Surface excitations and surface energy of the antiferromagnetic XXZ chain by the Bethe ansatz approach

TL;DR

This paper investigates boundary bound states and surface energy in the antiferromagnetic XXZ chain using Bethe ansatz, revealing critical fields and boundary excitations with results consistent with known limits.

Contribution

It provides an analytic expression for the surface energy and characterizes boundary bound states and excitations in the XXZ chain with boundary magnetic fields.

Findings

Boundary strings indicate boundary bound states at certain fields.

Surface energy is an analytic function of boundary magnetic field h.

Boundary excitations occur in pairs with bulk or boundary excitations at the chain ends.

Abstract

We study boundary bound states using the Bethe ansatz formalism for the finite $XXZ$ $(\Delta>1)$ chain in a boundary magnetic field $h$. Boundary bound states are represented by the ``boundary strings'' similar to those described in the work of H.Saleur, S.Skorik. We find that for certain values of $h$ the ground state wave function contains boundary strings, and from this infer the existence of two ``critical'' fields in agreement with the results of Jimbo et al. An expression for the vacuum surface energy in the thermodynamic limit is derived and found to be an analytic function of $h$. We argue that boundary excitations appear only in pairs with ``bulk'' excitations or with boundary excitations at the other end of the chain. We mainly discuss the case where the magnetic fields at the left and the right boundaries are antiparallel, but we also comment on the case of the parallel fields. In the Ising ($\Delta=\infty$) and isotropic ($\Delta=1$) limits our results agree with those previously known.