

This paper investigates the dynamic critical behavior of 3D ${ m Z}_2$-gauge spin models at continuous phase transitions, revealing significantly slower dynamics at topological transitions compared to Ising models, and identifying the universality class for non-topological transitions.

Contribution

It provides the first detailed characterization of the dynamic universality classes of 3D ${ m Z}_2$-gauge models at continuous transitions, including the critical exponent $z$ and the comparison with ungauged models.

Findings

Topological ${ m Z}_2$-gauge transitions have $z=2.55(6)$, indicating slower dynamics.

Non-topological transitions in ${ m Z}_2$-gauge XY models share the same dynamic universality class as ungauged XY systems.

The static critical behavior remains the same due to duality, despite differences in dynamics.

Abstract

We characterize the dynamic universality classes of a relaxational dynamics under equilibrium conditions at the continuous transitions of three-dimensional (3D) spin systems with a ${\mathbb Z}_2$-gauge symmetry. In particular, we consider the pure lattice ${\mathbb Z}_2$-gauge model and the lattice ${\mathbb Z}_2$-gauge XY model, which present various types of transitions: topological transitions without a local order parameter and transitions characterized by both gauge-invariant and non-gauge-invariant XY order parameters. We consider a standard relaxational (locally reversible) Metropolis dynamics and determine the dynamic critical exponent $z$ that characterizes the critical slowing down of the dynamics as the continuous transition is approached. At the topological ${\mathbb Z}_2$-gauge transitions we find $z=2.55(6)$. Therefore, the dynamics is significantly slower than in Ising systems -- $z\approx 2.02$ for the 3D Ising universality class -- although 3D ${\mathbb Z}_2$-gauge systems and Ising systems have the same static critical behavior because of duality. As for the nontopological transitions in the 3D ${\mathbb Z}_2$-gauge XY model, we find that their critical dynamics belong to the same dynamic universality class as the relaxational dynamics in ungauged XY systems, independently of the gauge-invariant or nongauge-invariant nature of the order parameter at the transition.