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arXiv:cond-mat/9510124 · Journal of Physics A Mathematical and General

The critical behaviour of Potts models with symmetry- breaking fields

Francisco Castilho Alcaraz, J. C. Xavier

Published Jul 7, 1996• 0 citations
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Abstract

The Q-state Potts model in two dimensions in the presence of external magnetic fields is studied. For general special choices of these magnetic fields produce effective models with smaller symmetry . The phase diagram of these models and their critical behaviour are explored by conventional finite-size scaling and conformal invariance. The possibility of multicritical behaviour, for finite values of the symmetry-breaking fields, in the cases where Q > 4 is also analysed. Our results indicate that for effective models with symmetry the multicritical point occurs at zero field. This last result is also corroborated by Monte Carlo simulations.

DOI
10.1088/0305-4470/29/13/009

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The Q-state Potts model in two dimensions in the presence of external magnetic fields is studied. For general special choices of these magnetic fields produce effective models with smaller symmetry . The phase diagram of these models and their critical behaviour are explored by conventional finite-size scaling and conformal invariance. The possibility of multicritical behaviour, for finite values of the symmetry-breaking fields, in the cases where Q > 4 is also analysed. Our results indicate that for effective models with symmetry the multicritical point occurs at zero field. This last result is also corroborated by Monte Carlo simulations.
The q-state Potts model on a square lattice and in the presence of an external magnetic field that couples to only one state is studied by using extensive Monte Carlo simulations. A hybrid algorithm that combines the single spin-flip Metropolis method with Wolff cluster updates is employed. Single-histogram reweighting techniques and the field-mixing approach are also used to obtain the corresponding universal probability distributions. By analyzing the finite-size scaling behavior of the order parameters and universal probability distributions, the phase diagram in the temperature-versus-external field plane is obtained in the thermodynamic limit for some values of state q≥5. For positive fields and higher values of q, there is a clear first-order transition line that starts at the zero-field transition point and ends at an isolated critical point, which is in the two-dimensional Ising universality class. As the value of state q decreases, the corresponding critical field also decreases. The present results strongly suggest that the critical field vanishes when q→5^{+}. However, for negative fields, another first-order transition line occurs without any critical or multicritical behavior for q≥6. The special case with q=5 undergoes a weak first-order transition at zero field and a line of second-order transitions for negative fields. This second-order transition line is in the same universality class as the q=4 Potts model.
The analysis of renormalisation group equations for the q-state Potts model in two dimensions is generalised to include the presence of a surface field H s besides the bulk one, H B . Combined with conformal invariance results, this allows prediction of a surface susceptibility behaviour delta 2/f delta H s delta H B approximately L -1/8 (1-C'lnL) 15/16 for a q=4 system of size L at the critical temperature. This prediction is checked by a Monte Carlo based finite-size scaling analysis, which also nicely reproduces the exact and conjectured magnetic surface exponents for q=2 and 3, respectively. For the Baxter-Wu model similar methods give results consistent with the q=4 behaviour without logarithmic corrections (c'=0).
We apply simulated tempering and magnetizing (STM) Monte Carlo simulations to the two-dimensional threestate Potts model in an external magnetic field in order to investigate the crossover scaling behaviour in the temperature-field plane at the Potts critical point and towards the Ising universality class for negative magnetic fields.Our data set has been generated by STM simulations of several square lattices with sizes up to 160×160 spins, supplemented by conventional canonical simulations of larger lattices at selected simulation points.We present careful scaling and finite-size scaling analyses of the crossover behaviour with respect to temperature, magnetic field and lattice size.
The effects of linear and quadratic symmetry-breaking perturbations on the continuous version of the three-state Potts model are analysed, using both Landau's theory of phase transitions and renormalisation-group techniques. Variation of the strength of the perturbations produces many different types of phase diagrams, featuring lambda lines, first-order lines, critical, tricritical and triple points and other complexities. Universal amplitude ratios characterising the multicritical points are calculated to first order in epsilon =4-d.
The phase diagram and critical behavior of the spin-1 Baxter-Wu model with a crystal field in two dimensions is explored by renormalization group, conventional finite-size scaling, and conformal invariance techniques. We found that the phase diagram of this model is qualitatively the same as that of the dilute 4-states Potts model, presenting a multicritical point for a finite value of the crystal field, in disagreement with previous work based on finite-size calculations. However, our results indicate that the critical exponents vary continuously along the second-order transition line, differently from the expected behavior of the dilute 4-states Potts model.
The phase diagram and the critical behavior of the spin-1 and the spin-3/2 two-dimensional Baxter-Wu model in a crystal field are studied by conventional finite-size scaling and conformal invariance theory. The phase diagram of this model, for the spin-1 case, is qualitatively the same as those of the diluted 4-states Potts model and the spin-1 Blume-Capel model. However, for the present case, instead of a tricritical point one has a pentacritical point for a finite value of the crystal field, in disagreement with previous work based on finite-size calculations. On the other hand, for the spin-3/2 case, the phase diagram is much richer and can present, besides a pentacritical point, an additional multicritical end point. Our results also support that the universality class of the critical behavior of the spin-1 and spin-3/2 Baxter-Wu model in a crystal field is the same as the pure Baxter-Wu model, even at the multicritical points.