Completing the solution for the $OSp(1|2)$ spin chain

TL;DR

This paper addresses the completeness of the Bethe ansatz solutions for the $OSp(1|2)$ quantum spin chain, clarifying symmetry differences and resolving longstanding issues with physical solutions and signs in the equations.

Contribution

It introduces a mechanism to achieve completeness for the non-graded version and clarifies symmetry and degeneracy structures in both versions.

Findings

Complete spectrum obtained for the non-graded model using the proposed mechanism.

Symmetry analysis shows $OSp(1|2)$ for graded and $SU(2)$ for non-graded models.

Resolved controversy over signs in Bethe equations and conditions for physical solutions.

Abstract

The periodic $OSp(1|2)$ quantum spin chain has both a graded and a non-graded version. Naively, the Bethe ansatz solution for the non-graded version does not account for the complete spectrum of the transfer matrix, and we propose a simple mechanism for achieving completeness. In contrast, for the graded version, this issue does not arise. We also clarify the symmetries of both versions of the model, and we show how these symmetries are manifested in the degeneracies and multiplicities of the transfer-matrix spectrum. While the graded version has $OSp(1|2)$ symmetry, the non-graded version has only $SU(2)$ symmetry. Moreover, we obtain conditions for selecting the physical singular solutions of the Bethe equations. This analysis solves a lasting controversy over signs in the Bethe equations.