Bethe Equations "on the Wrong Side of Equator"

TL;DR

This paper investigates Baxter's T-Q equations for spin chains, revealing a second solution beyond the usual polynomial one, which is crucial for understanding fusion relations and Bethe-Ansatz solutions beyond half the system size.

Contribution

It uncovers a second solution to Baxter's equations that extends the Bethe-Ansatz framework beyond traditional limits, enriching the theoretical understanding of spin chains.

Findings

Identification of a second solution to Baxter's T-Q equations.

The second solution's role in fusion relations.

Implications for Bethe-Ansatz beyond N/2.

Abstract

We analyse the famous Baxter's $T-Q$ equations for $XXX$ ($XXZ$) spin chain and show that apart from its usual polynomial (trigonometric) solution, which provides the solution of Bethe-Ansatz equations, there exists also the second solution which should corresponds to Bethe-Ansatz beyond $N/2$. This second solution of Baxter's equation plays essential role and together with the first one gives rise to all fusion relations.