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Journal of Physics A Mathematical and General

Critical relaxation of the two-dimensional three-state Potts model with conserved dynamics

M. Aydin, M. Cemal Yalabık

Published Feb 7, 1988• 4 citations
Publisher

Abstract

The dynamical critical exponent z is obtained using the finite-size scaling method for the two-dimensional three-state Potts model with conserved dynamics. It is observed that there is a change in the dynamical behaviour of the system at the size n=15. For finite lattices of sizes n<or=15, the value of z is predicted as z=2.0+or-0.1. For n>or=15, the asymptotic behaviour is observed and z is calculated as z=2.78+or-0.20.

DOI
10.1088/0305-4470/21/3/032

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The dynamical critical exponent z is obtained using the finite-size scaling method for the two-dimensional three-state Potts model with conserved dynamics. It is observed that there is a change in the dynamical behaviour of the system at the size n=15. For finite lattices of sizes n<or=15, the value of z is predicted as z=2.0+or-0.1. For n>or=15, the asymptotic behaviour is observed and z is calculated as z=2.78+or-0.20.
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.
We study the dynamic critical behavior of the two-dimensional random-bond three-state Potts model using large-scale Monte Carlo simulations with the Wolff and Swendsen-Wang cluster algorithms. At the disorder-dependent critical temperature, we compute integrated and exponential autocorrelation times and extract the dynamic critical exponent z via finite-size scaling. Critical slowing down is significantly weakened by bond randomness, with the dynamic exponent decreasing from z≈0.54 in the pure system to z≈0.24 at strong disorder. From the finite-size scaling of the Wolff cluster size, we obtain γ/ν≈1.73, where γ and ν are the susceptibility and correlation-length exponents, respectively, consistent with the universality class of the two-dimensional three-state Potts model. These results indicate that bond randomness strongly affects the critical dynamics but leaves the underlying static universality class unchanged. Furthermore, both static and dynamic observables, including the specific heat and the autocorrelation times, show a lack of self-averaging.
The critical relaxation of the four-state Potts model on L*L lattices with L<or=64 is studied by the Monte Carlo method. The decay of the magnetisation from the nonequilibrium initial state exhibits a crossover from a power-law relaxation for t<<t c (L) to an exponential one for t>>t c (L), where t c (L) is some crossover time. This crossover is discussed within the context of dynamic finite-size scaling theory. As a result, the dynamic critical exponent is estimated to be Z=2.94+or-0.29.
The static critical behavior of the three-dimensional weakly diluted Potts model with the state q = 3 on a simple cubic lattice has been investigated by the Monte Carlo method using the Wolff single-cluster algorithm. It is shown that at the spin concentrations p = 0.9 and 0.8 a second-order phase transition is observed in the three-dimensional weakly diluted Potts model with the state q = 3. On the basis of the finite-size scaling theory, we calculated the static critical exponents of the specific heat α, susceptibility γ, magnetization β, and the correlation-length exponent v .