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Exact solution of the $SU_{q}(n)$ invariant quantum spin chains

Hector J. de Vega, González-Ruiz, A

Published Sep 3, 1993• 80 citations
OpenAlex

Abstract

The Nested Bethe Ansatz is generalized to open boundary conditions. This is used to find the exact eigenvectors and eigenvalues of the $A_{n-1}$ vertex model with fixed open boundary conditions and the corresponding $SU_{q}(n)$ invariant hamiltonian. The Bethe Ansatz equations obtained are solved in the thermodynamic limit giving the vertex model free energy and the hamiltonian ground state energy including the corresponding boundary contributions.

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Exact solution of the SUq(n)SU_{q}(n)SUq​(n) invariant quantum spin chains
The Nested Bethe Ansatz is generalized to open boundary conditions. This is used to find the exact eigenvectors and eigenvalues of the An−1A_{n-1}An−1​ vertex model with fixed open boundary conditions and the corresponding SUq(n)SU_{q}(n)SUq​(n) invariant hamiltonian. The Bethe Ansatz equations obtained are solved in the thermodynamic limit giving the vertex model free energy and the hamiltonian ground state energy including the corresponding boundary contributions.
We present in a unified and detailed way the nested Bethe ansatz for open spin chains based on(super)algebras, with arbitrary representations (i.e., 'spins') on each site of the chain and diagonal boundary matrices (K + (u), K -(u)).The nested Bethe ansatz applies for a general K -(u), but a particular form of the K + (u) matrix.The construction extends and unifies the results already obtained for open spin chains based on the fundamental representation and for some particular superspin chains.We give the eigenvalues, Bethe equations and the explicit form of the Bethe vectors for the corresponding models.The Bethe vectors are also expressed using a trace formula.
We review recent results on the Bethe Ansatz solutions for the eigenvalues of the transfer matrix of an integrable open XXZ quantum spin chain using functional relations which the transfer matrix obeys at roots of unity. First, we consider a case where at most two of the boundary parameters {{alpha−\\alpha_-alpha−​,alpha+\\alpha_+alpha+​,beta−\\beta_-beta−​,beta+\\beta_+beta+​}} are nonzero. A generalization of the Baxter T−QT-QT−Q equation that involves more than one independent QQQ is described. We use this solution to compute the boundary energy of the chain in the thermodynamic limit. We conclude the paper with a review of some results for the general integrable boundary terms, where all six boundary parameters are arbitrary.