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Physical review. B, Condensed matter

Monte Carlo investigation of the one-dimensional dynamical Potts model

Alain S. Karma, Michael J. Nolan

Published Feb 1, 1983• 1 citation
Publisher

Abstract

Monte Carlo results are presented for a variety of one-dimensional dynamical $q$-state Potts models. Our calculations confirm the expected universal value $z=2$ for the dynamic scaling exponent. Our results also indicate that an increase in $q$ at fixed correlation length drives the dynamics into the scaling regime

DOI
10.1103/physrevb.27.1922

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Monte Carlo results are presented for a variety of one-dimensional dynamical qqq-state Potts models. Our calculations confirm the expected universal value z=2z=2z=2 for the dynamic scaling exponent. Our results also indicate that an increase in qqq at fixed correlation length drives the dynamics into the scaling regime
We study two-dimensional q -state random-bond Potts models for both q=8 and q=5 with a linearly varying temperature. By applying a successive Monte Carlo renormalization group procedure, both the static and dynamic critical exponents are obtained for randomness amplitudes (the strong to weak coupling ratio) of r_{0}=3 , 10, 15, and 20. The correlation length exponent nu increases with disorder from less than to larger than unity and this variation is justified by the good collapse of the specific heat near the critical region. The specific heat exponent is obtained by the usual hyperscaling relation alpha=2-dnu and thus indicates no possibility of the activated dynamic scaling. Both r_{0} and q have effects on the critical dynamics of the disordered systems, which can be seen from variations of the rate exponent, the hysteresis exponent, and the dynamic critical exponent. Implications of these results are discussed.
We present results of Monte Carlo simulations of random bond Potts models in two dimensions, for different numbers of Potts states q. We introduce a simple scheme which yields continuous self-dual distributions of the interactions. As expected, we find multifractal behavior of the correlation functions at the critical point and obtain estimates of the exponent ${\ensuremath{\eta}}_{n}$ for several moments n of the correlation functions, including typical $(\stackrel{\ensuremath{\rightarrow}}{n}0)$, average (n=1)(n=1)(n=1), and others. In addition, for q=8q=8q=8, we find that there is only a single correlation length exponent $\ensuremath{\nu}$ describing the correlation length away from criticality. This is numerically very close to the pure Ising value $\ensuremath{\nu}=1$.
The critical behavior of the three-dimensional random-bond Potts model for large state numbers with q=16 and q=32 is studied by the finite-time dynamics combining with Monte Carlo renormalization group method. The correlation length and dynamic exponents as well as the critical temperatures are calculated through reducing the influence of finite-size effects. According to the variations of critical exponents with disorder amplitude, we identify the asymptotic critical exponents with ν=0.424(14), z=3.03(12) and find that they are q-independent. The results further confirm that the intrinsic ν, instead of finite-size one, can escape the bound ν>2/d suggested previously.