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arXiv:cond-mat/9812149 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Dynamic simulations of the Kosterlitz-Thouless phase transition

Bo Zheng, M. Schulz, Steffen Trimper

Published Feb 1, 1999• 39 citations
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Abstract

Based on the short-time dynamic scaling form, a dynamic approach is proposed to numerically tackle the Kosterlitz-Thouless phase transition. Taking the two-dimensional $\mathrm{XY}$ model as an example, the exponential divergence of the spatial correlation length, the transition temperature ${T}_{\mathrm{KT}},$ and all critical exponents are computed. Compared with Monte Carlo simulations in equilibrium, we obtain data at temperatures closer to ${T}_{\mathrm{KT}}.$

DOI
10.1103/physreve.59.r1351

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Based on the short-time dynamic scaling form, a dynamic approach is proposed to numerically tackle the Kosterlitz-Thouless phase transition. Taking the two-dimensional XY\mathrm{XY}XY model as an example, the exponential divergence of the spatial correlation length, the transition temperature TKT,{T}_{\mathrm{KT}},TKT​, and all critical exponents are computed. Compared with Monte Carlo simulations in equilibrium, we obtain data at temperatures closer to TKT.{T}_{\mathrm{KT}}.TKT​.
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.
Convincing numerical evidence is obtained via extensive quantum Monte Carlo simulations on square lattices as large as 96\ifmmode\times\else\texttimes\fi{}96 that the spin-1/2 XY model undergoes a Kosterlitz-Thouless phase transition at kTc{\mathit{kT}}_{\mathit{c}}kTc​/J=0.350(4). Correlation length and in-plane susceptibility diverge at Tc{\mathit{T}}_{\mathit{c}}Tc​ precisely according to the form predicted by Kosterlitz and Thouless for the classical XY model. The specific heat increases very rapidly near Tc{\mathit{T}}_{\mathit{c}}Tc​ and exhibits a peak around kT/J=0.45.
With Monte Carlo methods we investigate the dynamic relaxation of the fully frustrated XY\mathrm{XY}XY model in two dimensions below and at the Kosterlitz-Thouless phase transition temperature. Special attention is given to the sublattice structure of the dynamic evolution. The short-time scaling behavior is found and universality confirmed. The critical exponent $\ensuremath{\theta}$ is measured for different temperatures and with different algorithms.