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.
By using topological current theory we study the inner topological structure of vortices a two-dimensional (2D) XY model and find the topological current relating to the order parameter field. A scalar field,
ψ, is introduced through the topological current theory. By solving the scalar field, the interaction energy of vortices in a 2D XY model is revisited. We study the dynamical evolution of vortices and present the branch conditions for generating, annihilating, crossing, splitting and merging of vortices. During the growth or annihilation of vortices, the dynamical scaling law of relevant length in a 2D XY model,
ξ(t)∝(t−t∗)1/z, is obtained in the neighborhood of the limit point, given the dynamic exponent
z=2. This dynamical scaling behavior is consistent with renormalization group theory, numerical simulations, and experimental results. Furthermore, it is found that during the crossing, splitting and merging of vortices, the dynamical scaling law of relevant length is
ξ(t)∝(t−t∗). However, if vortices are at rest during splitting or merging, the dynamical scaling law of relevant length is a constat.
The persistence exponents associated with the T = 0 quenching dynamics of the two-dimensional XY model and a two-dimensional uniaxial spin nematic model have been evaluated using a numerical simulation. The site persistence or the probability that the sign of a local spin component does not change starting from initial time t = 0 up to a certain time t , is found to decay as L ( t ) −θ ( L ( t ) is the linear domain length scale), with θ = 0.305(±0.020) for the two-dimensional XY model and 0.199(±0.009) for the two-dimensional uniaxial spin nematic model. We have also investigated the scaling (at the late time of phase ordering) associated with the correlated persistent sites in both models. The persistence correlation length was found to grow in the same way as L ( t ).
In conventional quench dynamics governed by the Kibble-Zurek mechanism (KZM), the defect density generally decays as a pure power law of the quench rate. However, the KZM scaling of the two-dimensional (2D) XY model with topological phase transitions features prominent logarithmic corrections. Nevertheless, it remains unclear whether such logarithmic scaling corrections emerge in discrete-spin systems that host two successive topological phase transitions under thermal quenches. This work investigates the J1-J2 antiferromagnetic Potts model and constructs its equilibrium phase diagram. Based on the temperature ranges of the paramagnetic phase, quasi-long-range ordered (QLRO) phase, long-range ordered phase, and zero-temperature ground state, we design four quench protocols with distinct temperature intervals. Our results demonstrate that quenches terminating in the QLRO phase exhibit logarithmically corrected KZM scaling of the excess energy density, consistent with the dynamical universality class of the 2D XY model. In contrast, quenches ending in the LRO phase, including both finite-temperature and zero-temperature protocols, follow conventional power-law scaling. Our results clearly uncover the characteristic scaling corrections of the J1-J2 Potts model and offer theoretical guidance for future experimental investigations of KZM via photonic simulation platforms.
With a large-scale Monte Carlo simulation, non-equilibrium dynamics of the two-dimensional fully frustrated XY model is investigated. We tackle the Kosterlitz-Thouless phase transition. Starting from an ordered initial state, we study the dynamic evolution of the magnetization as well as a specifically defined Binder cumulant. From the dynamic scaling ansatz, we extract the correlating time of the dynamics and the spatial correlation length of the equilibrium state. The dynamic exponent z is determined with relatively high accuracy. Especially, we suggest and demonstrate how one may directly measure the dynamic exponent z above TKT from the scaling fit of the Binder cumulant. These results indicate that the dynamic exponent z fluctuates around z=2, and this is consistent with that observed at temperatures below the transition temperature TKT.