Truncation identities for the small polaron fusion hierarchy

TL;DR

This paper investigates a one-dimensional integrable fermionic lattice model, deriving a hierarchy of transfer matrices, identifying conditions for hierarchy truncation, and formulating spectral solutions via Bethe ansatz for various boundary conditions.

Contribution

It introduces a fusion-based hierarchy of transfer matrices for the model, analyzes truncation at roots of unity, and discusses spectral solutions for non-diagonal boundary fields.

Findings

Hierarchy truncates at roots of unity, leading to finite functional equations.

Spectral problem expressed via TQ-equation solvable by Bethe ansatz.

Framework applicable to models with non-diagonal boundary conditions.

Abstract

We study a one-dimensional lattice model of interacting spinless fermions. This model is integrable for both periodic and open boundary conditions, the latter case includes the presence of Grassmann valued non-diagonal boundary fields breaking the bulk U(1) symmetry of the model. Starting from the embedding of this model into a graded Yang-Baxter algebra an infinite hierarchy of comuting transfer matrices is constructed by means of a fusion procedure. For certain values of the coupling constant related to anisotropies of the underlying vertex model taken at roots of unity this hierarchy is shown to truncate giving a finite set of functional equations for the spectrum of the transfer matrices. For generic coupling constants the spectral problem is formulated in terms of a TQ-equation which can be solved by Bethe ansatz methods for periodic and diagonal open boundary conditions. Possible approaches for the solution of the model with generic non-diagonal boundary fields are discussed.