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Journal of Physics C Solid State Physics

Critical behaviour of the dilute Ashkin-Teller-Potts model

Julia M. Yeomans, Robin B. Stinchcombe

Published Apr 10, 1980• 29 citations
Publisher

Abstract

A real-space rescaling technique (decimation) is used to study the critical properties of the bond-dilute, S state Ashkin-Teller-Potts model on the square lattice. Fixed points and critical exponents are obtained and critical curves plotted for values of S from 1 to 4. For S=1, 2 the dilute model exhibits the same critical behaviour as the pure system. However, for S=3, 4, there is crossover to a new dilute fixed point. The transition remains second order but exhibits new values of the critical exponents. This behaviour is in accord with the Harris criterion (1974).

DOI
10.1088/0022-3719/13/10/006

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Journal of Physics C Solid State Physics•Aug 14, 1978•1 citation
A real-space rescaling technique (decimation) is used to study the critical properties of the bond-dilute, S state Ashkin-Teller-Potts model on the square lattice. Fixed points and critical exponents are obtained and critical curves plotted for values of S from 1 to 4. For S=1, 2 the dilute model exhibits the same critical behaviour as the pure system. However, for S=3, 4, there is crossover to a new dilute fixed point. The transition remains second order but exhibits new values of the critical exponents. This behaviour is in accord with the Harris criterion (1974).
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.
The critical behavior of the sss-state Ashkin-Teller-Potts model on a square lattice is studied for s=3,4,and5s=3, 4, \mathrm{and} 5s=3,4,and5, using a variational renormalization-group transformation of the type introduced by Kadanoff. For the s=3s=3s=3 and s=4s=4s=4 models, the transformation correctly predicts a continuous phase transition. The calculated values of the critical exponents and the transition temperature are consistent with the results obtained from other approaches. For s=5s=5s=5, however, the transformation fails to detect the first-order transition that is known to occur.
A position space renormalisation group technique (decimation) is used to treat site- and bond-diluted Ising ferromagnets on square and triangular lattices. In every case the renormalisation equations lead to two fixed points, one corresponding to the pure Ising system and one to percolation. Linearising around the fixed points, eigenvalues and hence the critical exponents nu T , nu P (correlation length), alpha (specific heat) and phi (crossover) are obtained. Numerical iteration of the recursion relations provides the critical curves. Values of critical temperatures, critical concentrations, exponents, and of limiting slopes of critical curves are in very satisfactory agreement with all known exact or series results. Points on the critical curve flow into the Ising fixed point under the transformation. The sign of the pure Ising exponent alpha is consistent with this flow for the site-diluted cases but not for the bond-diluted cases.
The (1+1)-dimensional Ashkin-Teller model is studied by the renormalisation group method and by the finite lattice extrapolation technique. The possible phases and their boundaries in an extended space of couplings are determined from the degeneracy structure of the low-lying levels in the thermodynamic limit. The critical behaviour is investigated for ferromagnetic couplings using increasing block sizes in the renormalisation transformation. The line of fixed points and its bifurcation at the Potts fixed point is studied.
The critical behavior of a bond-disordered Ashkin-Teller model on a square lattice is investigated by intensive Monte Carlo simulations. A duality transformation is used to locate a critical plane of the disordered model. This critical plane corresponds to the line of critical points of the pure model, along which critical exponents vary continuously. Along this line the scaling exponent corresponding to randomness \ensuremath{\varphi}=(\ensuremath{\alpha}/\ensuremath{\nu}) varies continuously and is positive so that the randomness is relevant, and different critical behavior is expected for the disordered model. We use a cluster algorithm for the Monte Carlo simulations based on the Wolff embedding idea, and perform a finite size scaling study of several critical models, extrapolating between the critical bond-disordered Ising and bond-disordered four-state Potts models. The critical behavior of the disordered model is compared with the critical behavior of an anisotropic Ashkin-Teller model, which is used as a reference pure model. We find no essential change in the order parameters' critical exponents with respect to those of the pure model. The divergence of the specific heat C is changed dramatically. Our results favor a logarithmic type divergence at Tc{\mathit{T}}_{\mathit{c}}Tc​, C\ensuremath{\sim}lnL for the random-bond Ashkin-Teller and four-state Potts models and C\ensuremath{\sim}ln lnL for the random-bond Ising model.
The Legendre transformation, a modification of the decimation method used in the real space renormalisation group, is employed to study the critical behaviour of the two-dimensional S-state Potts model which exhibits a second-order phase transition for S<or=4. Thermal and magnetic critical exponents are calculated for the percolation (S to 1), Ising (S=2) and three- and four-state Potts models. Decimation estimates of the critical exponents and critical points for S=2, 3 and 4 are also presented.