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Physical review. B, Condensed matter

Phase diagram of a six-state chiral Potts model

H. Meyer, J C Anglès d'Auriac

Published Mar 1, 1995• 1 citation
Publisher

Abstract

We present the two-dimensional phase diagram of an isotropic six-state chiral Potts model with a noncyclic Boltzmann weight matrix. This model has been introduced as being preintegrable in the sense that a very constraining functional relation for the partition function is known. The phase diagram shows itself to be very rich. In one region the model almost decouples, like a (${\mathit{q}}_{\mathit{s}}$,${\mathit{q}}_{\mathit{t}}$) model, in a product of two simpler models; while in another region a mean-field calculation gives good results compared to our Monte Carlo simulations. All the transition lines, as well as their order, are numerically determined. Besides, using the above-mentioned functional relation, an exact form for one of these transition lines is proposed. The low-energy states are also analyzed showing a variety of zero-temperature behavior in the different regions of the phase diagram.

DOI
10.1103/physrevb.51.6379

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We present the two-dimensional phase diagram of an isotropic six-state chiral Potts model with a noncyclic Boltzmann weight matrix. This model has been introduced as being preintegrable in the sense that a very constraining functional relation for the partition function is known. The phase diagram shows itself to be very rich. In one region the model almost decouples, like a (qs{\mathit{q}}_{\mathit{s}}qs​,qt{\mathit{q}}_{\mathit{t}}qt​) model, in a product of two simpler models; while in another region a mean-field calculation gives good results compared to our Monte Carlo simulations. All the transition lines, as well as their order, are numerically determined. Besides, using the above-mentioned functional relation, an exact form for one of these transition lines is proposed. The low-energy states are also analyzed showing a variety of zero-temperature behavior in the different regions of the phase diagram.
In honor of Onsager’s ninetieth birthday, we like to review some exact results obtained so far in the chiral Potts models and to translate these results into language more transparent to physicists, so that experts in Monte Carlo calculations, high and low temperature expansions, and various other methods, can use them. We shall pay special attention to the interfacial tension ǫr between the k state and the k −r state. By examining the ground states, it is seen that the integrable line ends at a superwetting point, on which the relation ǫr = r ǫ1 is satisfied, so that it is energetically neutral to have one interface or more. We present also some partial results on the meaning of the integrable line for low temperatures where it lives in the non-wet regime. We make Baxter’s exact results more explicit for the symmetric case. By performing a Bethe Ansatz calculation with open boundary conditions we confirm a dilogarithm identity for the low-temperature expansion which may be new. We propose a new model for numerical studies. This model has only two variables and exhibits commensurate and incommensurate phase transitions and wetting transitions near zero temperature. It appears to be not integrable, except at one point, and at each temperature there is a point, where it is almost identical with the integrable chiral Potts model. KEY WORDS: Chiral Potts model; chiral clock model; star-triangle equations; Yang-Baxter equations; interfacial tension; wetting; superwetting;
Abstract We derive the phase diagram of the one-dimensional three-state Potts model with an additional mean-field interaction in the canonical ensemble. The free energy is obtained by mapping the model onto the spin-1 Blume–Emery–Griffiths model and solving it by using a Hubbard–Stratonovich transformation combined with the transfer matrix method. A complex structure with lines of first-order transitions, two triple points and a critical point appears at finite temperature. The phase diagram is two-dimensional, since there are two adjustable parameters, the nearest-neighbour coupling K and the temperature T . We show that the phase diagram does not present second-order phase transition lines, due to the fact that the order parameter is not a symmetry-breaking one. Quite remarkably, we are able to determine analytically one of the first-order phase transition lines. We also prove that, when the nearest-neighbour coupling K is large and negative, the first-order transition temperature becomes asymptotically independent of K .
In this paper we study the phase diagram of the five-state Potts antiferromagnet on the bisected-hexagonal lattice. This question is important since Delfino and Tartaglia recently showed that a second-order transition in a five-state Potts antiferromagnet is allowed, and the bisected-hexagonal lattice had emerged as a candidate for such a transition on numerical grounds. By using high-precision Monte Carlo simulations and two complementary analysis methods, we conclude that there is a finite-temperature first-order transition point. This one separates a paramagnetic high-temperature phase, and a low-temperature phase where five phases coexist. This phase transition is very weak in the sense that its latent heat (per edge) is two orders of magnitude smaller than that of other well-known weak first-order phase transitions.
The static critical properties of the six-state vector Potts model (clock model) on L×L square lattices are studied in zero field using Monte Carlo simulations. Data are obtained for L≤48 keeping between 2.4 and 4.0×104 MCS (Monte Carlo steps/site) for computing averages. The data are analyzed using a block distribution technique in which the critical temperature and exponents may be estimated from the size dependence of the cumulants of the order parameter. We see clear evidence for two phase transitions but our estimates for the transition temperatures are different from results obtained through other methods. For the lattice sizes considered the intermediate phase does not appear to be XY-like.
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.