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Journal of the Physical Society of Japan

Integral Representation of the Correlations around a Vertex of the Six-Vertex Model

Таичиро Такаги

Published Jul 15, 1996• 2 citations
Publisher

Abstract

We study exact solutions of the four-point correlation functions around a vertex of the anti-ferroelectric six-vertex model. After showing that an expression for the correlations is readily obtained by differentiating the known exact solution of the free energy, we present a new expression for the correlations derived through the recently developed vertex operator approach. It has the form of a double contour integral of a meromorphic function. To give a new verification of the vertex operator approach beyond the case of the one-point function, we prove the equality of the two expressions by calculating the contour integrals.

DOI
10.1143/jpsj.65.2009

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We study exact solutions of the four-point correlation functions around a vertex of the anti-ferroelectric six-vertex model. After showing that an expression for the correlations is readily obtained by differentiating the known exact solution of the free energy, we present a new expression for the correlations derived through the recently developed vertex operator approach. It has the form of a double contour integral of a meromorphic function. To give a new verification of the vertex operator approach beyond the case of the one-point function, we prove the equality of the two expressions by calculating the contour integrals.
We study analytic expressions for the averaged fraction of each vertex type of the six-vertex model extensively. Upon the critical line the expressions of the quantity in the anti-ferroelectric phase and in the gapless phase get simplified. This enables us to see more easily the equivalence of the two quantities of which one is derived by differentiating the free energy, and the other is by calculating a trace of the vertex operators in the theory of a quantum affine algebra. We also prove an extra liner relation satisfied by the fractions on the critical line.
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.
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.
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.