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

Journal of Physics A Mathematical and General

An exact solution of a multi-state generalisation of the six-vertex model

Ciriyam Jayaprakash, A Sinha

Published Oct 21, 1985• 0 citations
Publisher

Abstract

An exact solution of a 2*n generalisation (two states for horizontal arrows and n states for vertical arrows with total arrow number conservation) of the six-vertex model is given using the commuting transfer matrix method. In the symmetric case, the free energy is a sum of n-1 terms each of which is the free energy of a six-vertex model such that the phase transition into the frozen ferroelectric state of the 2*n model is identical to that in the zero-field six-vertex model. For the six-vertex model the commuting transfer matrix solution is extended to the case of arbitrary vertex weights and a derivation of the Baxter parametrisation is provided.

DOI
10.1088/0305-4470/18/15/024

Related papers

Exact solution of a ten-vertex model in two dimensions

Roberto Nicolau Onody, M. Karowski

A ten-vertex model, which is a multi-state generalisation of the six-vertex model, is formulated on a square lattice and solved exactly in the thermodynamic limit using the inverse scattering method. It is found that this system undergoes a first-order phase transition into a 'frozen' ferroelectric state.

Journal of Physics A Mathematical and General•Jan 11, 1983•3 citations

Phase diagram of the five-vertex model

M. Gulácsi, Henk van Beijeren, A.C. Levi

Within the ice-type models, the solution of the five-vertex model is obtained with the use of the Bethe ansatz. Since the allowed number of vertex types is odd, the arrow-reversal symmetry of the system is broken by construction. Due to this, the exact solution obtained and the phase diagram are very different from those of the symmetric six-vertex model. A connection to the asymmetric six-vertex model (of which the five-vertex model is an extreme case) is made. The different regions of the phase diagram are described and the transitions between them are analyzed. Several aspects of the phase diagram are unusual, i.e., the ordered phases (both ferroelectric and antiferroelectric) are frozen-in phases and the disordered phase is replaced by a ferrielectric phase. In the free-fermion case, the known results of the modified KDP model are recovered.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Apr 1, 1993•24 citations

Exact solution and interfacial tension of the six-vertex model with anti-periodic boundary conditions

Murray T. Batchelor, R. J. Baxter, M J O'Rourke +1 more

We consider the six-vertex model with anti-periodic boundary conditions across a finite strip. The row-to-row transfer matrix is diagonalized by the 'commuting transfer matrices' method. From the exact solution we obtain an independent derivation of the interfacial tension of the six-vertex model in the anti-ferroelectric phase. The nature of the corresponding integrable boundary condition on the XXZ spin chain is also discussed.

arXiv:hep-th/9502040•Journal of Physics A Mathematical and General•May 21, 1995•98 citations

Transfer Matrix of the Two-dimensional Ising Model

Giuseppe Mussardo

Abstract This chapter deals with the exact solution of the two-dimensional Ising model as it is achieved through the transfer matrix formalism. It discusses the crucial role played by the commutative properties of the transfer matrices, which lead to a functional equation for their eigenvalues. The exact free energy of the Ising model and its critical point can be identified by means of the lowest eigenvalue. The chapter covers Baxter's approach, the Yang–Baxter equation and its relation to the Boltzmann weights, the R-matrix, and discusses activity away from the critical point, the six-vertex model, as well as functional equations and symmetries.

•Mar 26, 2020•0 citations

Exact solution of an N-sublattice vertex model

R. Z. Bariev, Trong Tuong Truong, Loïc Turban

The exact diagonalization of the diagonal-to-diagonal transfer matrix of an N-sublattice vertex model with interactions of the vertex-arrow type between sublattices is carried out using the generalized nested Bethe-ansatz method. An exact expression for the free energy is obtained involving the solution of an integral equation which is studied numerically.

Journal of Physics A Mathematical and General•May 7, 1992•5 citations

Arrow-arrow correlations for the six-vertex model

Pierluigi Falco

The six-vertex model on a square lattice is "exactly solvable" because an exact formula for the free energy can be obtained by the Bethe ansatz. However, exact formulas for the correlations of local bulk observables, such as the orientation of the arrow at a given edge, are, in general, not available. In this Rapid Communication, we consider the isotropic "zero-field" six-vertex model at small Δ. We derive the long-distance asymptotic formula of arrow-arrow correlations, which display power law decays with one anomalous exponent. Our method is based on an interacting fermion representation of the six-vertex model and does not use any information obtained from the exact solution.

Physical Review E•Sep 16, 2013•5 citations
An exact solution of a 2n generalisation (two states for horizontal arrows and n states for vertical arrows with total arrow number conservation) of the six-vertex model is given using the commuting transfer matrix method. In the symmetric case, the free energy is a sum of n-1 terms each of which is the free energy of a six-vertex model such that the phase transition into the frozen ferroelectric state of the 2n model is identical to that in the zero-field six-vertex model. For the six-vertex model the commuting transfer matrix solution is extended to the case of arbitrary vertex weights and a derivation of the Baxter parametrisation is provided.
Within the ice-type models, the solution of the five-vertex model is obtained with the use of the Bethe ansatz. Since the allowed number of vertex types is odd, the arrow-reversal symmetry of the system is broken by construction. Due to this, the exact solution obtained and the phase diagram are very different from those of the symmetric six-vertex model. A connection to the asymmetric six-vertex model (of which the five-vertex model is an extreme case) is made. The different regions of the phase diagram are described and the transitions between them are analyzed. Several aspects of the phase diagram are unusual, i.e., the ordered phases (both ferroelectric and antiferroelectric) are frozen-in phases and the disordered phase is replaced by a ferrielectric phase. In the free-fermion case, the known results of the modified KDP model are recovered.
Abstract This chapter deals with the exact solution of the two-dimensional Ising model as it is achieved through the transfer matrix formalism. It discusses the crucial role played by the commutative properties of the transfer matrices, which lead to a functional equation for their eigenvalues. The exact free energy of the Ising model and its critical point can be identified by means of the lowest eigenvalue. The chapter covers Baxter's approach, the Yang–Baxter equation and its relation to the Boltzmann weights, the R-matrix, and discusses activity away from the critical point, the six-vertex model, as well as functional equations and symmetries.
The six-vertex model on a square lattice is "exactly solvable" because an exact formula for the free energy can be obtained by the Bethe ansatz. However, exact formulas for the correlations of local bulk observables, such as the orientation of the arrow at a given edge, are, in general, not available. In this Rapid Communication, we consider the isotropic "zero-field" six-vertex model at small Δ. We derive the long-distance asymptotic formula of arrow-arrow correlations, which display power law decays with one anomalous exponent. Our method is based on an interacting fermion representation of the six-vertex model and does not use any information obtained from the exact solution.