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arXiv:cond-mat/9311055 · Physical review. B, Condensed matter

Three-state Potts model on a triangular lattice

Hyunggyu Park

Published May 1, 1994• 13 citations

Abstract

We study the phase diagram of the three-state Potts model on a triangular lattice with general interactions (ferro- and/or antiferromagnetic) between nearest-neighbor spins. When the interactions along two lattice-vector directions are antiferromagnetic and infinitely strong, this model becomes equivalent to a six-vertex model and exhibits a first-order potassium dihydrogen phosphate (KDP) transition from an ordered phase into a critical phase. Comparing the excitations that occurred by relaxing the restriction of infinite-strength interactions and those in the eight-vertex model, we analytically obtain the critical index for those excitations and demonstrate the existence of a critical phase for the case of finite antiferromagnetic interactions in two directions and ferromagnetic interactions in the other direction. When the interactions are antiferromagnetic in all three directions, Monte Carlo simulations show that a first-order line emerges from the KDP point and separates completely an ordered phase and a disordered phase. Along the special line where all three antiferromagnetic interactions have the same strength, the cell-spin analysis reveals that the symmetry of the ground states is dual to the symmetry of the n=3 ferromagnetic cubic model which is known to exhibit a first-order phase transition.

DOI: 10.1103/physrevb.49.12881

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We study the phase diagram of the three-state Potts model on a triangular lattice with general interactions (ferro- and/or antiferromagnetic) between nearest-neighbor spins. When the interactions along two lattice-vector directions are antiferromagnetic and infinitely strong, this model becomes equivalent to a six-vertex model and exhibits a first-order potassium dihydrogen phosphate (KDP) transition from an ordered phase into a critical phase. Comparing the excitations that occurred by relaxing the restriction of infinite-strength interactions and those in the eight-vertex model, we analytically obtain the critical index for those excitations and demonstrate the existence of a critical phase for the case of finite antiferromagnetic interactions in two directions and ferromagnetic interactions in the other direction. When the interactions are antiferromagnetic in all three directions, Monte Carlo simulations show that a first-order line emerges from the KDP point and separates completely an ordered phase and a disordered phase. Along the special line where all three antiferromagnetic interactions have the same strength, the cell-spin analysis reveals that the symmetry of the ground states is dual to the symmetry of the n=3 ferromagnetic cubic model which is known to exhibit a first-order phase transition.

We study phase transitions of the Potts model on the centered-triangular lattice with two types of couplings, namely, K between neighboring triangular sites, and J between the centered and the triangular sites. Results are obtained by means of a finite-size analysis based on numerical transfer-matrix calculations and Monte Carlo simulations. Our investigation covers the whole (K,J) phase diagram, but we find that most of the interesting physics applies to the antiferromagnetic case K<0, where the model is geometrically frustrated. In particular, we find that there are, for all finite J, two transitions when K is varied. Their critical properties are explored. In the limits J→±∞ we find algebraic phases with infinite-order transitions to the ferromagnetic phase.

The phase transitions in the two-dimensional ferro- and antiferromagnetic Potts models with q = 3 states of spin on a triangular lattice are studied using cluster algorithms and the classical Monte Carlo method. Systems with linear sizes L = 20–120 are considered. The method of fourth-order Binder cumulants and histogram analysis are used to discover that a second-order phase transition occurs in the ferromagnetic Potts model and a first-order phase transition takes place in the antiferromagnetic Potts model. The static critical indices of heat capacity (α), magnetic susceptibility (γ), magnetization (β), and correlation radius index (ν) are calculated for the ferromagnetic Potts model using the finite-size scaling theory.

The three-state Potts model with antiferromagnetic nearest-neighbor (NN) and ferromagnetic next-nearest-neighbor (NNN) interaction is investigated within a mean-field theory. We find that the phase diagram contains two kinds of ordered phases, the so-called broken-sublattice-symmetry and permutationally-symmetric-sublattice phases, separated by a discontinuous phase transition line. The order-disorder transition is continuous for weak NNN interactions and becomes a discontinuous transition, via a tricritical point, when the strength of the NNN interaction is increased.

The phase transitions in 2D ferro- and antiferromagnetic Potts models with number of spin states q = 3 on a triangular lattice are investigated by the cluster and classical Monte Carlo methods. Systems with linear sizes L = 20–120 are considered. Fourth-order Binder cumulants and histogram data analysis are used to show that second- and first-order phase transitions are observed in the ferromagnetic and antiferromagnetic Potts models, respectively. The static critical indices are calculated for specific heat α, susceptibility γ, magnetization β, and correlation length ν on the basis of finite-size scaling theory for a ferromagnetic Potts model.

Abstract The Monte Carlo method has been used to study phase transitions and critical properties of the antiferromagnetic 3-state Potts model on a body-centered cubic lattice with interactions of the first (J1) and second (J2) nearest neighbors. The competition of exchange interactions in the range –2/3 < J2 < 0 was determined to result in a phase separation. In the low-temperature region, an ordered phase, lying inside the phase with broken sublattice symmetry, was found. The magnetic structures of the ground state corresponding to different regions of the phase diagram have been obtained. The analysis of the phase transition order has been carried out using the Binder cumulant method and the histogram method. It has been revealed that the interaction of the second nearest neighbors leads to a change in the type of phase transition. Static critical exponents have been calculated based on scaling relations. The critical exponents correspond to the universality class of the three-dimensional XY model.

The three-state Potts antiferromagnetic model on the triangular lattice (3PAFT) has a weak first-order transition in the pure case. The first-order nature follows from analytic arguments and while it has been observed in some simulations, the precision of (and internal disagreement between) these observations has left much to be desired. The 3PAFT is the only two-dimensional nearest-neighbour Potts model with an ordered phase that does not have an infinite degeneracy, that remains without an exact value for its critical point. This model is expected to exhibit interesting glassy behaviour and a possible crossover to a second-order transition upon dilution. A careful characterization of the nature of the pure transition is needed in order to explore the physics of the dilute model. We find clear indication of a first-order transition at T t =0.62731+or-0.00006, using a cluster method that can be extended to the dilute case and histogram reweighting analysis. Our estimate is two orders of magnitude more precise than previous simulations and falls within the error bounds of a reanalysis of existing series expansion (T t =0.628+or-0.004) reported on herein.