Coordinate-energy transformation and the one-point function for the Heisenberg-Ising XXZ spin-1/2 chain on the ring

TL;DR

This paper introduces a basis transformation that inverts the coordinate Bethe Ansatz for the Heisenberg-Ising XXZ spin-1/2 chain, providing evidence for its completeness and deriving a new formula for the one-point function.

Contribution

It presents a constructive approach to demonstrate the completeness of the Bethe Ansatz and derives an exact one-point function formula assuming completeness.

Findings

Numerical verification of the transformation for non-zero anisotropy and odd-length rings.

A new exact formula for the one-point function based on the Izergin-Korepin determinant.

Evidence supporting the completeness of the Bethe Ansatz in this context.

Abstract

We provide a basis transformation that inverts the coordinate Bethe Ansatz. It is widely believed that the Bethe Ansatz is complete, based on numerical evidence and combinatorial arguments. We present a constructive and comprehensive approach to show that the Bethe Ansatz is complete. The transformation formula is verified numerically with excellent accuracy for non-zero anisotropy parameters and rings of odd length. Additionally, assuming the Bethe Ansatz is complete, we derive a novel exact formula for the one-point function systems through special identities for the Izergin-Korepin determinant.