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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Nonequilibrium relaxation analysis of Kosterlitz-Thouless phase transition

Yukiyasu Ozeki, Keita Ogawa, Nobuyasu Ito

Published Feb 14, 2003• 41 citations
Publisher

Abstract

A simple and efficient numerical analysis is proposed for the Kosterlitz-Thouless (KT) phase transition. The nonequilibrium relaxation method is applied to it. The two-dimensional ferromagnetic XY models are investigated to show the efficiency. At the KT transition point as well as inside the KT phase, the nonequilibrium relaxation of magnetization from the all-aligned state shows an asymptotic power-law decay, m(t) approximately t(-lambda(T)). Only outside the KT phase, an asymptotic single exponential decay is observed. Using a standard scaling form m(t)=tau(-lambda)(-)m(t/tau) in this regime, where tau is the relaxation time at each temperature, we find a simple and efficient numerical estimation of the KT transition point and dynamical exponent. This method can be applied to various kinds of models which show the KT-like behavior.

DOI
10.1103/physreve.67.026702

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A simple and efficient numerical analysis is proposed for the Kosterlitz-Thouless (KT) phase transition. The nonequilibrium relaxation method is applied to it. The two-dimensional ferromagnetic XY models are investigated to show the efficiency. At the KT transition point as well as inside the KT phase, the nonequilibrium relaxation of magnetization from the all-aligned state shows an asymptotic power-law decay, m(t) approximately t(-lambda(T)). Only outside the KT phase, an asymptotic single exponential decay is observed. Using a standard scaling form m(t)=tau(-lambda)(-)m(t/tau) in this regime, where tau is the relaxation time at each temperature, we find a simple and efficient numerical estimation of the KT transition point and dynamical exponent. This method can be applied to various kinds of models which show the KT-like behavior.
The phase transition and critical properties for the RP^{2} model in two dimensions is investigated by means of the nonequilibrium relaxation method (NER) together with the dynamical scaling analysis. The relaxation of nematic order from the all-aligned state is observed by Monte Carlo simulations. The comparison of types of the asymptotic form of the relaxation time around the transition point is considered by the dynamical scaling analysis, which clearly discriminates the Kosterlitz-Thouless (KT)-type transition from the second-order one. Using the relaxation of fluctuation, the static critical exponent η and the dynamical one z, which are only the intrinsic exponents for the KT transition, are estimated at and below the KT transition temperature. The result shows similar behaviors with those observed in the KT phase for the ferromagnetic XY model in two dimensions, which has been recognized as a typical KT system, and reveals the confirmation of the present KT transition.
The critical exponents are estimated for the gauge glass model in two dimensions, in which only the Kosterlitz-Thouless (KT) phase appears in the low-temperature regime. The nonequilibrium relaxation method is applied to estimate the transition temperature and critical exponents: the static exponent η and the dynamical exponent z. Since the system exhibits criticality in the whole KT phase, we estimate the exponents on the boundary as well as inside the KT phase. The static exponent η depends on both the temperature and the strength of randomness, while the dynamical one z is almost constant throughout the KT phase, including the boundary.
The Kosterlitz-Thouless (KT) transition is investigated for gauge glass models in two dimensions by means of the nonequilibrium relaxation (NER) method. Two kinds of models, which have the same symmetry, are analyzed. Using the scaling analysis of the NER function on a large lattice with L=1000L=1000L=1000, we confirm the KT transition numerically for both models. This indicates the stability of the KT phase against a small disorder, which was previously claimed by perturbation expansion and renormalization group arguments.
Recently we showed that the critical nonequilibrium relaxation in the Swendsen-Wang algorithm is widely described by the stretched-exponential relaxation of physical quantities in the Ising or Heisenberg models. Here we make a similar analysis in the Berezinsky-Kosterlitz-Thouless phase transition in the two-dimensional (2D) XY model and in the first-order phase transition in the 2D q=5 Potts model and find that these phase transitions are described by the simple exponential relaxation and power-law relaxation of physical quantities, respectively. We compare the relaxation behaviors of these phase transitions with those of the second-order phase transition in the three- and four-dimensional XY models and in the 2D q-state Potts models for 2≤q≤4 and show that the species of phase transitions can be clearly characterized by the present analysis. We also compare the size dependence of relaxation behaviors of the first-order phase transition in the 2D q=5 and 6 Potts models and propose a quantitative criterion on "weakness" of the first-order phase transition.
Critical properties of the fully frustrated XY models on square and triangular lattices are investigated by means of the nonequilibrium relaxation (NER) method. We examine the validity of the conclusion on the universality class of the chiral transition previously obtained by the NER method, in which that belongs to a non-Ising-type one. To clarify it, we analyze the NER of fluctuations for longer Monte Carlo steps on lager lattices than those performed previously. The calculation is made at the chiral transition temperatures estimated carefully by the use of recently improved dynamical scaling analysis. The result indicates that the asymptotic behavior of the time-dependent exponents, which should converge to the critical exponents, show the same tendency as those obtained previously in shorter times. We also apply the improved dynamical scaling analysis to the estimation of Kosterlitz-Thouless (KT) transitions and confirm the existence of the double transitions for the chiral and KT phases with more reliable estimations.