Vertex-face Correspondence Research Articles

Conserved currents in the six-vertex and trigonometric solid-on-solid models

We construct quasi-local conserved currents in the six-vertex model with anisotropy parameter η by making use of the quantum-group approach of Bernard and Felder. From these currents, we construct parafermionic operators with spin that obey a discrete-integral condition around lattice plaquettes embedded into the complex plane. These operators are identified with primary fields in a c = 1 compactified free Boson conformal field theory. We then consider a vertex-face correspondence that takes the six-vertex model to a trigonometric SOS model, and construct SOS operators that are the image of the six-vertex currents under this correspondence. We define corresponding SOS parafermionic operators with spins s = 1 and that obey discrete integral conditions around SOS plaquettes embedded into the complex plane. We consider in detail the cyclic-SOS case corresponding to the choice , with coprime. We identify our SOS parafermionic operators in terms of the screening operators and primary fields of the associated conformal field theory.

Determinant representations of scalar products for the open XXZ chain with non-diagonal boundary terms

The determinant representation of the scalar products of the Bethe states of the open XXZ spin chain with non-diagonal boundary terms is studied. Using the vertex-face correspondence, we transfer the problem into the corresponding trigonometric solid-on-solid (SOS) model with diagonal boundary terms. With the help of the Drinfeld twist or factorizing F-matrix, we obtain the determinant representation of the scalar products of the Bethe states of the associated SOS model. By taking the on shell limit, we obtain the determinant representations (or Gaudin formula) of the norms of the Bethe states.

Fusion of Baxter’s Elliptic R-Matrix and the Vertex-Face Correspondence

The matrix elements of the 2 × 2 fusion of Baxter’s elliptic R-matrix, R(2,2)(u), are given explicitly. Then a gauge equivalence between R(2,2)(u) and Fateev’s R-matrix for the 21-vertex model is shown. This part is based on an unpublished note by Jimbo. We then derive the crossing symmetry formula for R(2,2)(u). We also consider the fusion of the vertex-face correspondence relation and derive a crossing symmetry relation between the fusion of the intertwining vectors and their dual vectors. Communicated by Vincent Rivasseau

The vertex-face correspondence and correlation functions of the fusion eight-vertex model: I: The general formalism

By making use of the vertex-face correspondence, we give an algebraic analysis formulation of correlation functions of the k × k fusion eight-vertex model in terms of the corresponding fusion SOS model. Here k ∈ Z > 0 . A general formula for correlation functions is derived as a trace over the space of states of lattice operators such as the corner-transfer matrices, the half-transfer matrices (vertex operators) and the tail operator. We give a realization of these lattice operators as well as the space of states as objects in the level k representation theory of the elliptic algebra U q , p ( sl ˆ 2 ) .

Vertex–face correspondence of Boltzmann weights related tosl(m|n)

In this work, we present a vertex–face correspondence between an elliptic R-operator and Boltzmann weights related to the Lie superalgebra sl(mn).

Bootstrap equations and correlation functions for the Heisenberg XYZ antiferromagnet

We present two kinds of integral solutions to the quantum Knizhnik–Zamolodchikov equations for the 2n-point correlation functions of the Heisenberg XYZ antiferromagnet. Our first integral solution can be obtained from those for the cyclic SOS model by using the vertex–face correspondence. By the construction, the sum with respect to the local height variables k0, k1, ..., k2n of the cyclic SOS model remains other than the n-fold integral in the first solution. In order to perform these summations, we improve this to find the second integral solution of (r + 1)n-fold integral for r ∊ >1, where r is a parameter of the XYZ model. Furthermore, we discuss the relations among our formula, Lashkevich–Pugai's formula and that of Shiraishi.

Free field construction for correlation functions of the eight-vertex model

A free field representation for the type I vertex operators and the corner transfer matrices of the eight-vertex model is proposed. The construction uses the vertex-face correspondence, which makes it possible to express correlation functions of the eight-vertex model in terms of correlation functions of the SOS model with a non-local insertion. This new non-local insertion admits a free field representation in terms of Lukyanov's screening operator. The spectrum of the corner transfer matrix and the Baxter-Kelland formula for the average staggered polarization have been reproduced.

Bosonization of vertex operators for the -symmetric Belavin model

Based on the bosonization of vertex operators for the face model by Asai, Jimbo, Miwa and Pugai, using vertex-face correspondence we obtain vertex operators for the -symmetric Belavin model, which are constructed by deformed boson oscillators.

YANG-BAXTER EQUATION FOR BROKEN $\mathbb{Z}_N^{\otimes n-1}$ MODELS

We construct a factorized representation of the [Formula: see text]-Sklyanin algebra from the vertex-face correspondence. Using this representation, we obtain a new solvable model which gives an [Formula: see text]-generalization of the broken ℤN model. We further prove the Yang-Baxter equation for this model.