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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Monte Carlo simulation of critical dynamic spin systems

Lothar Schülke, Bo Zheng

Published Jan 1, 1998• 0 citations
Publisher

Abstract

Universal short-time behavior of the critical dynamics has been confirmed numerically for some simple statistical models. The critical initial increase of the magnetization has been observed and the new dynamic exponent (theta) has been measured. We report further about a new kind of measurement of all the static exponents and the dynamic exponent z from short- time dynamics.

DOI
10.1117/12.299577

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Universal short-time behavior of the critical dynamics has been confirmed numerically for some simple statistical models. The critical initial increase of the magnetization has been observed and the new dynamic exponent (theta) has been measured. We report further about a new kind of measurement of all the static exponents and the dynamic exponent z from short- time dynamics.
Monte Carlo simulations of the short-time critical dynamics are reviewed. The short-time universal scaling behavior of the dynamic Ising model and Potts model are discussed in detail, while extension and application to more complex systems as the XY model, the fully frustrated XY model and other dynamic systems are also presented. The investigation of the universal behavior of the short-time dynamics not only enlarges the fundamental knowledge on critical phenomena but also, more interestingly, provides possible new ways to determine not only the new critical exponents θ and θ1, but also the traditional dynamic critical exponent z as well as all static critical exponents.
Critical scaling and universality in short-time dynamics for spin models on a twodimensional triangular lattice are investigated by using Monte Carlo simulation. Emphasis is placed on the dynamic evolution from fully ordered initial states to show that universal scaling exists already in the short-time regime in form of power-law behavior of the magnetization and Binder cumulant. The results measured for the dynamic and static critical exponents, θ, z, β and ν, confirm explicitly that the Potts models on the triangular lattice and square lattice belong to the same universality class. Our critical scaling analysis strongly suggests that the simulation for the dynamic relaxation can be used to determine numerically the universality.
Dynamic relaxation of the XY-model quenched from a high temperature state to the critical temperature or below is investigated with Monte Carlo methods. When a non-zero initial magnetization is given, in the short-time regime of the dynamic evolution the critical initial increase of the magnetization is observed. The dynamic exponent is directly determined. The results show that the exponent varies with respect to the temperature. Furthermore, it is demonstrated that this initial increase of the magnetization is universal, i.e. independent of the microscopic details of the initial configurations and the algorithms.
In this work the two-dimensional Ising model with nearest- and next-nearest-neighbor interactions is revisited. We obtain the dynamic critical exponents z and θ from short-time Monte Carlo simulations. The dynamic critical exponent z is obtained from the time behavior of the ratio [Formula: see text], whereas the non-universal exponent θ is estimated from the time correlation of the order parameter <M(0)M(t)> ~ tθ, where M(t) is the order parameter at instant t, d is the dimension of the system and <(⋯)> is the average of the quantity (⋯) over different samples. We also obtain the static critical exponents β and ν by investigating the time behavior of the magnetization.
Comprehensive Monte Carlo simulations of the short-time dynamic behaviour are reported for the three-dimensional Ising model at criticality. Besides the exponent of the critical initial increase and the dynamic exponent z , the static critical exponents and as well as the critical temperature are determined from the power-law scaling behaviour of observables at the beginning of the time evolution. States of very high temperature as well as of zero temperature are used as initial states for the simulations.