Remarks towards the spectrum of the Heisenberg spin chain type models

TL;DR

This paper reviews the Bethe ansatz solutions for Heisenberg spin chain models, including close and open chains, with detailed constructions of Bethe vectors and special modifications like the small polaron model.

Contribution

It provides a comprehensive analysis of Bethe vectors for various Heisenberg spin chain models, including inhomogeneous and fermionic realizations, and discusses open chain generalizations.

Findings

Explicit Bethe vectors for two-component and inhomogeneous models

Bethe vectors for fermionic realizations of XXX and XXZ models

Discussion on open Hecke chain models

Abstract

The integrable close and open chain models can be formulated in terms of generators of the Hecke algebras. In this review paper, we describe in detail the Bethe ansatz for the XXX and the XXZ integrable close chain models. We find the Bethe vectors for two--component and inhomogeneous models. We also find the Bethe vectors for the fermionic realization of the integrable XXX and XXZ close chain models by means of the algebraic and coordinate Bethe ansatz. Special modification of the XXZ closed spin chain model ("small polaron model") is consedered. Finally, we discuss some questions relating to the general open Hecke chain models.