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arXiv:2501.09575 · Physical review. B./Physical review. B

Critical relaxational dynamics at the continuous transitions of three-dimensional spin models with Z 2 gauge symmetry

Claudio Bonati, Andrea Pelissetto C, Ettore Vicari

Published Mar 11, 2025• 8 citations
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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 ${\mathbb{Z}}_{2}$ 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---$z\ensuremath{\approx}2.02$ 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 non-gauge-invariant nature of the order parameter at the transition.

DOI
10.1103/physrevb.111.115129

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We characterize the dynamic universality classes of a relaxational dynamics under equilibrium conditions at the continuous transitions of three-dimensional (3D) spin systems with Z2{\mathbb{Z}}_{2}Z2​ gauge symmetry. In particular, we consider the pure lattice Z2{\mathbb{Z}}_{2}Z2​ gauge model and the lattice Z2{\mathbb{Z}}_{2}Z2​ 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 zzz that characterizes the critical slowing down of the dynamics as the continuous transition is approached. At the topological Z2{\mathbb{Z}}_{2}Z2​ gauge transitions we find z=2.55(6)z=2.55(6)z=2.55(6). Therefore, the dynamics is significantly slower than in Ising systems---z\ensuremath≈2.02z\ensuremath{\approx}2.02z\ensuremath≈2.02 for the 3D Ising universality class---although 3D Z2{\mathbb{Z}}_{2}Z2​ gauge systems and Ising systems have the same static critical behavior because of duality. As for the nontopological transitions in the 3D Z2{\mathbb{Z}}_{2}Z2​ 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 non-gauge-invariant nature of the order parameter at the transition.
We investigate the critical relaxational dynamics of the three-dimensional (3D) lattice Z_{N} gauge models with N=6 and N=8, whose equilibrium critical behavior at their topological transitions belongs to the inverted XY (IXY) universality class [this is also the universality class of the continuous transitions of the 3D lattice U(1) gauge Higgs models with a one-component complex scalar field], which is connected to the standard XY universality class by a nonlocal duality relation of the partition functions. Specifically, we consider the purely relaxational dynamics realized by a locally reversible Metropolis dynamics, as commonly used in Monte Carlo simulations. To determine the corresponding dynamic exponent z, we focus on the out-of-equilibrium critical relaxational flows arising from instantaneous quenches to the critical point, which are analyzed within an out-of-equilibrium finite-size scaling framework. We obtain the estimate z=2.59(3). A numerical analysis of the equilibrium critical dynamics give consistent, but less accurate, results. This dynamic exponent is expected to characterize the critical slowing down of the purely relaxational dynamics of all topological transitions that belong to the 3D IXY universality class. We note that this result implies that the critical relaxational dynamics of the 3D IXY universality class is slower than that of the standard 3D XY universality class, whose relaxational dynamic exponent z≈2.02 is significantly smaller, although they share the same length-scale critical exponent ν≈0.6717.
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We study the phase diagram and critical behaviors of three-dimensional lattice Z2{\mathbb{Z}}_{2}Z2​-gauge NNN-vector models, in which an NNN-component real field is minimally coupled with Z2{\mathbb{Z}}_{2}Z2​-gauge link variables. These models are invariant under global O(N\mathrm{O}(NO(N) and local Z2{\mathbb{Z}}_{2}Z2​ transformations. They present three phases characterized by the spontaneous breaking of the global O(N\mathrm{O}(NO(N) symmetry and by the different topological properties of the Z2{\mathbb{Z}}_{2}Z2​-gauge correlations. We address the nature of the three transition lines separating the three phases. The theoretical predictions are supported by numerical finite-size scaling analyses of Monte Carlo data for the N=2N=2N=2 model. In this case, continuous transitions can be observed along both transition lines where the NNN-component spins order, in the regimes of small and large inverse gauge coupling KKK. Even though these continuous transitions belong to the same XYXYXY universality class, their critical modes turn out to be different. When the gauge variables are disordered (small KKK), the relevant order-parameter field is a gauge-invariant bilinear combination of the vector field. On the other hand, when the gauge variables are ordered (large KKK), the order-parameter field is the gauge-dependent NNN-vector field, whose critical behavior can only be probed by using a stochastic gauge fixing that reduces the gauge freedom.
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