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arXiv:2501.09975

Out-of-equilibrium critical dynamics of the three-dimensional ${\mathbb Z}_2$ gauge model along critical relaxational flows

Claudio Bonati, H. Panagopoulos, Ettore Vicari

Published Jan 17, 2025• 0 citations
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Abstract

We address the out-of-equilibrium critical dynamics of the three-dimensional lattice ${\mathbb Z}_2$ gauge model, and in particular the critical relaxational flows arising from instantaneous quenches to the critical point, driven by purely relaxational (single-spin-flip Metropolis) upgradings of the link ${\mathbb Z}_2$ gauge variables. We monitor the critical relaxational dynamics by computing the energy density, which is the simplest local gauge-invariant quantity that can be measured in a lattice gauge theory. The critical relaxational flow of the three-dimensional lattice ${\mathbb Z}_2$ gauge model is analyzed within an out-of-equilibrium finite-size scaling framework, which allows us to compute the dynamic critical exponent $z$ associated with the purely relaxational dynamics of the three-dimensional ${\mathbb Z}_2$ gauge universality class. We obtain $z=2.610(15)$, which significantly improves earlier results obtained by other methods, in particular those obtained by analyzing the equilibrium critical dynamics.

DOI
10.48550/arxiv.2501.09975

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Out-of-equilibrium critical dynamics of the three-dimensional Z2{\mathbb Z}_2Z2​ gauge model along critical relaxational flows
We address the out-of-equilibrium critical dynamics of the three-dimensional lattice Z2{\mathbb Z}_2Z2​ gauge model, and in particular the critical relaxational flows arising from instantaneous quenches to the critical point, driven by purely relaxational (single-spin-flip Metropolis) upgradings of the link Z2{\mathbb Z}_2Z2​ gauge variables. We monitor the critical relaxational dynamics by computing the energy density, which is the simplest local gauge-invariant quantity that can be measured in a lattice gauge theory. The critical relaxational flow of the three-dimensional lattice Z2{\mathbb Z}_2Z2​ gauge model is analyzed within an out-of-equilibrium finite-size scaling framework, which allows us to compute the dynamic critical exponent zzz associated with the purely relaxational dynamics of the three-dimensional Z2{\mathbb Z}_2Z2​ gauge universality class. We obtain z=2.610(15)z=2.610(15)z=2.610(15), which significantly improves earlier results obtained by other methods, in particular those obtained by analyzing the equilibrium critical dynamics.
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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.
Three-dimensional Z2{\mathbb Z}_2Z2​-gauge NNN-vector models
We study the phase diagram and critical behaviors of three-dimensional lattice Z2{\mathbb Z}_2Z2​-gauge NNN-vector models, in which an NNN-component real field is minimally coupled with a Z2{\mathbb Z}_2Z2​-gauge link variables. These models are invariant under global O(NNN) and local Z2{\mathbb Z}_2Z2​ transformations. They present three phases characterized by the spontaneous breaking of the global O(NNN) symmetry and by the different topological properties of the Z2{\mathbb Z}_2Z2​-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 spins order, in the regime 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.
Abstract Wegner’s Z 2 gauge model is the earliest formulation of pure lattice gauge theory and predicts the topological nature of the confinement–deconfinement transition. In three dimensions ( D = 3 ), the equilibrium critical behavior of the model is understood in terms of geometrically defined objects, namely loop excitations and Fortuin–Kasteleyn clusters. This work investigates the critical quench dynamics of this model from a geometric perspective, following quenches from both a high-temperature percolation phase and the zero-temperature ground state. Using time-dependent finite-size scaling analysis, we find that the critical non-equilibrium relaxation of the percolation order parameter is governed by a dynamical exponent z p ∼ 2.6 , consistent with that associated with the energy density, z c . Importantly, the value of z p is robust with respect to the initial quench condition and the choice of geometrical objects. Furthermore, we provide a detailed characterization of the kinetics of different geometrical objects during the evolution from the percolation phase. Notably, we observe that the quench dynamics obeys dynamic scaling in terms of a growing lengthscale, ξ p ( t ) ∼ t 1 / z p , despite the absence of a local order parameter.
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