Completeness of ``Good'' Bethe Ansatz Solutions of a Quantum Group Invariant Heisenberg Model

TL;DR

This paper proves the completeness of 'good' Bethe ansatz solutions for a quantum group invariant Heisenberg model, showing they account for all relevant states within a restricted path framework, a first for such anisotropic models.

Contribution

It provides the first proof of completeness for 'good' Bethe states in an anisotropic quantum invariant Heisenberg model, linking solutions to restricted paths on the Bratteli diagram.

Findings

'Good' Bethe states correspond to roots in the first periodicity strip.

Completeness is established via Bethe's string counting technique.

All 'good' states are accounted for by restricted paths on the Bratteli diagram.

Abstract

The $sl_q(2)$-quantum group invariant spin 1/2 XXZ-Heisenberg model with open boundary conditions is investigated by means of the Bethe ansatz. As is well known, quantum groups for $q$ equal to a root of unity possess a finite number of ``good'' representations with non-zero q-dimension and ``bad'' ones with vanishing q-dimension. Correspondingly, the state space of an invariant Heisenberg chain decomposes into ``good'' and ``bad'' states. A ``good'' state may be described by a path of only ``good'' representations. It is shown that the ``good'' states are given by all ``good'' Bethe ansatz solutions with roots restricted to the first periodicity strip, i.e. only positive parity strings (in the language of Takahashi) are allowed. Applying Bethe's string counting technique completeness of the ``good'' Bethe states is proven, i.e. the same number of states is found as the number of all restricted path's on the $sl_q(2)$-Bratteli diagram. It is the first time that a ``completeness" proof for an anisotropic quantum invariant reduced Heisenberg model is performed.