Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/9705062 · Journal of Physics A Mathematical and General

Variational method and duality in the 2D square Potts model

Leonardo Angelini, M. Pellicoro, I Sardella, M Villani

Published Nov 21, 1997• 0 citations
arXivHTMLPDF

Abstract

The ferromagnetic q-states Potts model on a square lattice is analysed, for q > 4, through an elaborate version of the operatorial variational method. In the variational approach proposed in this paper, the duality relations are exactly satisfied, involving at a more fundamental level, a duality relationship between variational parameters. Besides some exact predictions, the approach is very effective in the numerical estimates over a wide range of temperature and can be systematically improved.

DOI
10.1088/0305-4470/30/22/008

Related papers

Duality and Fisher zeros in the two-dimensional Potts model on a square lattice

Marco Astorino, Fabrizio Canfora

A phenomenological approach to the ferromagnetic two-dimensional (2D) Potts model on square lattice is proposed. Our goal is to present a simple functional form that obeys the known properties possessed by the free energy of the q-state Potts model. The duality symmetry of the 2D Potts model together with the known results on its critical exponent α allows us to fix consistently the details of the proposed expression for the free energy. The agreement of the analytic ansatz with numerical data in the q=3 case is very good at high and low temperatures as well as at the critical point. It is shown that the q>4 cases naturally fit into the same scheme and that one should also expect a good agreement with numerical data. The limiting q=4 case is shortly discussed.

arXiv:0901.2711•Physical Review E•May 28, 2010•6 citations

Duality and Regge trajectories in the 2D Potts model on square lattice.

Marco Astorino, Fabrizio Canfora

A unified approach to the ferromagnetic two dimensional Potts model on square lattice is proposed. The compatibility with the still to be found solutions of the three (and higher) dimensional Ising model allows one to write down an explicit analytic ansatz for the free energy in terms of few q-dependent Regge trajectories. The duality symmetry of the 2D Potts model together with the known results on its critical exponent α allow to fix a priori in an analytic way all but one Regge trajectory. The agreement of proposed analytic ansatz with Monte Carlo data and observations in the q = 3 case is remarkable in all the range of temperatures. It is shown that the q> 4 cases naturally fit into the same scheme and that one should also expect a good agreement with Monte Carlo data and observations. The limiting q = 4 case is shortly discussed.

•Jan 1, 2009•0 citations

Estimates of the critical exponent β for the Potts model using a variational approximation

S B Kelland

The q-state Potts model on a square lattice is transformed into a staggered ice model. A variational principle for the largest eigenvalue of the transfer matrix of this model is used to develop a set of matrix equations. In the limit of these matrices becoming infinite, the equations determine the partition function and order parameter of the Potts model exactly. We have solved the equations numerically for finite matrices, obtaining estimates of these quantities and, as a result, the critical exponent β.

Canadian Journal of Physics•Aug 1, 1976•14 citations

Duality in two-dimensional Z(N)-symmetric spin models on a finite lattice

A. I. Bugriĭ, V. N. Shadura

We propose a method for deriving duality relations for two-dimensional inhomogeneous Z ( N )-symmetric models on a finite square lattice wound around a torus. The method is used to obtain duality relations for the vector Potts model, the Berezinskii-Villain Z ( N )-model, the Ashkin-Teller model, and the 8-vertex model on a lattice obliquely wound around a torus, as well as an exact relation linking the partition functions of the latter two models.

Journal of Experimental and Theoretical Physics•Jan 1, 1998•0 citations

Duality in 2D Spin Models on Torus

A. I. Bugrij, V. N. Shadura

Method of derivation of the duality relations for two-dimensional Z(N)-symmetric spin models on finite square lattice wrapped on the torus is proposed. As example, exact duality relations for the nonhomogeneous Ising model (N=2) and the Z(N)-Berezinsky-Villain model are obtained.

arXiv:hep-th/9706045•Jun 6, 1997•1 citation

Critical temperature of an anisotropic two-dimensional Potts model

Kaoru Yasumura

The phase transition temperature T c of the q-state Potts model with ferromagnetic interactions J x and antiferromagnetic ones J y in the two respective directions of a 2D lattice is calculated by the high-temperature series expansion (HTSE) of the appropriate staggered susceptibility as well a by the cluster variate approximation (CVA). For q=3 the HTSE result in the range of mod J y mod /J x <or approximately=1.5 shows a good coincidence with the Kosterlitz-Thouless-type phase transition temperature calculated on the prescription by Ostlund which is given by 1/16 pi =exp(-2 beta c J x ) coth(- beta c J y /2), beta =1/kT c . The CVA result shows similar behaviour of T c to that of the Migdal-Kadanoff renormalisation group method obtained by Kinzel et al, but it is considered to be less reliable than the HTSE result.

Journal of Physics A Mathematical and General•Oct 1, 1987•8 citations
The ferromagnetic q-states Potts model on a square lattice is analysed, for q > 4, through an elaborate version of the operatorial variational method. In the variational approach proposed in this paper, the duality relations are exactly satisfied, involving at a more fundamental level, a duality relationship between variational parameters. Besides some exact predictions, the approach is very effective in the numerical estimates over a wide range of temperature and can be systematically improved.
A phenomenological approach to the ferromagnetic two-dimensional (2D) Potts model on square lattice is proposed. Our goal is to present a simple functional form that obeys the known properties possessed by the free energy of the q-state Potts model. The duality symmetry of the 2D Potts model together with the known results on its critical exponent α allows us to fix consistently the details of the proposed expression for the free energy. The agreement of the analytic ansatz with numerical data in the q=3 case is very good at high and low temperatures as well as at the critical point. It is shown that the q>4 cases naturally fit into the same scheme and that one should also expect a good agreement with numerical data. The limiting q=4 case is shortly discussed.
A unified approach to the ferromagnetic two dimensional Potts model on square lattice is proposed. The compatibility with the still to be found solutions of the three (and higher) dimensional Ising model allows one to write down an explicit analytic ansatz for the free energy in terms of few q-dependent Regge trajectories. The duality symmetry of the 2D Potts model together with the known results on its critical exponent α allow to fix a priori in an analytic way all but one Regge trajectory. The agreement of proposed analytic ansatz with Monte Carlo data and observations in the q = 3 case is remarkable in all the range of temperatures. It is shown that the q> 4 cases naturally fit into the same scheme and that one should also expect a good agreement with Monte Carlo data and observations. The limiting q = 4 case is shortly discussed.
The q-state Potts model on a square lattice is transformed into a staggered ice model. A variational principle for the largest eigenvalue of the transfer matrix of this model is used to develop a set of matrix equations. In the limit of these matrices becoming infinite, the equations determine the partition function and order parameter of the Potts model exactly. We have solved the equations numerically for finite matrices, obtaining estimates of these quantities and, as a result, the critical exponent β.
The phase transition temperature T c of the q-state Potts model with ferromagnetic interactions J x and antiferromagnetic ones J y in the two respective directions of a 2D lattice is calculated by the high-temperature series expansion (HTSE) of the appropriate staggered susceptibility as well a by the cluster variate approximation (CVA). For q=3 the HTSE result in the range of mod J y mod /J x <or approximately=1.5 shows a good coincidence with the Kosterlitz-Thouless-type phase transition temperature calculated on the prescription by Ostlund which is given by 1/16 pi =exp(-2 beta c J x ) coth(- beta c J y /2), beta =1/kT c . The CVA result shows similar behaviour of T c to that of the Migdal-Kadanoff renormalisation group method obtained by Kinzel et al, but it is considered to be less reliable than the HTSE result.