Algebraic Bethe ansatz for the XXZ Heisenberg spin chain with triangular boundaries and the corresponding Gaudin model

N Manojlović,I Salom

doi:10.1016/j.nuclphysb.2017.07.017

Abstract

The implementation of the algebraic Bethe ansatz for the XXZ Heisenberg spin chain in the case, when both reflection matrices have the upper-triangular form is analyzed. The general form of the Bethe vectors is studied. In the particular form, Bethe vectors admit the recurrent procedure, with an appropriate modification, used previously in the case of the XXX Heisenberg chain. As expected, these Bethe vectors yield the strikingly simple expression for the off-shell action of the transfer matrix of the chain as well as the spectrum of the transfer matrix and the corresponding Bethe equations. As in the XXX case, the so-called quasi-classical limit gives the off-shell action of the generating function of the corresponding trigonometric Gaudin Hamiltonians with boundary terms.

Highlights

The quantum inverse scattering method (QISM) is an approach to construct and solve quantum integrable systems [1,2,3]
As opposed to the case of the XXX Heisenberg spin chain where the general reflection matrices could be put into the upper triangular form without any loss of generality [10,12], here the triangularity of the reflection matrices has to be imposed as extra conditions on the parameters of the reflection matrices K∓(λ) (3.4) and (3.5)
As opposed to the case of the XXX Heisenberg spin chain where the general reflection matrices could be put into the upper triangular form without any loss of generality [10,12], here the triangularity of the reflection matrices has to be imposed as extra conditions on the respective parameters

Summary

Introduction

The quantum inverse scattering method (QISM) is an approach to construct and solve quantum integrable systems [1,2,3]. Obtain the off-shell action of the generating function of the trigonometric Gaudin Hamiltonians with boundary terms, on the corresponding Bethe vectors In his approach, Gaudin defined these models as a quasi-classical limit of the integrable quantum chains [26,27]. As we will show below, the quasi-classical expansion of the Bethe vectors we have defined for the XXZ Heisenberg spin chain yields the Bethe vectors of the corresponding Gaudin model The importance of these Bethe vectors stems from the striking simplicity of the off-shell action of the generating function of the trigonometric Gaudin Hamiltonians with boundary terms. Detailed presentation of the illustrative example of the Bethe vector 3(μ1, μ2, μ3), including its general form and some important identities, are given in Appendix C

Inhomogeneous XXZ Heisenberg spin chain

Reflection equation

Inhomogeneous XXZ Heisenberg spin chain with boundary terms

Algebraic Bethe Ansatz

Corresponding Gaudin model

Conclusions