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arXiv:1001.0693 · Physical Review E

Kibble-Zurek mechanism and infinitely slow annealing through critical points

Giulio Biroli, Leticia F. Cugliandolo, Alberto Sicilia

Published May 6, 2010• 83 citations
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Abstract

We revisit the Kibble-Zurek mechanism by analyzing the dynamics of phase ordering systems during an infinitely slow annealing across a second-order phase transition. We elucidate the time and cooling rate dependence of the typical growing length, and we use it to predict the number of topological defects left over in the symmetry broken phase as a function of time, both close and far from the critical region. Our results extend the Kibble-Zurek mechanism and reveal its limitations.

DOI
10.1103/physreve.81.050101

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We revisit the Kibble-Zurek mechanism by analyzing the dynamics of phase ordering systems during an infinitely slow annealing across a second-order phase transition. We elucidate the time and cooling rate dependence of the typical growing length, and we use it to predict the number of topological defects left over in the symmetry broken phase as a function of time, both close and far from the critical region. Our results extend the Kibble-Zurek mechanism and reveal its limitations.
The Kibble-Zurek mechanism (KZM) successfully predicts the density of topological defects deposited by the phase transitions, but it is not clear why. Its key conjecture is that, near the critical point of the second-order phase transition, critical slowing down will result in a period when the system is too sluggish to follow the potential that is changing faster than its reaction time. The correlation length at the freeze-out instant t^\hat tt^ when the order parameter catches up with the post-transition broken symmetry configuration is then decisive, determining when the mosaic of broken symmetry domains locks in topological defects. To understand why the KZM works so well we analyze Landau-Ginzburg model and show why temporal evolution of the order parameter plays such a key role. The analytical solutions we obtain suggest novel, hitherto unexplored, experimentally accessible observables that can shed light on symmetry breaking dynamics while testing the conjecture on which the KZM is based.
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.
The crossing of a continuous phase transition gives rise to the formation of topological defects described by the Kibble-Zurek mechanism (KZM) in the limit of slow quenches. The KZM predicts a universal power-law scaling of the defect density as a function of the quench time. We focus on the deviations from KZM experimentally observed in rapid quenches and establish their universality. While KZM scaling holds below a critical quench rate, for faster quenches the defect density and the freeze-out time become independent of the quench rate and exhibit a universal power-law scaling with the final value of the control parameter. These predictions are verified in several paradigmatic scenarios in both the classical and quantum domains.
The Kibble-Zurek mechanism predicts a universal scaling behavior of the defect production when a system is slowly quenched across a critical phase transition point. Here, we discover that the universal scaling behavior across a higher-order topological phase transition does not conform to the traditional Kibble-Zurek mechanism. To explain the anomalous scaling exponent, we develop a solvable Landau-Zener model that takes into account the role of topological edge band that dominates the phase transition. For a two-dimensional boundary-obstructed higher-order topological system, the Kibble-Zurek scaling must be modified to adopt the effective dimension of topological edge band instead of the real physical dimension. We also find that across the boundary-obstructed higher-order topological phase transition, boundary conditions can drastically modify the scaling behaviors. For comparison, we investigate the slow quench dynamics across the bulk-obstructed phase transitions and a single multicritical point, which obeys the Kibble-Zurek mechanism with physical dimension.
The Kibble-Zurek mechanism demands an initial adiabatic stage before an impulse stage to have a frozen correlation length that generates topological defects in a cooling phase transition. Here we study such a driven critical dynamics but with an initial condition that is near the critical point and that is far away from equilibrium. In this case, there is no initial adiabatic stage at all and thus adiabaticity is broken. However, we show that there again exists a finite length scale arising from the driving that divides the evolution into three stages. A relaxation--finite-time-scaling--adiabatic scenario is then proposed in place of the adiabatic-impulse-adiabatic scenario of the original Kibble-Zurek mechanism. A unified scaling theory, which combines finite-time scaling with critical initial slip, is developed to describe the universal behavior and is confirmed with numerical simulations of a two-dimensional classical Ising model.
Abstract Kibble-Zurek scaling is the scaling of the density of topological defects formed via the Kibble-Zurek mechanism with respect to the rate at which a system is cooled across a continuous phase transition. Recently, the density of the topological defects formed via the Kibble-Zurek mechanism was estimated for a system cooled through a first-order phase transition rather than conventional continuous transitions. Here we address the problem of whether such defects generated across a first-order phase transition exhibit Kibble-Zurek scaling similar to the case in continuous phase transitions. We show that any possible Kibble-Zurek scaling for the topological defects can only be a very rough approximation due to an intrinsic field responsible for the scaling. However, complete universal scaling for other properties does exist.