Jordan blocks and the Bethe ansatz III: Class 5 model and its symmetries

TL;DR

This paper investigates the non-diagonalisable structure of the transfer matrix and Hamiltonian in a specific integrable model, revealing differences in their Jordan block structures despite their commutation.

Contribution

It introduces a symmetry-based method to construct Jordan blocks for the transfer matrix in the Class 5 model, extending understanding beyond the Algebraic Bethe Ansatz.

Findings

Transfer matrix and Hamiltonian are not diagonalisable.

Jordan blocks are constructed using model symmetries.

Hamiltonian and transfer matrix have different Jordan block structures.

Abstract

We study the Hilbert space of the Class 5 model described in arXiv:1904.12005. Despite being integrable, neither its transfer matrix nor its Hamiltonian are diagonalisable, meaning that the usual Algebraic Bethe Ansatz does not provide the full Hilbert space. Instead, we make use of the symmetries of the model to construct the Jordan blocks of the transfer matrix. We also show that the Hamiltonian and the transfer matrix, despite commuting, do not have the same Jordan block structure.