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Chinese Physics B

Drinfeld twist and the domain wall partition function of the eight-vertex model

Kun Hao, Xi Chen, Kangjie Shi, Wen‐Li Yang

Published Jan 1, 2011• 1 citation
Publisher

Abstract

With the help of the F -basis provided by the Drinfeld twist or factorising F -matrix for the spatial optical soliton model associated with the eight-vertex model, we calculate the partition function for the eight-vertex model on an N × N square lattice with domain wall boundary condition.
DOI
10.1088/1674-1056/20/1/010303

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The six-vertex model on an N × N square lattice with domain wall boundary conditions is considered. A Fredholm determinant representation for the partition function of the model is given. The kernel of the corresponding integral operator is of the so-called integrable type, and involves classical orthogonal polynomials. From this representation, a ‘reconstruction’ formula is proposed, which expresses the partition function as the trace of a suitably chosen quantum operator, in the spirit of corner transfer matrix and vertex operator approaches to integrable spin models.
The partition function of the six-vertex model on a square lattice with domain wall boundary conditions (DWBC) is rewritten as a Hermitian one-matrix model or a discretized version of it (similar to sums over Young diagrams), depending on the phase. The expression is exact for finite lattice size, which is equal to the size of the corresponding matrix. In the thermodynamic limit, the matrix integral is computed using traditional matrix model techniques, thus providing a complete treatment of the bulk free energy of the six-vertex model with DWBC in the different phases. In particular, in the antiferroelectric phase, the bulk free energy and a subdominant correction are given exactly in terms of elliptic theta functions.
We introduce and study the domain wall boundary partition function of the integrable six-vertex model with triangular boundary. We first formulate the domain wall boundary partition function with triangular boundary by using the Uq(sl2)U_q(sl_2)Uq​(sl2​) RRR-matrix and a special class of the triangular KKK-matrix. By using its graphical representation, we make the Izergin-Korepin analysis with the help of the Yang-Baxter relation and the reflection equation to give a characterization of the partition function. The explict form of the symmetric function representing the partition function is presented by showing that it satisfies all the required properties.