Internal parity symmetry and degeneracy of Bethe Ansatz strings in the isotropic heptagonal magnetic ring

TL;DR

This paper investigates the degeneracy of Bethe eigenfunctions in the isotropic XXX model on a heptagonal ring, revealing how internal parity symmetry leads to doubly degenerate energy levels and explicit eigenfunction construction.

Contribution

It provides an explicit algebraic Bethe Ansatz construction of degenerate eigenfunctions and links degeneracy to internal parity symmetry of Bethe strings.

Findings

Degeneracy occurs at the center of the Brillouin zone in the three-deviation sector.

Eigenfunctions with opposite quasimomenta form orthogonal pairs due to internal parity symmetry.

Strings of different lengths are distinguishable, enabling the generation of orthogonal eigenfunctions.

Abstract

The exact Bethe eigenfunctions for the heptagonal ring within the isotropic XXX model exhibit a doubly degenerated energy level in the three-deviation sector at the centre of the Brillouin zone. We demonstrate an explicit construction of these eigenfunctions by use of algebraic Bethe Ansatz, and point out a relation of degeneracy to parity conservation, applied to the configuration of strings for these eigenfunctions. Namely, the internal structure of the eigenfunctions (the 2-string and the 1-string, with opposite quasimomenta) admits generation of two mutually orthogonal eigenfunctions due to the fact that the strings which differ by their length are distinguishable objects.