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arXiv:1607.00617 · Europhysics Letters (EPL)

Critical dynamics of classical systems under slow quench

Priyanka, Kavita Jain

Published Oct 1, 2016• 7 citations
arXivHTMLPDF

Abstract

We study the slow quench dynamics of a one-dimensional nonequilibrium lattice gas model which exhibits a phase transition in the stationary state between a fluid phase with homogeneously distributed particles and a jammed phase with a macroscopic hole cluster. Our main result is that in the critical region ( i.e. , at the critical point and in its vicinity) where the dynamics are assumed to be frozen in the standard Kibble-Zurek argument, the defect density exhibits an algebraic decay in the inverse annealing rate with an exponent that can be understood using critical coarsening dynamics. However, in a part of the critical region in the fluid phase, the standard Kibble-Zurek scaling holds. We also find that when the slow quench occurs deep into the jammed phase, the defect density behavior is explained by the rapid quench dynamics in this phase.

DOI
10.1209/0295-5075/116/26003

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We study the slow quench dynamics of a one-dimensional nonequilibrium lattice gas model which exhibits a phase transition in the stationary state between a fluid phase with homogeneously distributed particles and a jammed phase with a macroscopic hole cluster. Our main result is that in the critical region ( i.e. , at the critical point and in its vicinity) where the dynamics are assumed to be frozen in the standard Kibble-Zurek argument, the defect density exhibits an algebraic decay in the inverse annealing rate with an exponent that can be understood using critical coarsening dynamics. However, in a part of the critical region in the fluid phase, the standard Kibble-Zurek scaling holds. We also find that when the slow quench occurs deep into the jammed phase, the defect density behavior is explained by the rapid quench dynamics in this phase.
Abstract While a large number of studies have focused on the nonequilibrium dynamics of a system when it is quenched instantaneously from a disordered phase to an ordered phase, such dynamics have been relatively less explored when the quench occurs at a finite rate. Here, we study the slow quench dynamics in two paradigmatic models of classical statistical mechanics, a one-dimensional kinetic Ising model and a mean-field zero-range process, when the system is annealed slowly to the critical point. Starting from the time evolution equations for the spin–spin correlation function in the Ising model and the mass distribution in the zero-range process, we derive the Kibble–Zurek scaling laws. We then test a recent proposal that critical coarsening, which is ignored in the Kibble–Zurek argument, plays a role in the nonequilibrium dynamics close to the critical point. We find that the defect density in the Ising model and a scaled mass distribution in the zero-range process decay linearly to their respective values at the critical point with the time remaining until the end of the quench provided the final quench point is approached sufficiently fast, and sublinearly otherwise. As the linear scaling for the approach to the critical point also holds when a system following an instantaneous quench is allowed to coarsen for a finite time interval, we conclude that critical coarsening captures the scaling behavior in the vicinity of the critical point if the annealing is not too slow.
Using Monte Carlo simulations, we investigate the dynamical properties of the Baxter-Wu model under slow quenches, i.e., tuning the temperature slowly. For the cooling process, the scaling behavior of the excess defect density in the critical region aligns well with the predictions of the Kibble-Zurek (KZ) mechanism. However, the scaling behavior of the excess defect density after exiting the impulse regime does not follow from a simple interplay between the KZ mechanism and the coarsening dynamics; the system undergoes a decay close to a power-law form with an exponent that is significantly different from the coarsening exponent observed in instantaneous quenching. For the heating process, we show that if the system starts from its ground state, the relevant exponents describing the KZ mechanism are identical to those in the cooling scenario. Furthermore, in the heating process, we find that the system does not directly enter the adiabatic regime after leaving the impulse regime but instead passes through a crossover regime with an exponential decay of the excess defect density. If the initial state is ordered but not the ground state of the system, the defect density exhibits a good scaling behavior, but the relevant exponents do not conform to the predictions of the KZ mechanism.
In the course of a nonequilibrium continuous phase transition, the dynamics ceases to be adiabatic in the vicinity of the critical point as a result of the critical slowing down (the divergence of the relaxation time in the neighborhood of the critical point). This enforces a local choice of the broken symmetry and can lead to the formation of topological defects. The Kibble–Zurek mechanism (KZM) was developed to describe the associated nonequilibrium dynamics and to estimate the density of defects as a function of the quench rate through the transition. During recent years, several new experiments investigated the formation of defects in phase transitions induced by a quench both in classical and quantum mechanical systems. At the same time, some established results were called into question. We review and analyze the Kibble–Zurek mechanism focusing in particular on this surge of activity, and suggest possible directions for further progress.
Universal scaling laws govern the density of topological defects generated while crossing an equilibrium continuous phase transition. The Kibble-Zurek mechanism (KZM) predicts the dependence on the quench time for slow quenches. By contrast, for fast quenches, the defect density scales universally with the amplitude of the quench. We show that universal scaling laws apply to dynamic phase transitions driven by an oscillating external field. The difference in the energy response of the system to a periodic potential field leads to energy absorption, spontaneous breaking of symmetry, and its restoration. We verify the associated universal scaling laws, providing evidence that the critical behavior of nonequilibrium phase transitions can be described by time-average critical exponents combined with the KZM. Our results demonstrate that the universality of critical dynamics extends beyond equilibrium criticality, facilitating the understanding of complex nonequilibrium systems.
We propose a theory to explain the experimental observed deviation from the Kibble-Zurek mechanism (KZM) scaling in rapidly quenched critical phase transition dynamics. There is a critical quench rate τQc1τ_{Q}^{c1}τQc1​ above it the KZM scaling begins to appear. Smaller than τQc1τ_Q^{c1}τQc1​, the defect density nnn is a constant independent of the quench rate but depends on the final temperature TfT_fTf​ as n∝LdεTfdνn \propto L^d ε_{T_f} ^{d ν}n∝LdεTf​dν​, the freeze out time t^\hat{t}t^ admits the scaling law t^∝εTf−νz\hat{t} \propto ε_{T_f}^{-νz}t^∝εTf​−νz​ where ddd is the spatial dimension, εTf=(1−Tf/Tc)ε_{T_f}= (1-T_f/T_c)εTf​​=(1−Tf​/Tc​) is the dimensionless reduced temperature, LLL is the sample size, ννν and zzz are spatial and dynamical critical exponents. Quench from TcT_cTc​, the critical rate is determined by the final temperature TfT_fTf​ as $τ_Q^{c1} \propto ε_{T_f}^{-(1+z ν)} $. All the scaling laws are verified in a rapidly quenched superconducting ring via the AdS/CFT correspondence.
We consider the finite-time quench dynamics in the quantum transverse field Ising model which exhibits a second order phase transition from a paramagnetic to a ferromagnetic phase, as the transverse magnetic field is decreased. These dynamics have been thoroughly investigated in previous studies when the critical point is crossed during the quench; here, we quench the system from deep in the paramagnetic phase to just above the critical field so that the system remains in the gapped phase throughout the quench duration. On linearly quenching the infinitely large system, we find that the behavior of mean longitudinal defect density and mean transverse magnetization at the end of the quench falls into three distinct scaling regimes as the quench time is increased. For sufficiently small quench times, these observables remain roughly constant, but for larger quench times, a crossover occurs from the Kibble-Zurek scaling law to the quadratic quench rate law when the Kibble-Zurek time is of the order of relaxation time at the final quench field. These results are shown analytically using power series and uniform asymptotic expansions of the exact solution of the model, and also compared with an adiabatic perturbation theory in the third regime. We find that the above mentioned scaling regimes hold for quenches within the ferromagnetic phase also, and provide a general scaling argument for crossover from the Kibble-Zurek regime to an adiabatic regime for slow quenches within a gapped phase.