The open XXZ and associated models at q root of unity

TL;DR

This paper analyzes the open XXZ model at q root of unity, exploring associated models, boundary conditions, and spectrum, providing explicit Hamiltonians, non-local charges, and Bethe states for various related integrable systems.

Contribution

It offers explicit Hamiltonians, boundary charge expressions, and spectrum analysis for the open XXZ model and related models at q root of unity, including new insights into boundary effects.

Findings

Explicit local Hamiltonian for spin 1/2 XXZ chain with boundary degrees of freedom

Derived boundary non-local charges for sine-Gordon and q harmonic oscillator models

Identified spectrum and Bethe states for models with specific boundary conditions

Abstract

The generalized open XXZ model at $q$ root of unity is considered. We review how associated models, such as the $q$ harmonic oscillator, and the lattice sine-Gordon and Liouville models are obtained. Explicit expressions of the local Hamiltonian of the spin ${1 \over 2}$ XXZ spin chain coupled to dynamical degrees of freedom at the one end of the chain are provided. Furthermore, the boundary non-local charges are given for the lattice sine Gordon model and the $q$ harmonic oscillator with open boundaries. We then identify the spectrum and the corresponding Bethe states, of the XXZ and the q harmonic oscillator in the cyclic representation with special non diagonal boundary conditions. Moreover, the spectrum and Bethe states of the lattice versions of the sine-Gordon and Liouville models with open diagonal boundaries is examined. The role of the conserved quantities (boundary non-local charges) in the derivation of the spectrum is also discussed.