Jordan blocks and the Bethe ansatz I: The eclectic spin chain as a limit

TL;DR

This paper introduces a method to derive generalized eigenvectors of non-diagonalisable matrices via perturbation and applies it to analyze the eclectic spin chain using the Bethe Ansatz, revealing new spectral insights.

Contribution

It develops a perturbation-based procedure to obtain generalized eigenvectors and demonstrates its application to the eclectic spin chain with the Bethe Ansatz.

Findings

Successfully computed a subset of the spectrum of the eclectic spin chain.

Showed that the Bethe Ansatz contains sufficient information to reconstruct generalized eigenvectors.

Established a link between the twisted spin chain's Bethe Ansatz and the eclectic spin chain's spectral properties.

Abstract

We present a procedure to extract the generalised eigenvectors of a non-diagonalisable matrix by considering a diagonalisable perturbation of it and computing the non-diagonalisable limit of its eigenvectors. As an example of this process, we compute a subset of the spectrum of the eclectic spin chain by means of the Nested Coordinate Bethe Ansatz. This allows us to show that the Bethe Ansatz of the finitely twisted spin chain contains enough information to reconstruct the generalised eigenvectors of the eclectic spin chain.