Exact spectrum of the XXZ open spin chain from theq-Onsager algebra representation theory

Abstract

The transfer matrix of the XXZ open spin-½ chain with general integrable boundary conditions and generic anisotropy parameter(q is not a rootof unity and |q| = 1) is diagonalized using the representation theory of theq-Onsager algebra. Similarly to the Ising and superintegrable chiral Potts models, the completespectrum is expressed in terms of the roots of a characteristic polynomial of degreed = 2N. The complete family of eigenstates are derived in terms of rational functions defined on adiscrete support which satisfy a system of coupled recurrence relations. In the special caseof linear relations between left and right boundary parameters for which Bethe-typesolutions are known to exist, our analysis provides an alternative derivation of the results ofNepomechie et al and Cao et al. In the latter case the complete family of eigenvalues andeigenstates splits into two sets, each associated with a characteristic polynomial of degreed < 2N. Numerical checks performed for small values ofN support the analysis.