Bethe states of the XXZ spin-[formula omitted] chain with arbitrary boundary fields

Xin Zhang,Yuan-Yuan Li,Junpeng Cao,Wen-Li Yang,Kangjie Shi,Yupeng Wang

doi:10.1016/j.nuclphysb.2015.01.022

Abstract

Based on the inhomogeneous T−Q relation constructed via the off-diagonal Bethe Ansatz, the Bethe-type eigenstates of the XXZ spin-12 chain with arbitrary boundary fields are constructed. It is found that by employing two sets of gauge transformations, proper generators and reference state for constructing Bethe vectors can be obtained respectively. Given an inhomogeneous T−Q relation for an eigenvalue, it is proven that the resulting Bethe state is an eigenstate of the transfer matrix, provided that the parameters of the generators satisfy the associated Bethe Ansatz equations.

Highlights

In this paper we focus on constructing the Bethe-type eigenstates (Bethe states) of the quantum boundary fields, defined by the Hamiltonian N −1 H=σjxσjx+1 + σjyσjy+1 + cosh ησjzσjz+1 + h1 · σ1 + hN · σN j=1 =σjxσjx+1 + σjyσjy+1 + cosh ησjzσjz+1 ++ sinh sinh η α− cosh β−
When the boundary parameters obey a constraint [8, 9], which is already in U(1)-symmetry-broken case, the associated Bethe states were constructed [9] within the framework of the generalized algebraic Bethe Ansatz [25, 34]
It should be emphasized that constructing the Bethe state of U(1)-symmetry-broken models had challenged for many years because of the lacking of the inhomogeneous T − Q relations such as (2.25)

Summary

Introduction

In this paper we focus on constructing the Bethe-type eigenstates (Bethe states) of the quantum boundary fields, defined by the Hamiltonian. When the boundary parameters obey a constraint [8, 9], which is already in U(1)-symmetry-broken case, the associated Bethe states were constructed [9] within the framework of the generalized algebraic Bethe Ansatz [25, 34]. Very recently, based on the inhomogeneous T − Q relation and small sites analysis of the model with triangular boundaries, the corresponding Bethe states are conjectured [35] and proven in [36]. In this paper we study the Bethe states of the transfer matrix for the quantum based on the inhomogeneous T − Q relation of the eigenvalues obtained by ODBA. Based on the chosen parameters of the resulting transformations, the Bethe-type eigenstates of the transfer matrix are constructed. Some useful formulae and technical proofs are given in Appendices A-C respectively

ODBA solution

Gauge transformations and the associated operators

Conclusions

The proof of the reference state