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arXiv:cond-mat/0201285 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Simulation of Potts models with realqand no critical slowing down

Ferdinando Gliozzi

Published Jul 23, 2002• 34 citations
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Abstract

A Monte Carlo algorithm is proposed to simulate the ferromagnetic q-state Potts model for any real q>0. A single update is a random sequence of disordering and deterministic moves, one for each link of the lattice. A disordering move attributes a random value to the link, regardless of the state of the system, while in a deterministic move this value is a state function. The relative frequency of these moves depends on the two parameters q and beta=1/kT. The algorithm is not affected by critical slowing down and the dynamical critical exponent z is exactly vanishing. We simulate in this way a three-dimensional Potts model in the range 2<q<3 for estimating the critical value q(c) where the thermal transition changes from second order to first order, and find q(c)=2.620+/-0.005.

DOI
10.1103/physreve.66.016115

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A Monte Carlo algorithm is proposed to simulate the ferromagnetic q-state Potts model for any real q>0. A single update is a random sequence of disordering and deterministic moves, one for each link of the lattice. A disordering move attributes a random value to the link, regardless of the state of the system, while in a deterministic move this value is a state function. The relative frequency of these moves depends on the two parameters q and beta=1/kT. The algorithm is not affected by critical slowing down and the dynamical critical exponent z is exactly vanishing. We simulate in this way a three-dimensional Potts model in the range 2<q<3 for estimating the critical value q(c) where the thermal transition changes from second order to first order, and find q(c)=2.620+/-0.005.
The three-state Potts model with nearest ferromagnetic interaction J on the simple cubic lattice, exposed to a field which randomly favours one of the states, is studied by finite size scaling analysis of Monte Carlo simulations. The consequences of the hyperscaling violations for the finite size scaling description are worked out, and it is shown that at criticality the order parameter distribution develops a trivial character (superposition of three delta functions), unlike the nontrivial distributions in models where hyperscaling holds. The numerical data are compatible with this interpretation, and are also compatible with a description in terms of only two independent static critical exponents, as proposed by Schwartz et al . Slow relaxation and the need of averaging over a large number of random field samples restricts the range of linear dimensions L to , and in view of crossover problems only rough estimates of critical exponents are obtained.
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.
Three-state Potts ferromagnets on the simple cubic lattice in random fields are studied by extensive Monte Carlo simulation. The implications of the possible hyperscaling violation to finite-size scaling are discussed. Evidence for a second-order transition is obtained, as well as rough estimates for critical exponents. A qualitative phase diagram is proposed compatible with the predictions of Blankschtein et al. on the random field-induced shift of the tricritical curve q = q c ( d ) in the plane of variables dimensionality d and number of states q .
The phase transition in the q -state Potts model with homogeneous ferromagnetic couplings is strongly first order for large q, while it is rounded in the presence of quenched disorder. Here we study this phenomenon on different two-dimensional lattices by using the fact that the partition function of the model is dominated by a single diagram of the high-temperature expansion, which is calculated by an efficient combinatorial optimization algorithm. For a given finite sample with discrete randomness the free energy is a piecewise linear function of the temperature, which is rounded after averaging, however, the discontinuity of the internal energy at the transition point (i.e., the latent heat) stays finite even in the thermodynamic limit. For a continuous disorder, instead, the latent heat vanishes. At the phase transition point the dominant diagram percolates and the total magnetic moment is related to the size of the percolating cluster. Its fractal dimension is found d(f) = ( 5 + square root of 5)/4 and it is independent of the type of the lattice and the form of disorder. We argue that the critical behavior is exclusively determined by disorder and the corresponding fixed point is the isotropic version of the so-called infinite randomness fixed point, which is realized in random quantum spin chains. From this mapping we conjecture the values of the critical exponents as beta=2- d(f), beta(s) =1/2, and nu=1.