On the completeness of solutions of Bethe's equations

TL;DR

This paper investigates the solutions of Bethe's equations for the Heisenberg spin chain, proposing a conjecture on their count, verifying it numerically for small chains, and discussing implications for thermodynamics.

Contribution

It formulates a conjecture on the number of solutions with distinct roots and verifies it using homotopy continuation methods for chains up to length 14.

Findings

Numerical solutions agree with the conjecture.

Homotopy continuation effectively finds all solutions for small chains.

Discusses implications for thermodynamic calculations.

Abstract

We consider the Bethe equations for the isotropic spin-1/2 Heisenberg quantum spin chain with periodic boundary conditions. We formulate a conjecture for the number of solutions with pairwise distinct roots of these equations, in terms of numbers of so-called singular (or exceptional) solutions. Using homotopy continuation methods, we find all such solutions of the Bethe equations for chains of length up to 14. The numbers of these solutions are in perfect agreement with the conjecture. We also discuss an indirect method of finding solutions of the Bethe equations by solving the Baxter T-Q equation. We briefly comment on implications for thermodynamical computations based on the string hypothesis.