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Journal of the Physical Society of Japan

Exact Landau Free-Energy of SolvableN-State Vertex Model

Yasuhiro Akutsu

Published Jul 15, 1989• 16 citations
Publisher

Abstract

For a solvable N -state vertex model in the low-temperature antiferroelectric phase, exact per-site free-energy f is calculated as a function of the order-parameter p (`vertical polarization'), based on the quantum inverse scattering method. The Bethe-ansatz integral equation is solved as an expansion with respect to p , which leads to the Gruber-Mullins-Pokrovsky-Talapov-type expansion, f ( p )= f (0)+ a ·| p |+ b ·| p | 3 + O (| p | 4 ). It is found that the coefficients a and b are identical to those of the six-vertex model ( N =2 case). This fact gives support for the universal Gaussian curvature jump for the N -state vertex model interpreted as a model of interface roughening.

DOI
10.1143/jpsj.58.2219

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For a solvable N -state vertex model in the low-temperature antiferroelectric phase, exact per-site free-energy f is calculated as a function of the order-parameter p (`vertical polarization'), based on the quantum inverse scattering method. The Bethe-ansatz integral equation is solved as an expansion with respect to p , which leads to the Gruber-Mullins-Pokrovsky-Talapov-type expansion, f ( p )= f (0)+ a ·| p |+ b ·| p | 3 + O (| p | 4 ). It is found that the coefficients a and b are identical to those of the six-vertex model ( N =2 case). This fact gives support for the universal Gaussian curvature jump for the N -state vertex model interpreted as a model of interface roughening.
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.
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.
Some features of integrable lattice models are reviewed for the case of the six-vertex model.By the Bethe ansatz method we derive the free energy of the six-vertex model.Then, from the expression of the free energy we show analytically the critical singularity near the phase transition in the anti-ferroelectric regime, where the essential singularity similar to the Kosterlitz-Thouless transition appears.We discuss the connection of the six-vertex model to the conformal field theory with c = 1.We also introduce various exactly solvable models defined on two-dimensional lattices such as the chiral Potts model and the IRF models.We show that the six-vertex model has rich mathematical structures such as the quantum groups and the braid group.The graphical approach is emphasized in this review.We explain the meaning of the Yang-Baxter equation by its diagram.Furthermore, we can understand the defining relation of the algebraic Bethe ansatz by the graphical representation.We can thus easily translate formulas of the algebraic Bethe ansatz into those of the statistical models.As an illustration, we show explicitly how we can derive Baxter's expressions from those of the algebraic Bethe ansatz.
Some features of integrable lattice models are reviewed for the case of the six-vertex model.By the Bethe ansatz method we derive the free energy of the six-vertex model.Then, from the expression of the free energy we show analytically the critical singularity near the phase transition in the anti-ferroelectric regime, where the essential singularity similar to the Kosterlitz-Thouless transition appears.We discuss the connection of the six-vertex model to the conformal field theory with c = 1.We also introduce various exactly solvable models defined on two-dimensional lattices such as the chiral Potts model and the IRF models.We show that the six-vertex model has rich mathematical structures such as the quantum groups and the braid group.The graphical approach is emphasized in this review.We explain the meaning of the Yang-Baxter equation by its diagram.Furthermore, we can understand the defining relation of the algebraic Bethe ansatz by the graphical representation.We can thus easily translate formulas of the algebraic Bethe ansatz into those of the statistical models.As an illustration, we show explicitly how we can derive Baxter's expressions from those of the algebraic Bethe ansatz.
Some features of integrable lattice models are reviewed for the case of the six-vertex model. By the Bethe ansatz method we derive the free energy of the six-vertex model. Then, from the expression of the free energy we show analytically the critical singularity near the phase transition in the anti-ferroelectric regime, where the essential singularity similar to the Kosterlitz-Thouless transition appears. We discuss the connection of the six-vertex model to the conformal field theory with c=1. We also introduce various exactly solvable models defined on two-dimensional lattices such as the chiral Potts model and the IRF models. We show that the six-vertex model has rich mathematical structures such as the quantum groups and the braid group. The graphical approach is emphasized in this review. We explain the meaning of the Yang-Baxter equation by its diagram. Furthermore, we can understand the defining relation of the algebraic Bethe ansatz by the graphical representation. We can thus easily translate formulas of the algebraic Bethe ansatz into those of the statistical models. As an illustration, we show explicitly how we can derive Baxter's expressions from those of the algebraic Bethe ansatz.