Analytical Bethe Ansatz for Quantum-Algebra-Invariant Spin Chains

TL;DR

This paper develops an analytical Bethe Ansatz method to exactly solve quantum spin chains with quantum-algebra symmetry, determining their spectra and revealing doubled Bethe Ansatz equations for open chains.

Contribution

It generalizes the analytical Bethe Ansatz to quantum-algebra-invariant open spin chains, providing exact solutions for their transfer matrix spectra.

Findings

Exact spectra of $U_q [(su(2)]$-invariant spin chains determined.

Bethe Ansatz equations for open chains are doubled compared to closed chains.

Quantum-algebra symmetry is crucial for solving these models.

Abstract

We have recently constructed a large class of open quantum spin chains which have quantum-algebra symmetry and which are integrable. We show here that these models can be exactly solved using a generalization of the analytical Bethe Ansatz (BA) method. In particular, we determine in this way the spectrum of the transfer matrices of the $U_q [(su(2)]$-invariant spin chains associated with $A^{(1)}_1$ and $A^{(2)}_2$ in the fundamental representation. The quantum-algebra invariance of these models plays an essential role in obtaining these results. The BA equations for these open chains are ``doubled'' with respect to the BA equations for the corresponding closed chains.