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arXiv:1201.6423 · Journal of Physics A Mathematical and Theoretical

Dynamic relaxation of topological defect at Kosterlitz–Thouless phase transition

X P Qin, B Zheng, N J Zhou

Published Aug 8, 2011• 1 citation
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Abstract

With Monte Carlo methods we study the dynamic relaxation of a vortex state at the Kosterlitz-Thouless phase transition of the two-dimensional XY model. A local pseudo-magnetization is introduced to characterize the symmetric structure of the dynamic systems. The dynamic scaling behavior of the pseudo-magnetization and Binder cumulant is carefully analyzed, and the critical exponents are determined. To illustrate the dynamic effect of the topological defect, similar analysis for the the dynamic relaxation with a spin-wave initial state is also performed for comparison. We demonstrate that a limited amount of quenched disorder in the core of the vortex state may alter the dynamic universality class. Further, theoretical calculations based on the long-wave approximation are presented.

DOI
10.1088/1751-8113/44/34/345005

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With Monte Carlo methods we study the dynamic relaxation of a vortex state at the Kosterlitz-Thouless phase transition of the two-dimensional XY model. A local pseudo-magnetization is introduced to characterize the symmetric structure of the dynamic systems. The dynamic scaling behavior of the pseudo-magnetization and Binder cumulant is carefully analyzed, and the critical exponents are determined. To illustrate the dynamic effect of the topological defect, similar analysis for the the dynamic relaxation with a spin-wave initial state is also performed for comparison. We demonstrate that a limited amount of quenched disorder in the core of the vortex state may alter the dynamic universality class. Further, theoretical calculations based on the long-wave approximation are presented.
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.
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.
Abstract A Monte Carlo simulation of the nonequilibrium critical behavior is carried out for the two-dimensional structurally disordered XY model during its evolution from a high-temperature initial state. The features of the effect of structural disorder on the time dependence of the correlation length ξ(t) of the system and the dynamical scaling of the spin-spin autocorrelation function CSS(t, tw) are revealed. A direct calculation of the dynamic behavior of the correlation length ξ(t) of the two-dimensional structurally disordered XY model is carried out, and it is shown that this model, just as the pure model, exhibits a time dependence with a logarithmic correction ξ(t) ∝ (t/lnt)1/2, which is associated with the nonequilibrium annihilation of vortices and antivortices in the forming vortex pairs. Based on the analysis of the time dependence of the correlation length ξ(t) and the magnetization cumulant g2(t), it is shown that the two-dimensional XY model with the spin concentration p = 0.7 is so close to the spin percolation threshold pc that the influence of the attraction of the percolation fixed point becomes crucial for the relaxation properties of the system. However, the features of the critical dynamics of systems with spin concentrations of p = 0.9 and p = 0.8 are determined by the attraction of the pure fixed point. Temperature and concentration dependences of the Fisher critical exponent η(p, T) are determined using the scaling properties of the calculated two-time dependence of the spin-spin autocorrelation function CSS(t, tw). The scaling functions of the two-time dependence of the spin-spin autocorrelation function CSS(t, tw) are calculated using the dynamic dependence of the correlation length ξ(t) obtained as a result of simulation, and the values of the decay exponents λC(p, T) of the scaling functions in the long-time regime are determined, which are in good agreement with the relation λC(p, T) = 1 + η(p, T) and prove that the dynamical scaling is realized for the nonequilibrium characteristics of structurally disordered systems.