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Journal of Physics C Solid State Physics

Quenched bond disorder in the Potts model: single-bond effective interaction approximation

Loïc Turban

Published Jan 10, 1980• 18 citations
Publisher

Abstract

An effective-medium approximation is proposed for thermal phase transitions. For quenched bond disorder in the Potts model, the exact critical line is obtained when s=1; for s=2 (Ising), some exact results in the dilute limit and the correct behaviour for T c (p) near p c are recovered. The critical concentrations are in good agreement with the known values, in both two and three dimensions. For s>4 (first-order transition) the phase diagram shows a two-phase region.

DOI
10.1088/0022-3719/13/1/004

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An effective-medium approximation is proposed for thermal phase transitions. For quenched bond disorder in the Potts model, the exact critical line is obtained when s=1; for s=2 (Ising), some exact results in the dilute limit and the correct behaviour for T c (p) near p c are recovered. The critical concentrations are in good agreement with the known values, in both two and three dimensions. For s>4 (first-order transition) the phase diagram shows a two-phase region.
A systematic cluster extension of the effective-interaction approximation for quenched-bond disorder in random Ising systems is proposed. For bond dilution, exact results for the critical concentrations are obtained in the single-bond approximation for the self-dual square lattice and in the three-bonds approximation for the dual triangular and honeycomb lattices. Qualitative changes in the critical line are observed in the frustration model when going from the single-bond to the three-bonds approximations.
We study by extensive Monte Carlo simulations the effect of random bond dilution on the phase transition of the three-dimensional four-state Potts model that is known to exhibit a strong first-order transition in the pure case. The phase diagram in the dilution-temperature plane is determined from the peaks of the susceptibility for sufficiently large system sizes. In the strongly disordered regime, numerical evidence for softening to a second-order transition induced by randomness is given. Here a large-scale finite-size scaling analysis, made difficult due to strong crossover effects presumably caused by the percolation fixed point, is performed.
We study by Monte Carlo simulations the influence of bond dilution on the three-dimensional Ising model. This paradigmatic model in its pure version displays a second-order phase transition with a positive specific heat critical exponent α\alphaα . According to the Harris criterion disorder should hence lead to a new fixed point characterized by new critical exponents. We have determined the phase diagram of the diluted model, starting from the pure model limit down to the neighbourhood of the percolation threshold. For the estimation of critical exponents, we have first performed a finite-size scaling study, where we concentrated on three different dilutions to check the stability of the disorder fixed point. We emphasize in this work the great influence of the cross-over phenomena between the pure, disorder and percolation fixed points which lead to effective critical exponents dependent on the concentration. In a second set of simulations, the temperature behaviour of physical quantities has been studied in order to characterize the disorder fixed point more accurately. In particular this allowed us to estimate ratios of some critical amplitudes. In accord with previous observations for other models this provides stronger evidence for the existence of the disorder fixed point since the amplitude ratios are more sensitive to the universality class than the critical exponents. Moreover, the question of non-self-averaging at the disorder fixed point is investigated and compared with recent results for the bond-diluted q = 4 Potts model. Overall our numerical results provide evidence that, as expected on theoretical grounds, the critical behaviour of the bond-diluted model is indeed governed by the same universality class as the site-diluted model.
In order to study the influence of quenched disorder on second-order phase transitions, high-temperature series expansions of the susceptibility and the free energy are obtained for the quenched bond-diluted Ising model in d=3−−5d=3--5d=3−−5 dimensions. They are analyzed using different extrapolation methods tailored to the expected singularity behaviors. In d=4d=4d=4 and 5 dimensions we confirm that the critical behavior is governed by the pure fixed point up to dilutions near the geometric bond percolation threshold. The existence and form of logarithmic corrections for the pure Ising model in d=4d=4d=4 are confirmed, and our results for the critical behavior of the diluted system are in agreement with the type of singularity predicted by renormalization group considerations. In three dimensions we find large crossover effects between the pure Ising, percolation, and random fixed points. We estimate the critical exponent of the susceptibility to be $\ensuremath{\gamma}=1.305(5)$ at the random fixed point.
The critical properties of systems with quenched bond disorder are determined from a fixed distribution, under renormalization group, of the random bonds. Full fixed distributions with all moments are obtained numerically by histograms and, to a good approximation, in terms of $\ensuremath{\Gamma}$ distributions. For such systems, the specific-heat exponent $\ensuremath{\alpha}$ does not equal the crossover exponent $\ensuremath{\varphi}$ at random criticality. We derive a new relation between $\ensuremath{\alpha}$ and $\ensuremath{\varphi}$, which invokes characteristics of the fixed distribution. The difference between $\ensuremath{\alpha}$ and $\ensuremath{\varphi}$ is noted for nnn-vector models in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and for Potts models on hierarchical lattices solved exactly. In general, stable random critical behavior with positive $\ensuremath{\alpha}$ appears to be possible. We develop a general treatment of quenched disorder and illustrate it by calculating specific-heat curves. It is suggested that the critical exponents of the three- and four-state random-bond Potts models in two dimensions are $\ensuremath{\nu}\ensuremath{\simeq}1.06 \mathrm{and} 1.19$.