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arXiv:2311.13074 · International Symposium on Lattice Field Theory (LATTICE)

Non-equilibrium dynamics of topological defects in the 3d O(2) model

Edgar López-Contreras, Jaime Fabián Nieto Castellanos, Elías Natanael Polanco-Euán, Wolfgang Bietenholz A

Published Dec 27, 2023• 0 citations
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Abstract

We present a study of the 3d O(2) non-linear $\sigma$-model on the lattice, which exhibits topological defects in the form of vortices. They tend to organize into vortex lines that bear close analogies with global cosmic strings. Therefore, this model serves as a testbed for studying the dynamics of topological defects. It undergoes a second order phase transition, hence it is appropriate for investigating the Kibble-Zurek mechanism. In this regard, we explore the persistence of topological defects when the temperature is rapidly reduced from above to below the critical temperature; this cooling (or “quenching”) process takes the system out of thermal equilibrium. We probe a wide range of inverse cooling rates $\tau_\mathrm{Q}$ and final temperatures, employing distinct Monte Carlo algorithms. The results consistently show that the density of persisting topological defects follows a power-law in $\tau_\mathrm{Q}$, in agreement with Zurek’s conjecture. On the other hand, at this point, our results do not confirm Zurek’s prediction for the exponent in this power-law, but its final test is still under investigation.

DOI
10.22323/1.453.0346

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We present a study of the 3d O(2) non-linear σ\sigmaσ-model on the lattice, which exhibits topological defects in the form of vortices. They tend to organize into vortex lines that bear close analogies with global cosmic strings. Therefore, this model serves as a testbed for studying the dynamics of topological defects. It undergoes a second order phase transition, hence it is appropriate for investigating the Kibble-Zurek mechanism. In this regard, we explore the persistence of topological defects when the temperature is rapidly reduced from above to below the critical temperature; this cooling (or “quenching”) process takes the system out of thermal equilibrium. We probe a wide range of inverse cooling rates τQ\tau_\mathrm{Q}τQ​ and final temperatures, employing distinct Monte Carlo algorithms. The results consistently show that the density of persisting topological defects follows a power-law in τQ\tau_\mathrm{Q}τQ​, in agreement with Zurek’s conjecture. On the other hand, at this point, our results do not confirm Zurek’s prediction for the exponent in this power-law, but its final test is still under investigation.
The Kibble mechanism plays a prominent role in the theory of the early Universe, as an explanation of the possible formation of cosmic strings. Zurek suggested the analogous effect in liquid helium under rapid cooling, and he conjectured - together with del Campo and Kibble - a scaling-law for the relation between the density of remnant topological defects and the cooling rate, when a system passes through its critical temperature. Such scaling-laws were indeed observed in condensed matter experiments. Here we test the validity of Zurek's scaling-law in a different framework. We numerically study the behavior of the Ising model (with classical spins) in 1, 2 and 3 dimensions under rapid cooling. This model does not have topological defects, so we consider the dynamics of domains instead, i.e. we measure their evolution during a cooling process down to zero temperature. For several Markov chain Monte Carlo algorithms, we consistently observe scaling-laws along the lines of Zurek's conjecture, in all dimensions under consideration, which shows that this feature holds more generally than expected. It is highly remarkable that even the exponents of these scaling-laws are consistent for three different algorithms, which hints at a physical meaning.
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.
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 formation of topological defects during continuous phase transitions exhibits nonequilibrium universality. While the Kibble-Zurek mechanism (KZM) predicts universal scaling of point-like defect numbers under slow driving, the statistical properties of extended defects remain largely unexplored across both slow and fast protocols. We investigate vortex string formation in a three-dimensional holographic superfluid. For slow quenches, the vortex string number follows KZM scaling, while for rapid quenches, it exhibits complementary universal scaling governed by the final temperature. Beyond the vortex string number, the loop-length distribution reveals a richer structure: individual loops follow the first-return statistics of three-dimensional random walks, P(ℓ)∼ℓ−5/2P(\ell) \sim \ell^{-5/2}P(ℓ)∼ℓ−5/2. While the total vortex length distribution remains Gaussian, its cumulants obey universal scaling laws with varying power-law exponents, and thus differ markedly from those observed in point-defect systems, indicating distinct statistical features of extended topological defects.
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 predictions can lead to results that are inconsistent with a more careful KZM 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.
Topological defects that form in a nonequilibrium second-order phase transition are presumably seeded by fluctuations of the order parameter in the vicinity of the critical point. Motivated by this conjecture that underlies the Kibble-Zurek mechanism (KZM), we investigate whether machine learning can anticipate their locations from the fluctuations in the "impulse regime," the time interval when the evolution of the order parameter cannot keep up with the conditions imposed by the quench. Going beyond the conventional KZM focus on defect density, we show that a recurrent neural network can predict locations of topological defects from short-time dynamical data deep within the impulse regime. We thus demonstrate that defects are sown in the immediate vicinity of the critical point. The seeds of defects-fluctuations imprinted on the evolving order parameter-are exponentially small near the critical point, but become amplified by the evolution and play a dominant role in breaking symmetry. This occurs before the freeze-out time t[over ^] that concludes the impulse regime, and well before the order parameter assumes its final symmetry-broken configuration. Furthermore, we find that the predictive power of the machine learning also exhibits power-law scaling consistent with KZM.