No 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.

The Kosterlitz-Thouless ͑KT͒ transition is investigated for gauge glass models in two dimensions by means of the nonequilibrium relaxation ͑NER͒ 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 = 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.The gauge glass ͑GG͒ model is a classical spin system with quenched disorder and has attracted much attention. It describes the thermodynamics of various systems such as disordered magnets with random Dzyaloshinskii-Moriya interaction, 1 Josephson-junction arrays with positional disorder in a magnetic field, 2 vortex glasses, 3 and crystal systems on disordered substrates. 4 In three dimensions, the spin-glass ͑SG͒ transition for a strongly disordered regime has been confirmed in the GG systems theoretically, 5,6 as well as experimentally. 7 In two dimensions, there is a controversy about the existence of the SG-like phase in the strongly disordered regime. Some numerical simulations suggested the quasi-long-range order, 8,9 while experimental observation 10 and other numerical simulations 6,11-14 deny it. Although the long-range SG order is denied in two dimensions following the Marmine-type argument, 15 there is a possibility of a phase in which the SG correlation decays in a power law. 16 In weakly disordered regime, there has been another controversy about the existence of the reentrant transition from the Kosterlitz-Thouless ͑KT͒ phase 17 to the non-KT one. Earlier works with real-space renormalization group ͑RG͒ analysis suggest reentrance. 1,2,18,19 The analysis has been modified and provides the absence of it. [20][21][22] The same results are obtained by Monte Carlo simulations. 4,6,9,13,23,24 and the RG analyses. 20,25 With all these studies, the instability of the KT phase against a small disorder is pointed out by the perturbation expansion and the RG analysis. 26,27 However, it is denied by numerical simulations 4,6,9,13 and other RG analyses. 9,20-22 Analytically, the gauge theory, which has provided several exact relations in Ising SG models, shows the absence of reentrance if the KT phase appears in a finite disordered regime. 16 The same result is also derived from a dynamical point of view obtained by the dynamical gauge theory. 28 While the results of gauge theory are plausible, it is necessary to assume the stability of the KT phase. Investigation of the phase diagram of these random XY models is still at a primitive stage as compared to the case of the Ising model.In the present study, we apply the nonequilibrium relaxation ͑NER͒ analysis to the GG models in two dimensions in order to clarify the stability of the KT phase against a small disorder. The NER method has been an efficient numerical technique for analyzing equilibrium phase transitions. It pro-vides the critical temperatu...

No abstract

The frustrated system is one of main subjects in critical phenomena because of its rich phase diagrams and possibility of new universality classes. The equilibrium Monte Carlo simulation is a useful technique to study critical phenomena and has provided many helpful informations in statistical mechanics. Although it works even in frustrated systems, it sometimes suffers difficulties in the analysis due to large fluctuation and degeneracy, which restrict the available system sizes too small.In standard Monte Carlo simulations, the total amount of calculation time is consist of equilibration and averaging, and is, in usual, proportional to the equilibration time τ eq , since the averaging time is taken several times of τ eq . The equilibration from a fixed non-equilibrium state is achieved when the spin correlation ξ(t) grows up to the scale of the correlation length ξ eq in equilibrium or the system size L. In the critical regime, since ξ eq becomes larger than sizes simulated, τ eq becomes longer as L is larger. In frustrated systems, the fluctuation and degeneracy reveal a slow dynamics, that also makes τ eq very large. Therefore, available system sizes are reduced. In small systems, the correction to scaling is necessary to discuss critical phenomena accurately which makes estimations for the critical point and critical exponents complicated. The development and improvement of simulation technique in last decades have been mainly devoted to overcome such difficulties.Recently, efficient Monte Carlo technique is proposed to study critical phenomena using non-equilibrium relaxation (NER) process. 1-11) One may observe the relaxation of the order parameter (e.g., the magnetization in the ferromagnetic case) in the thermalization process from the complete ordered state. It provides the critical erature and the dynamic critical exponent accurately. This analysis has been used successfully to study various such as ferromagnetic (FM) Ising dtemp problems mo 2808 els, 2-5) XY models, 6) spin glass models 7-10) and quantum spin systems. 11) Since only the process in equilibration is observed, one can simulate quite large systems; as large as the memory permits. Recently, the method is extended to estimate various exponents using quantities of fluctuations (susceptibility, specific heat and so on). 4, 5)The critical exponents can be determined by asymptotic powers of such quantities or their combinations. The correlation length ξ(t) in the non-equilibrium process, which grows up to ξ eq , is finite transiently even at the critical point. The finite-size effect does not appear while ξ(t) L. If one may prepare systems with sizes larger than the maximum correlation length observed in the simulation, any finite-size effect is not necessary to be taken into account for the estimation of the critical point and the critical exponent. Therefore, the NER method is useful to study systems in which strong fluctuations appear or sufficient sizes can not be prepared on account of large equilibration time. It is appropriate much to st...