Rational Q-systems at Root of Unity I. Closed Chains

TL;DR

This paper develops a rational Q-system approach for solving Bethe ansatz equations of the XXZ spin chain at roots of unity, systematically identifying all physical solutions and constructing the full Hilbert space.

Contribution

It introduces a new rational Q-system with constraints to eliminate FM string solutions and provides a method to generate the entire solution space at roots of unity.

Findings

Successfully eliminates FM string solutions

Constructs the complete Hilbert space including primitive and descendant solutions

Validates the approach with extensive numerical checks and applies to 6-vertex model

Abstract

The solution of Bethe ansatz equations for XXZ spin chain with the parameter $q$ being a root of unity is infamously subtle. In this work, we develop the rational $Q$-system for this case, which offers a systematic way to find all physical solutions of the Bethe ansatz equations at root of unity. The construction contains two parts. In the first part, we impose additional constraints to the rational $Q$-system. These constraints eliminate the so-called Fabricius-McCoy (FM) string solutions, yielding all primitive solutions. In the second part, we give a simple procedure to construct the descendant tower of any given primitive state. The primitive solutions together with their descendant towers constitute the complete Hilbert space. We test our proposal by extensive numerical checks and apply it to compute the torus partition function of the 6-vertex model at root of unity.