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International Journal of Modern Physics C

NONEQUILIBRIUM RELAXATION METHOD AND ITS APPLICATIONS

Nobuyasu Ito, Yukiyasu Ozeki

Published Dec 1, 1999• 16 citations
Publisher

Abstract

Nonequilibrium relaxation (NER) process turns out to be an useful method to study phase transition and critical phenomena. The NER method and its applications are reviewed. Characteristic behavior of the local exponent is analyzed with a simple model.
DOI
10.1142/s0129183199001273

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The nonequilibrium relaxation (NER) process of fluctuations of physical quantities is studied simulationally. It is shown that the NER method, which is a convenient technique for studying the phase and transition, is useful not only for identifying the phase and locating the transition point, but also for estimating both static and dynamical exponents. As an example, the cubic-lattice ferromagnetic Ising model is analyzed. The transition inverse-temperature is estimated to be K c =0.2216595(15). The exponents of correlation length, magnetization, and specific heat are estimated to be ν=0.635(5), β= 0.325(5) and α= 0.14(2), respectively. The dynamical exponent z is estimated to be 2.055(10).
It is shown that the non-equilibrium relaxation (NER) process of the order-parameter and energy is useful to study the phase transition.By observing the log-derivatives of those decay functions, the critical point is located finely and the critical exponents are estimated accurately.The major advantage of NER method is it's simplicity; not only the behavior in the large-size limit can be observed easily, but also it is not necessary to pursue the equilibrium of the system to get information of the final equilibrium state.Applications are expanding from basic systems like Ising ferromagnets, to spin-glass, quantal system, chiral and KT transitions, and so on.Now the NER process of fluctuations is also expected to share those advantages.It is found that the combinations of the NER functions of some fluctuations provide the way to determine static and dynamics critical exponents directly.
The nonequilibrium relaxation (NER) method is a numerical technique to analyse equilibrium phase transitions. One can estimate the transition point and critical exponents calculating relaxations of order parameter and fluctuations in NER processes from nonequilibrium initial states to the equilibrium ones. It is applied to the second-order transition as well as the first-order one; the precise estimation of transition temperature for the latter case is achieved with the mixed phase initialization technique. Since the equilibration is not necessary in the simulation, one can analyse systems with large sizes which can be recognized as in the thermodynamic limit up to the observation time. This leads to the analysis being more efficient for slowly relaxing systems with frustrations and randomness. It can be extended to the Kosterlitz–Thouless transition and the spin glass transition cases by the use of the finite-time scaling analysis. It is also extended to quantum systems.
The non-equilibrium relaxation method (NER) is applied to a frustrated spin system with competitive interactions. The critical point and critical exponents are estimated accurately for the Blume-Emery-Griffiths model in three dimensions. It is applied to the region where successive transitions of PM-FM-PM-SQ phases occur; we denote paramagnetic (PM), ferromagnetic (FM) and staggered quadrupole (SQ) phases. With the analysis for relaxation of fluctuations, we can estimate various critical exponents. The result indicates that all transitions belong to the Ising universality class in three dimensions. It is found that the NER method is useful to analyze the critical phenomena for complex systems in which equilibrium simulation is even difficult.
Although the nonequilibrium-relaxation (NER) method has been widely used in Monte Carlo studies on phase transitions in classical spin systems, such studies have been quite limited in quantum phase transitions. The reason is that the relaxation process based on cluster-update quantum Monte Carlo (QMC) algorithms, which are now standards in Monte Carlo studies on quantum systems, has been considered "too fast" for such analyses. Recently, the present authors revealed that the NER process in classical spin systems based on cluster-update algorithms is characterized by stretched-exponential critical relaxation, rather than conventional power-law relaxation in local-update algorithms. In the present article, we show that this is also the case in quantum phase transitions analyzed with the cluster-update QMC. As the simplest example of isotropic quantum spin models that exhibit quantum phase transitions, we investigate the Néel-dimer quantum phase transition in the two-dimensional S=1/2 columnar-dimerized antiferromagnetic Heisenberg model with the continuous-time loop algorithm, and we confirm stretched-exponential critical relaxation consistent with the three-dimensional classical Heisenberg model in the Swendsen-Wang algorithm.
Kinetic models for nonequilibrium phase transitions have been extensively studied via mean-field theory, series expansions, and Monte Carlo simulation. In this paper we present a different approach based on studying the normal-mode solutions of the master equation for the system to extract information on the static and dynamic behavior of the system in the vicinity of the phase transition. This method provides insight into the physics of the transition and offers an alternative technique for analyzing kinetic phase transitions. We demonstrate the method by extracting the critical behavior of a simple model from finite-size scaling.