Three-state Chiral Potts Model Research Articles

Identifying the Huse-Fisher universality class of the three-state chiral Potts model

Using the corner-transfer matrix renormalization group approach, we revisit the three-state chiral Potts model on the square lattice, a model proposed in the eighties to describe commensurate-incommensurate transitions at surfaces, and with direct relevance to recent experiments on chains of Rydberg atoms. This model was suggested by Huse and Fisher to have a chiral transition in the vicinity of the Potts point, a possibility that turned out to be very difficult to definitely establish or refute numerically. Our results confirm that the transition changes character at a Lifshitz point that separates a line of Pokrosky-Talapov transitions far enough from the Potts point from a line of direct continuous order-disorder transition close to it. Thanks to the accuracy of the numerical results, we have been able to base the analysis entirely on effective exponents to deal with the crossovers that have hampered previous numerical investigations. The emerging picture is that of a new universality class with exponents that do not change between the Potts point and the Lifshitz point, and that appear to be consistent with those of a self-dual version of the model, namely correlation lengths exponents νx=2/3 in the direction of the asymmetry and νy=1 perpendicular to it, an incommensurability exponent β¯=2/3, a specific heat exponent that keeps the value α=1/3 of the three-state Potts model, and a dynamical exponent z=3/2. These results are in excellent agreement with experimental results obtained on reconstructed surfaces in the nineties, and shed light on recent Kibble-Zurek experiments on the period-3 phase of chains of Rydberg atoms.

New symmetries of the chiral Potts model

In this paper a hitherto unknown spectrum generating algebra, consisting of two coupled Temperley–Lieb algebras, is found in the three-state chiral Potts model. From this, we can construct new Onsager integrable models. One realization is in terms of a staggered isotropic XY spin chain. Further we investigate the importance of the algebra for the existence of mutually commuting conserved charges. This leads us to a natural generalization of the boost operator, which generates the charges.

Algebraic geometry of the three-state chiral Potts model

For more than a decade now, the chiral Potts model in statistical mechanics has attracted much attention. A number of mathematical physicists have written quite extensively about it. The solutions give rise to a curve over ℂ, and much effort has gone into studying the curve and its Jacobian.

Dynamical phase transitions in the square lattice three-state chiral Potts model

We study, through the damage spreading method, the dynamical transitions in the square lattice three-state chiral Potts model within the heat-bath dynamics. We clearly see five dynamical phases, among which only four of them can be associated with the static ones (ferromagnetic, chiral, floating and paramagnetic). The additional phase is a dynamical chaotic regime with unusual features which recovers, for a zero chiral field, the new phase reported recently for the standard Potts model. The floating regime has been observed for all values of non-null chiral field, similarly to some predictions for its corresponding equilibrium phase.

CHIRAL POTTS MODEL: CORNER TRANSFER MATRICES AND PARAMETRIZATIONS

We consider the star-triangle relation and the form of its solutions. We present some simple parametrizations of the weight functions of the three-state chiral Potts model. This model does not have the “difference property”: we discuss the resulting difficulties in attempting to use the corner transfer matrix method for this model.

Corner transfer matrices of the chiral Potts model. II. The triangular lattice

We consider a two-dimensional edge-interaction model satisfying the star-triangle relations. For the triangular lattice, the corner transfer matrices are functions of three rapidities: we show that they possess various factorization properties and satisfy certain equations. We indicate how these equations can be solved for the Ising model. We then consider the three-state chiral Potts model and obtain low-temperature solutions to the equations. The conjectured formula for the order parameter (the spontaneous magnetization) is verified to one more order in a series expansion.

Three-state chiral Potts models on the triangular lattice: a Monte Carlo study

The phase transition of the three-state chiral Potts model is investigated by Monte Carlo simulations performed on two types of isotropic triangular lattices of various linear sizes up to 128. These two types of model are shown to have different phase diagrams. In addition, the critical antifferomagnetic point is analyzed in detail.

Three-state chiral Potts models in two dimensions: integrability and symmetry

Symmetry and integrability relations for the three-state chiral Potts model on two-dimensional lattices are reviewed. Our detailed and systematic analysis leads to new integrability varieties for specific chiral Potts models including new integrable points for the standard antiferromagnetic Potts model. We also describe a three-state vertex-model representation of the chiral Potts model, and write down the associated fundamental invariants for three-coordinated lattices.

On the phase diagram of the chiral Potts model

The authors study the phase diagram of the isotropic three-state chiral Potts model. Monte Carlo simulations performed on twisted square lattices with different sizes up to 64*64 indicate different types of critical curves. In particular, a floating phase seems to occur. Finite-size scaling analyses are performed at some specific critical points. The phase diagram they propose is discussed using exact results.

Mean-field theory evidence for a Devil's staircase in the three-state chiral Potts model

The author studies the mean-field theory of the three-state chiral Potts model with isotropic next-nearest-neighbour interactions. Numerical analysis of the mean-field equations shows evidence for an infinite cascade of phases arranged in a Devil's staircase. This result agrees with earlier predictions of the low-temperature expansion technique.

Dynamics of interfacial wetting near the roughening transition

Theoretical and computer simulation results are presented for the effect of fluctuations on the dynamics of layer growth during complete wetting. For example, in two dimensions the authors predict a crossover in the dynamics of the layer thickness W, from Lipowsky's result, W approximately t1/4, for all non-zero temperatures in a lattice-gas system, to W approximately (ln t)1/2 at zero temperature (the roughening transition temperature) for intermediate times. Simulations of wetting in the three-state chiral Potts model give W approximately tpsi , where Psi is 0.25+or-0.03 for temperatures T>or=0.6 Tc, where Tc is the critical temperature, while the effective exponent Psi monotonically decreases with T below this temperature.

Chiral Potts model on a Cayley tree with complete and incomplete devil's staircase

The three-state chiral Potts model on a Cayley tree is analysed in the limit of infinitely large coordination number. The fractal dimensionalities of the wavenumber against chiral field curves are computed. It is shown that they change from a complete to an incomplete devil's staircase as the temperature is raised. Commensurate phase boundaries are determined analytically for high and low temperatures, and the discommensurations are shown to result from tangent bifurcations.

Floating phase in the chiral Potts model. A Monte Carlo renormalization-group analysis

Layers of atoms or molecules adsorbed on surfaces may exhibit floating critical phases. To investigate this phenomenon we have developed a Monte Carlo renormalization-group scheme and performed calculations on the three-state chiral Potts model. The results indicate a continuous transition from a floating phase to a liquid phase at a temperature much higher than that found by series-expansion methods.