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arXiv:1310.1600 · International Journal of Modern Physics A

Universality of phase transition dynamics: Topological defects from symmetry breaking

Adolfo del Campo, Wojciech H. Zurek

Published Mar 20, 2014• 363 citations
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Abstract

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.

DOI
10.1142/s0217751x1430018x

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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.
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 (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.
The Kibble-Zurek mechanism (KZM) describes the nonequilibrium dynamics and topological defect formation in systems undergoing second-order phase transitions. KZM has found applications in fields such as cosmology and condensed matter physics. However, it is generally not suitable for describing first-order phase transitions. It has been demonstrated that transitions in systems like superconductors or charged superfluids, typically classified as second order, can exhibit weakly first-order characteristics when the influence of fluctuations is taken into account. Moreover, the order of the phase transition (i.e., the extent to which it becomes first rather than second order) can be tuned. We explore quench-induced formation of topological defects in such tunable phase transitions and propose that their density can be predicted by combining KZM with nucleation theory.
Kibble-Zurek mechanism (KZM) uses critical scaling to predict density of topological defects and other excitations created in second order phase transitions. We point out that simply inserting asymptotic critical exponents deduced from the immediate vicinity of the critical point to obtain KZM predictions can lead to results that are inconsistent with a more careful KZM-like analysis based on causality -- on the comparison of the relaxation time of the order parameter with the time "distance" from the critical point. As a result, scaling of quench-generated excitations with quench rates can exhibit behavior that is locally (i.e., in the neighborhood of any given quench rate) well approximated by the power law, but with exponents that depend on that rate, and that are quite different from the naive prediction based on the critical exponents relevant for asymptotically long quench times. Kosterlitz-Thouless scaling (that governs e.g. Mott insulator to superfluid transition in the Bose-Hubbard model in one dimension) is investigated as an example of this phenomenon.
The crossing of a continuous phase transition results in the formation of topological defects with a density predicted by the Kibble-Zurek mechanism (KZM). We characterize the spatial distribution of pointlike topological defects in the resulting nonequilibrium state and model it using a Poisson point process in arbitrary spatial dimensions with KZM density. Numerical simulations in a one-dimensional ${\ensuremath{\phi}}^{4}$ theory unveil short-distance defect-defect corrections stemming from the kink excluded volume, while in two spatial dimensions, our model accurately describes the vortex spacing distribution in a strongly coupled superconductor indicating the suppression of defect-defect spatial correlations.