Bethe's Quantum Numbers And Rigged Configurations

TL;DR

This paper introduces a new method to determine Bethe's quantum numbers, called rigged configurations, for solutions of the Bethe ansatz equations in the spin-1/2 Heisenberg model, validated for chains of length 12.

Contribution

It presents a novel approach to compute quantum numbers for Bethe ansatz solutions using string sum behaviors, confirmed for all solutions of length 12.

Findings

Method successfully determines quantum numbers for all solutions of length 12

Rigged configurations align with Bethe ansatz solutions

Behavior of quantum number sums within strings is consistent

Abstract

We propose a method to determine the quantum numbers, which we call the rigged configurations, for the solutions to the Bethe ansatz equations for the spin-1/2 isotropic Heisenberg model under the periodic boundary condition. Our method is based on the observation that the sums of Bethe's quantum numbers within each string behave particularly nicely. We confirm our procedure for all solutions for length 12 chain (totally 923 solutions).