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arXiv:cond-mat/9705233

Generalized Dynamic Scaling for Critical Magnetic Systems

Bo Zheng

Published May 22, 1997• 128 citations
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Abstract

The short-time behaviour of the critical dynamics for magnetic systems is investigated with Monte Carlo methods. Without losing the generality, we consider the relaxation process for the two dimensional Ising and Potts model starting from an initial state with very high temperature and arbitrary magnetization. We confirm the generalized scaling form and observe that the critical characteristic functions of the initial magnetization for the Ising and the Potts model are quite different.

DOI
10.48550/arxiv.cond-mat/9705233

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The short-time behaviour of the critical dynamics for magnetic systems is investigated with Monte Carlo methods. Without losing the generality, we consider the relaxation process for the two dimensional Ising and Potts model starting from an initial state with very high temperature and arbitrary magnetization. We confirm the generalized scaling form and observe that the critical characteristic functions of the initial magnetization for the Ising and the Potts model are quite different.
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The time evolution of the three-dimensional critical Ising model relaxing from a non-equilibrium initial state is studied by means of Monte Carlo simulation. We observe the characteristic initial increase of the (spatially) averaged magnetization predicted by Janssen et al. The exponent theta ' that governs the initial behaviour is determined, and the dependence of the long-time linear decay on the initial magnetization, m 0 , is analysed. Our simulation corroborates earlier results derived from continuum models.
The critical relaxation of the magnetization in a two-dimensional Ising model with quenched random non-magnetic impurities has been studied by numerical simulation. A squared lattice of size 400 2 with spin concentrations p=1.0; 0.95, 0,9, 0.85, 0.8, 0.75 and 0.7 was considered. The dynamical critical exponent z was determined by the Monte Carlo method combined with the dynamical renormalization group method. The following values were received for z(p): z(1)=2.24+or-0.07, z(0.95)=2.24+or-0.06, z(0.9)=2.24+or-0.06, z(0.85)=2.38+or-0.05, z(0.8)=2.51+or-0.06, z(0.75)=2.66+or-0.07 and z(0.7)=2.88+or-0.06. A singular dynamic scaling behaviour with z=A' mod ln(p-p c ) mod +B' was found for systems with spin concentrations p<or=0.85 near the percolation threshold p c , where A'=0.56+or-0.07 and B'=1.62+or-0.07 are temperature-independent constants.
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.
In this work we have considered the Taylor series expansion of the dynamic scaling relation of the magnetization with respect to small initial magnetization values in order to study the dynamic scaling behaviour of 2- and 3-dimensional Ising models. We have used the literature values of the critical exponents and of the new dynamic exponent x0 to observe the dynamic finite-size scaling behaviour of the time evolution of the magnetization during early stages of the Monte Carlo simulation. For 3-dimensional Ising Model we have also presented that this method opens the possibility of calculating z and x0 separately. Our results show good agreement with the literature values. Measurements done on lattices with different sizes seem to give very good scaling.