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arXiv:2001.07950 · Journal of Mathematical Physics

Topologically induced metastability in a periodic XY chain

Clément Cosco, Assaf Shapira

Published Apr 1, 2021• 2 citations
arXivHTMLPDF

Abstract

Non-trivial topological behavior appears in many different contexts in statistical physics, perhaps the most known one being the Kosterlitz–Thouless phase transition in the two-dimensional XY model. We study the behavior of a simpler, one dimensional, XY chain with periodic boundary and strong interactions, but rather than concentrating on the equilibrium measure, we try to understand its dynamics. The equivalent of the Kosterlitz–Thouless transition in this one-dimensional case happens when the interaction strength scales like the size of system N, yet we show that a sharp transition for the dynamics occurs at the scale of log N. When the interactions are weaker than a certain threshold, topological phases could not be observed over long times, while for interactions that are stronger than that threshold, topological phases become metastable, surviving for diverging time scales.

DOI
10.1063/5.0004606

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Non-trivial topological behavior appears in many different contexts in statistical physics, perhaps the most known one being the Kosterlitz–Thouless phase transition in the two-dimensional XY model. We study the behavior of a simpler, one dimensional, XY chain with periodic boundary and strong interactions, but rather than concentrating on the equilibrium measure, we try to understand its dynamics. The equivalent of the Kosterlitz–Thouless transition in this one-dimensional case happens when the interaction strength scales like the size of system N, yet we show that a sharp transition for the dynamics occurs at the scale of log N. When the interactions are weaker than a certain threshold, topological phases could not be observed over long times, while for interactions that are stronger than that threshold, topological phases become metastable, surviving for diverging time scales.
The Berezinskii-Kosterlitz-Thouless (BKT) transition is the paradigmatic example of a topological phase transition without symmetry breaking, where a quasiordered phase, characterized by a power-law scaling of the correlation functions at low temperature, is disrupted by the proliferation of topological excitations above the critical temperature T_{BKT}. In this Letter, we consider the effect of long-range decaying couplings ∼r^{-2-σ} on the BKT transition. After pointing out the relevance of this nontrivial problem, we discuss the phase diagram, which is far richer than the corresponding short-range one. It features-for 7/4<σ<2-a quasiordered phase in a finite temperature range T_{c}<T<T_{BKT}, which occurs between a symmetry broken phase for T<T_{c} and a disordered phase for T>T_{BKT}. The transition temperature T_{c} displays unique universal features quite different from those of the traditional, short-range XY model. Given the universal nature of our findings, they may be observed in current experimental realizations in 2D atomic, molecular, and optical quantum systems.
We study the two-dimensional generalized XY model that depends on an integer qqq by the Monte Carlo method. This model was recently proposed by Romano and Zagrebnov. We find a single Kosterlitz-Thouless (KT) transition for all values of qqq, in contrast with the previous speculation that there may be two transitions, one a regular KT transition and another a first-order transition at a higher temperature. We show the universality of the KT transitions by comparing the universal finite-size scaling behaviors at different values of qqq without assuming a specific universal form in terms of the KT transition temperature TKTT_{\rm KT}TKT​.
We study the two-dimensional XY\mathrm{XY}XY model with quenched random phases by Monte Carlo simulation and finite-size scaling analysis. We determine the phase diagram of the model and study its critical behavior as a function of disorder and temperature. If the strength of the randomness is less than a critical value, ${\ensuremath{\sigma}}_{c},$ the system has a Kosterlitz-Thouless (KT) phase transition from the paramagnetic phase to a state with quasi-long-range order. Our data suggest that the latter exists down to T=0T=0T=0 in contradiction with theories that predict the appearance of a low-temperature reentrant phase. At the critical disorder ${T}_{\mathrm{KT}}\ensuremath{\rightarrow}0$ and for $\ensuremath{\sigma}>{\ensuremath{\sigma}}_{c}$ there is no quasiordered phase. At zero temperature there is a phase transition between two different glassy states at ${\ensuremath{\sigma}}_{c}.$ The functional dependence of the correlation length on $\ensuremath{\sigma}$ suggests that this transition corresponds to the disorder-driven unbinding of vortex pairs.
Topology plays a fundamental role in our understanding of many-body physics, from vortices and solitons in classical field theory to phases and excitations in quantum matter. Topological phenomena are intimately connected to the distribution of information content that, differently from ordinary matter, is now governed by nonlocal degrees of freedom. However, a precise characterization of how topological effects govern the complexity of a many-body state, i.e., its partition function, is presently unclear. In this paper, we show how topology and complexity are directly intertwined concepts in the context of classical statistical mechanics. We concretely present a theory that shows how the Kolmogorov complexity of a classical partition function sampling carries unique, distinctive features depending on the presence of topological excitations in the system. We confront two-dimensional Ising, Heisenberg, and XY models on several topologies and study the corresponding samplings as high-dimensional manifolds in configuration space, quantifying their complexity via the intrinsic dimension. While for the Ising and Heisenberg models the intrinsic dimension is independent of the real-space topology, for the XY model it depends crucially on temperature: across the Berezkinskii-Kosterlitz-Thouless (BKT) transition, complexity becomes topology dependent. In the BKT phase, it displays a characteristic dependence on the homology of the real-space manifold, and, for g-torii, it follows a scaling that is solely genus dependent. We argue that this behavior is intimately connected to the emergence of an order parameter in data space, the conditional connectivity, which displays scaling behavior. Our approach paves the way for an understanding of topological phenomena emergent from many-body interactions from the perspective of Kolmogorov complexity.
Non-equilibrium critical dynamics of the two-dimensional XY model is investigated with Hamiltonian equations of motion. Critical relaxation starting from both ordered and random states is carefully analyzed, and the short- time dynamic scaling behavior is revealed. Logarithmic corrections to scaling are detected for relaxation with a random initial state, while power-law corrections to scaling are observed for relaxation with an ordered initial state. The static exponent η and dynamic exponent z are determined around and below the Kosterlitz-Thouless phase transition temperature. Our results show that the deterministic dynamics described by Hamiltonian equations is in the same universality class as the stochastic dynamics described by Monte Carlo algorithms and Langevin equations.
We investigate the out of equilibrium dynamics of the two-dimensional XY model when cooled across the Berezinskii–Kosterlitz–Thouless (BKT) phase transition using different protocols. We focus on the evolution of the growing correlation length and the density of topological defects (vortices). By using Monte Carlo simulations we first determine the time and temperature dependence of the growing correlation length after an infinitely rapid quench from above the transition temperature to the quasi-long-range order region. The functional form is consistent with a logarithmic correction to the diffusive law and it serves to validate dynamic scaling in this problem. This analysis clarifies the different dynamic roles played by bound and free vortices. We then revisit the Kibble–Zurek mechanism in thermal phase transitions in which the disordered state is plagued with topological defects. We provide a theory of quenching rate dependence in systems with the BKT-type transition that goes beyond the equilibrium scaling arguments. Finally, we discuss the implications of our results for a host of physical systems with vortex excitations, including planar ferromagnets and liquid crystals.