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Physical Review E

Numerical studies on critical properties of the Kosterlitz-Thouless phase for the gauge glass model in two dimensions

Yukiyasu Ozeki, Satoshi Yotsuyanagi, Takuya Sakai, Yuki Echinaka

Published Feb 18, 2014• 6 citations
Publisher

Abstract

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.

DOI
10.1103/physreve.89.022122

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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.
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 q-state clock gauge glass model is studied to see the effect of discreteness on the Kosterlitz-Thouless (KT) transition and the ferromagnetic (FM) critical phenomenon in random systems. The nonequilibrium relaxation analysis is applied. In two dimensions, the successive transitions of paramagnetic (PM), KT, and FM phases are investigated along the Nishimori line for q=6, 8, 10, 12, 14, 16, and 1024 (recognized as infinity) cases. For the upper critical temperature, it is found that the transition temperature is almost the same as in the continuous case for all q values. The lower transition temperature is found to be proportional to 1/q2. In three dimensions, the critical behavior of the PM-FM transition is studied along the Nishimori line for q=6, 8, 16, and 1024 cases. It is found that the spin discreteness is irrelevant, and the transition belongs to the same universality class as in the (continuous) XY case.
The ferromagnetic critical exponents are estimated along the phase boundary for the gauge glass model in three dimensions. The nonequilibrium relaxation method is applied to estimate the transition temperature and critical exponents. Together with the previously obtained results for the ± J Ising model in two and three dimensions, universality classes are examined for these disordered models. The result reveals a direct numerical confirmation of the Harris criterion for typical models with positive, zero and negative values of α 0 , the exponent for the specific heat in the regular case. In addition, we show a nonuniversal behavior of the dynamical exponent z with respect to the randomness for these models irrespective of α 0 .
Monte Carlo simulations of the two-dimensional XY model are performed in a square geometry with various boundary conditions (BC). Using conformal mappings we deduce the exponent η σ ( T ) of the order parameter correlation function and its surface analogue η ∥ ( T ) as a function of the temperature in the critical (low-temperature) phase of the model. The temperature dependence of both exponents is obtained numerically with a good accuracy up to the Kosterlitz–Thouless transition temperature. The bulk exponent follows from simulations of correlation functions with periodic boundary conditions or order parameter profiles with open boundary conditions and with symmetry breaking surface fields. At very low temperatures, η σ ( T ) is found in pretty good agreement with the linear temperature-dependence of Berezinskii's spin-wave approximation. We also show some evidence that there are no noticeable logarithmic corrections to the behaviour of the order parameter density profile at the Kosterlitz–Thouless (KT) transition temperature, while these corrections exist for the correlation function. At the KT transition the value η σ ( T KT ) = 1/4 is accurately recovered. The exponent associated with the surface correlations is similarly obtained after a slight modification of the boundary conditions: the correlation function is computed with free BC, and the profile with mixed fixed–free BC. It exhibits a monotonic behaviour with temperature, starting linearly according to the spin-wave approximation and increasing up to a value η ∥ ( T KT ) ≃ 1/2 at the Kosterlitz–Thouless transition temperature. The thermal exponent η ε ( T ) is also computed and we give some evidence that it keeps a constant value in agreement with the marginality condition of the temperature field below the KT transition.