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Phase Transitions

Critical behavior of the three-dimensional random-bond Potts model with many states

Wanjie Xiong, Chudong Xu, Zizheng Guo

Published May 29, 2013• 0 citations
Publisher

Abstract

The critical behavior of the three-dimensional random-bond Potts models with many states ( and ) is studied by the finite-time dynamic method. We evaluate the static and dynamic critical exponents as well as the critical temperatures. The asymptotic critical behavior for is identified with , and but the exponents obtained for with are the only effective ones. Our results support that is positive and the intrinsic can escape the bound suggested by previous work. By comparing with the results for various , we infer that the asymptotic critical behavior of the three-dimensional random-bond Potts model with many states is -dependent.

DOI
10.1080/01411594.2013.797981

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The critical behavior of the three-dimensional random-bond Potts models with many states ( and ) is studied by the finite-time dynamic method. We evaluate the static and dynamic critical exponents as well as the critical temperatures. The asymptotic critical behavior for is identified with , and but the exponents obtained for with are the only effective ones. Our results support that is positive and the intrinsic can escape the bound suggested by previous work. By comparing with the results for various , we infer that the asymptotic critical behavior of the three-dimensional random-bond Potts model with many states is -dependent.
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.
The short-time critical dynamics of the two-dimensional eight-state random-bond Potts model is investigated with large-scale Monte Carlo simulations. Dynamic relaxation starting from a disordered and an ordered state is carefully analyzed. The continuous phase transition induced by disorder is studied, and both the dynamic and static critical exponents are estimated. The static exponent beta/nu shows little dependence on the disorder amplitude r, while the dynamic exponent z and static exponent 1/nu vary with the strength of disorder.
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.
We have investigated the critical behavior of a three-dimensional random-bond Ising model for a series of the disorder strength by a finite-time scaling combining with Monte Carlo renormalization-group method in the presence of a linearly varying temperature. The method enables us to estimate a lot of critical exponents of both static and dynamic nature independently as well as the critical temperatures. The static exponents obtained agree well with most existing results, verify both the hyperscaling and the Rushbrooke scaling laws and their combined scaling law, which in turn validate their asymptotic nature, and corroborate the universality of the relevant random fixed point with respect to the forms of disorder. The dynamic critical exponent z is estimated to 2.114(51), which is compatible with those obtained from experiments and renormalization-group analyses. The exponents at low and high disorder strengths do not satisfy all scaling laws and are argued to be crossover exponents that reflect crossover from the random fixed point to the pure and the percolation fixed point. They also indicate that the exponents that were previously suggested to be a distinct universality class for strong disorder strength in the literature may be just crossover. Our results demonstrate the effectiveness of the finite-time scaling method.