© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Journal of Physics A Mathematical and General

Three-state Potts antiferromagnet on the triangular lattice

Joan Adler, Achi Brandt, Wolfhard Janke E, S Shmulyian

Published Sep 21, 1995• 37 citations

Abstract

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.

DOI: 10.1088/0305-4470/28/18/005

Related papers

Three-state Potts model on a triangular lattice

Hyunggyu Park

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.

arXiv:cond-mat/9311055•Physical review. B, Condensed matter•May 1, 1994•13 citations

Phase transitions in the two-dimensional antiferromagnetic Potts model on a triangular lattice

Akai Kurbanovich Murtazaev, Albert Babaevich Babaev

Phase transitions and thermodynamic properties in the two-dimensional three-state antiferromagnetic Potts model on a triangular lattice are investigated using the Monte Carlo method and the histogram analysis of the data. It is shown that pronounced first-order phase transitions are observed in this model for systems with rather large linear dimensions ( L > 120). No first-order PTs are observed for systems with L < 120.

Bulletin of the Russian Academy of Sciences Physics•Oct 1, 2013•5 citations

Phase transitions in the two-dimensional ferro- and antiferromagnetic potts models on a triangular lattice

Akai Kurbanovich Murtazaev, Albert Babaevich Babaev

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.

Journal of Experimental and Theoretical Physics•Dec 1, 2012•17 citations

Monte Carlo study of the phase transitions in 2D ferro- and antiferromagnetic Potts models on a triangular lattice

Akai Kurbanovich Murtazaev, Albert Babaevich Babaev

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.

Bulletin of the Russian Academy of Sciences Physics•Jan 1, 2014•2 citations

Three-state Potts model on the maple leaf lattice

Z. F. Wang, B. W. Southern

We use Monte Carlo methods to study the three-state Potts model on a two-dimensional lattice with fivefold coordination. The lattice consists of triangular and hexagonal units and has properties intermediate between the triangular and kagom\'e lattices. The critical behavior of both ferromagnetic and antiferromagnetic cases is considered. As expected, the critical behavior of the ferromagnetic model is independent of the lattice details and lies in the universality class of the three-state ferromagnetic Potts model. A critical end point is found at the coexistence between the ferromagnetic and antiferromagnetic phases, and the transition to the antiferromagnetic state is always first order.

Physical review. B, Condensed matter•Sep 18, 2003•7 citations

\(\sqrt{3}\times\sqrt{3}\) Structure on the Triangular Lattice

Yukio Saitō, Ganji Tabe

An Ising model with nearest neighbor antiferromagnetic and next nearest neighbor ferromagnetic interactions is studied by the phenomenological renormalization method. The phase transition to the \(\sqrt{3}\times\sqrt{3}\) structure is of the second order for weak fields, and the critical exponents \(y_{\text{T}}{\cong}1.2\) and \(\eta{\cong}0.30\) are consistent with the conjectured values of the 3-state Potts model. Beyond a tricritical point the transition becomes of the first order for strong fields. The tricritical exponent \(y_{\text{T,t}}{\cong}1.73\) and \(\eta_{\text{t}}{\cong}0.21\) are also in agreement with the 3-state Potts tricritical values y T,t 3P =12/7 and η t 3P =4/21. The XY-like phase transition at zero fields, which was obtained by the Monte Carlo simulations, is not found in the present calculation.

Journal of the Physical Society of Japan•Aug 15, 1985•12 citations

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.

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.

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 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.

We use Monte Carlo methods to study the three-state Potts model on a two-dimensional lattice with fivefold coordination. The lattice consists of triangular and hexagonal units and has properties intermediate between the triangular and kagom\'e lattices. The critical behavior of both ferromagnetic and antiferromagnetic cases is considered. As expected, the critical behavior of the ferromagnetic model is independent of the lattice details and lies in the universality class of the three-state ferromagnetic Potts model. A critical end point is found at the coexistence between the ferromagnetic and antiferromagnetic phases, and the transition to the antiferromagnetic state is always first order.

\sqrt{3}\times\sqrt{3}

Structure on the Triangular Lattice

An Ising model with nearest neighbor antiferromagnetic and next nearest neighbor ferromagnetic interactions is studied by the phenomenological renormalization method. The phase transition to the

\sqrt{3}\times\sqrt{3}

structure is of the second order for weak fields, and the critical exponents

y_{\text{T}}{\cong}1.2

and

\eta{\cong}0.30

are consistent with the conjectured values of the 3-state Potts model. Beyond a tricritical point the transition becomes of the first order for strong fields. The tricritical exponent

y_{\text{T,t}}{\cong}1.73

and

\eta_{\text{t}}{\cong}0.21

are also in agreement with the 3-state Potts tricritical values y T,t 3P =12/7 and η t 3P =4/21. The XY-like phase transition at zero fields, which was obtained by the Monte Carlo simulations, is not found in the present calculation.