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

Determination of dynamical critical exponents from hysteresis scaling

Guangping Zheng, J. X. Zhang

Published Aug 1, 1998• 12 citations
Publisher

Abstract

A method is proposed to determine the dynamical critical exponent. The method is based on the scaling for dynamical hysteresis resulted from a linearly swept field. We prove that in model $A$ dynamical hysteresis scaling at critical temperature is universal. The nearest-neighbor Ising models are used to demonstrate such concepts and the dynamical critical exponents can be determined accurately. We also propose a universal relation between static and dynamical critical exponents in Ising class in single-spin-flip dynamics.

DOI
10.1103/physreve.58.r1187

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A method is proposed to determine the dynamical critical exponent. The method is based on the scaling for dynamical hysteresis resulted from a linearly swept field. We prove that in model AAA dynamical hysteresis scaling at critical temperature is universal. The nearest-neighbor Ising models are used to demonstrate such concepts and the dynamical critical exponents can be determined accurately. We also propose a universal relation between static and dynamical critical exponents in Ising class in single-spin-flip dynamics.
Hysteresis is observed at second order phase transitions. Universal scaling formulæ for the areas of hysteresis loops are written down. Critical exponents are defined, and related to other exponents for static and dynamic critical phenomena. These relations are verified with Langevin dynamics in both the critical and tricritical mean-field models. A finite-size scaling relation is tested in the two-dimensional Ising model with heat-bath dynamics. (Shar file; 2 figures created through dvips directives from 4 PS files included)
Dynamical phase transitions in the Ising model on hypercubic lattices are considered. Under a linearly swept magnetic field, the hysteresis loop that characterizes the field-driven first-order phase transition is studied carefully. Using the Glauber dynamics, we find that, in the mean-field approximation, the energy dissipation of this phase transition or the hysteresis loop area A of the M-H curve can be scaled with respect to the sweep rate h of magnetic field in the form with a = 2 and b = 2/3. However, b varies when fluctuations and spin correlations are taken into account. Monte Carlo simulation is used to obtain the scaling relation for A in two-, three- and four-dimensional Ising models and we obtain the exponents and respectively. These exponents are obviously different from those obtained by scaling A as for any temperatures in Ising models under a sinusoidal field. Finally we point out that, in the concept of universality, field-driven first-order phase transitions in the Ising model in different dimensions belong to different universal classes due to the spin fluctuation and correlation below the Curie temperature.
We extend the Monte Carlo renormalization-group technique to investigate the scaling behavior in the vicinity of the critical point of the two-dimensional Ising model in the presence of linearly changing temperatures, instead of some static temperatures. We find that it is possible in such a dynamically driven, nonequilibrium case to measure both the dynamic and static critical exponents. Two additional critical exponents associated with the temperature sweep rate are presented, and scaling laws relating them to other known exponents are obtained. Also a scaling form of the order parameter is extracted.
Recent results concerning non-universality in the dynamical critical behavior of the one-dimensional Ising model, are generalised to several formulations of the Potts model, using arguments based on domain wall motion. By applying real-space renormalisation group methods, the authors also study the dynamical Ising model, with single flips, both for a period distribution of nearest-neighbor exchanges and for the random chain (with nonzero couplings). They show that the critical dynamical exponent is z=1+J M /J m where J M (J m ) is the greatest (smallest) of the exchanges.
Dynamical scaling study for the estimation of dynamical exponent zzz of three-dimensional XY spin glass model
To analyze the ±J\pm J±J XY spin-glass in three dimensions, we verified a method aimed at obtaining a high-precision dynamical exponent zzz from the correlation length in the nonequilibrium relaxation process. The obtained zzz yielded consistent and highly accurate results in previous studies for relatively well-studied models---specifically, the three-dimensional (3D) ferromagnetic Ising model and the 3D ±J\pm J±J Ising model. Building on these previous studies, we used this method and the dynamical scaling method to analyze the 3D ±J\pm J±J XY model and obtained highly precise critical temperatures and exponents. The estimated transition temperatures satisfy $T_{\mathrm{SG}}