Separation of variables for integrable spin-boson models

TL;DR

This paper develops a novel separation of variables approach for integrable spin-boson models using the functional Bethe ansatz, enabling spectrum analysis under various boundary conditions.

Contribution

It introduces a new half-infinite Sklyanin lattice framework for bosonic representations and derives TQ equations for spectrum computation in spin-boson models.

Findings

Polynomial solutions to TQ equations are found for twisted and certain open boundaries.

Spectrum of the transfer matrix and its quasi-classical limit are computed.

A two-parameter family of Bethe equations is derived for generic open boundaries.

Abstract

We formulate the functional Bethe ansatz for bosonic (infinite dimensional) representations of the Yang-Baxter algebra. The main deviation from the standard approach consists in a half infinite 'Sklyanin lattice' made of the eigenvalues of the operator zeros of the Bethe annihilation operator. By a separation of variables, functional TQ equations are obtained for this half infinite lattice. They provide valuable information about the spectrum of a given Hamiltonian model. We apply this procedure to integrable spin-boson models subject to both twisted and open boundary conditions. In the case of general twisted and certain open boundary conditions polynomial solutions to these TQ equations are found and we compute the spectrum of both the full transfer matrix and its quasi-classical limit. For generic open boundaries we present a two-parameter family of Bethe equations, derived from TQ equations that are compatible with polynomial solutions for Q. A connection of these parameters to the boundary fields is still missing.