Where are the roots of the Bethe Ansatz equations?

TL;DR

This paper analyzes the roots of Bethe Ansatz equations for the XXZ model by transforming them into polynomial systems, revealing root distributions, addressing completeness issues, and connecting to Salem's polynomials.

Contribution

It introduces a polynomial reformulation of Bethe Ansatz equations, providing new insights into root distributions and their relation to Salem's polynomials, and discusses completeness and singularities.

Findings

Root locations and multiplicities in the complex plane

Disproof of the string hypothesis for roots

Connection between BAE and Salem's polynomials

Abstract

Changing the variables in the Bethe Ansatz Equations (BAE) for the XXZ six-vertex model we had obtained a coupled system of polynomial equations. This provided a direct link between the BAE deduced from the Algebraic Bethe Ansatz (ABA) and the BAE arising from the Coordinate Bethe Ansatz (CBA). For two magnon states this polynomial system could be decoupled and the solutions given in terms of the roots of some self-inversive polynomials. From theorems concerning the distribution of the roots of self-inversive polynomials we made a thorough analysis of the two magnon states, which allowed us to find the location and multiplicity of the Bethe roots in the complex plane, to discuss the completeness and singularities of Bethe's equations, the ill-founded string-hypothesis concerning the location of their roots, as well as to find an interesting connection between the BAE with Salem's polynomials.