Fusion Hierarchy and Finite-Size Corrections of $U_q[sl(2)]$ Invariant Vertex Models with Open Boundaries

TL;DR

This paper analyzes fused six-vertex models with open boundaries, extending Bethe ansatz solutions, revealing fusion rules, and deriving finite-size corrections to connect to conformal field theory characteristics.

Contribution

It generalizes the Bethe ansatz solution to fused models with open boundaries and uncovers finite-size effects and fusion rules related to quantum group symmetry.

Findings

Eigenvalues satisfy $su$(2) fusion rules

Finite-size corrections yield conformal data

Functional relations truncate at special parameters

Abstract

The fused six-vertex models with open boundary conditions are studied. The Bethe ansatz solution given by Sklyanin has been generalized to the transfer matrices of the fused models. We have shown that the eigenvalues of transfer matrices satisfy a group of functional relations, which are the $su$(2) fusion rule held by the transfer matrices of the fused models. The fused transfer matrices form a commuting family and also commute with the quantum group $U_q[sl(2)]$. In the case of the parameter $q^h=-1$ ($h=4,5,\cdots$) the functional relations in the limit of spectral parameter $u\to \i\infty$ are truncated. This shows that the $su$(2) fusion rule with finite level appears for the six vertex model with the open boundary conditions. We have solved the functional relations to obtain the finite-size corrections of the fused transfer matrices for low-lying excitations. From the corrections the central charges and conformal weights of underlying conformal field theory are extracted. To see different boundary conditions we also have studied the six-vertex model with a twisted boundary condition.