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Journal of Physics Condensed Matter

Monte Carlo investigation of the three-dimensional random-field three-state Potts model

Klaus D. Eichhorn, Kurt Binder

Published Jul 8, 1996• 76 citations
Publisher

Abstract

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.

DOI
10.1088/0953-8984/8/28/005

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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.
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 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 .
We present a Monte Carlo simulation of the d = 3 Ising model in a random magnetic field with values ±H on each lattice site. Calculations have been performed on cubic lattices with linear dimension L = 6, 8, 10 and 12, and several values of H. A finite-size scaling analysis of the data for H = 0.5, 0.75 and 1.0 is consistent with hyperscaling, and a correlation exponent v = 0.7 ± 0.1. For H = 1.50, 1.75 and 2.0 some of the L = 12 data show a discontinuity as a function of the inverse temperature β. The implications of these results are discussed.
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.
A Monte Carlo method has been used to study a simple cubic Ising ferromagnet in a random quenched magnetic field. The Hamiltonian for this model is ℋ==JΣ(ij)σiσj−ΣiHiσi, where σi,σj=±1, J is the nearest-neighbor interaction constant, and the field Hi=tH is fixed at each site with ti=±1 at random and Σti=0. L×L×L lattices with periodic boundary conditions have been studied for a range of H and T. As expected we find a ferromagnetic ordered state which for small H undergoes a second order phase change to the paramagnetic state with increasing temperature. A finite size scaling analysis of the preliminary data suggests that the critical exponent β is substantially smaller (β∼0.2) than the usual 3-dim Ising value of 0.31. Results obtained for small lattices indicate that below kT/J-2 the transition becomes first order suggesting that a tricritical point appears on the critcal field curve.
For small lattices with linear dimension L ranging from L=3 to L=8 we obtain the distribution function P(q) of the overlap q between two real replicas of the three-state Potts-glass model with symmetric nearest-neighbor interaction with a Gaussian distribution. A finite-size-scaling analysis suggests a zero-temperature transition to occur with an exponentially diverging correlation length ${\ensuremath{\xi}}_{\mathrm{SG}}$\ensuremath{\sim}exp(C/${\mathit{T}}^{\mathrm{\ensuremath{\sigma}}}$). This implies that d=3 is the lower critical dimension.