We address the out-of-equilibrium critical dynamics of the three-dimensional lattice 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 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 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 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.
We address the out-of-equilibrium critical dynamics of the three-dimensional lattice 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 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 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 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.
Out-of-equilibrium critical dynamics of the three-dimensional Z2 gauge model along critical relaxational flows
We address the out-of-equilibrium critical dynamics of the three-dimensional lattice Z2 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 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 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 Z2 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.
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.
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 gauge symmetry. In particular, we consider the pure lattice Z2 gauge model and the lattice 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 z that characterizes the critical slowing down of the dynamics as the continuous transition is approached. At the topological Z2 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 Z2 gauge systems and Ising systems have the same static critical behavior because of duality. As for the nontopological transitions in the 3D 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.
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.
We study the out-of-equilibrium behavior of statistical systems along critical relaxational flows arising from instantaneous quenches of the temperature T to the critical point T_{c}, starting from equilibrium conditions at time t=0. In the case of soft quenches, i.e., when the initial temperature T is assumed sufficiently close to T_{c} (to keep the system within the critical regime), the critical modes develop an out-of-equilibrium finite-size-scaling (FSS) behavior in terms of the rescaled time variable Θ=t/L^{z}, where t is the time interval after quenching, L is the size of the system, and z is the dynamic exponent associated with the dynamics. However, the realization of this picture is less clear when considering the energy density, whose equilibrium scaling behavior (corresponding to the starting point of the relaxational flow) is generally dominated by a temperature-dependent regular background term or mixing with the identity operator. These issues are investigated by numerical analyses within the three-dimensional lattice N-vector models, for N=3 and 4, which provide examples of critical behaviors with negative values of the specific-heat critical exponent α, implying that also the critical behavior of the specific heat gets hidden by the background term. The results show that, after subtraction of its asymptotic critical value at T_{c}, the energy density develops an asymptotic out-of-equilibrium FSS in terms of Θ as well, whose scaling function appears singular in the small-Θ limit.
We study the purely relaxational dynamics (model A) at criticality in three-dimensional disordered Ising systems whose static critical behaviour belongs to the randomly diluted Ising universality class. We consider the site-diluted and bond-diluted Ising models, and the ± J Ising model along the paramagnetic–ferromagnetic transition line. We perform Monte Carlo simulations at the critical point using the Metropolis algorithm and study the dynamic behaviour in equilibrium at various values of the disorder parameter. The results provide a robust evidence of the existence of a unique model-A dynamic universality class which describes the relaxational critical dynamics in all considered models. In particular, the analysis of the size dependence of suitably defined autocorrelation times at the critical point provides the estimate z = 2.35(2) for the universal dynamic critical exponent. We also study the off-equilibrium relaxational dynamics following a quench from to T c . In agreement with the field-theory scenario, the analysis of the off-equilibrium dynamic critical behaviour gives an estimate of z that is perfectly consistent with the equilibrium estimate z = 2.35(2).