On the NLIE of (inhomogeneous) open spin-1 XXZ chain with general integrable boundary terms

Abstract

Starting from the T -Q equations of the open spin-1 XXZ quantum spin chain with general integrable boundary terms, for values of the boundary parameters which satisfy a certain constraint, we derive a set of nonlinear integral equations (NLIEs) for the inhomogeneous open spin-1 XXZ chain. By taking the continuum limit of these NLIEs, and working in analogy with the open spin-1 XXZ chain with diagonal boundary terms, we compute the boundary and the Casimir energies of the corresponding supersymmetric sine-Gordon (SSG) model. We also present an analytical result for the effective central charge in the ultraviolet (UV) limit. 1 email: rmurgan@gustavus.edufor the open spin-1 XXZ spin chain with general integrable boundary terms are reviewed. This is followed by the derivation of a set of NLIEs for the inhomogeneous open spin-1 XXZ chain in section 3. We then give a brief review of the SSG model following [41]. Proceeding by analogy with the SG model with general integrable boundary conditions [14,27] and the SSG model with Dirichlet boundary conditions [46], we consider the continuum limit of the NLIEs in section 4. In section 5, we compute the boundary and Casimir energies for the ground state. We analyze the UV limit of the Casimir energy and give an analytical result for the effective central charge in section 6. Finally, we conclude the paper with a brief discussion of our results and some open problems in section 7.

A note on the IR limit of the NLIEs of boundary supersymmetric sine-Gordon model

“…Since the set of NLIEs that describe BSSG models and the NLIEs for the inhomogeneous open spin-1 XXZ quantum spin chains are closely related, in this section, we briefly review some crucial results on the open spin-1 XXZ quantum spin chain which include the Hamiltonian of the open spin-1 XXZ chain, the T -Q equations for the inhomogeneous model, important auxiliary functions and the NLIEs. The readers are urged to refer to [15,54] for more details on these materials.…”

“…The supersymmetric counterpart, the SSG model [41]- [54], have also received equally extensive attention. Initially, a set of NLIEs for the close spin-1 XXZ chain, which is related to the SSG model, were derived in [55,56] from the T -Q equations of close spin-1 XXZ spin chain, by exploiting the analyticity of the transfer matrix eigenvalues described by these equations.…”

“…Initially, a set of NLIEs for the close spin-1 XXZ chain, which is related to the SSG model, were derived in [55,56] from the T -Q equations of close spin-1 XXZ spin chain, by exploiting the analyticity of the transfer matrix eigenvalues described by these equations. Such a method have been later used to derive the desired NLIEs describing SSG models with periodic boundary conditions [52], the Dirichlet [53] and also for a more general boundary condition [54]. These NLIEs were used to calculate finite-size properties of ground and excited states.…”

“…Our motivation for the present work comes from these efforts. Utilizing solutions for the open spin-1 XXZ chain with general integrable boundary terms (rather than the diagonal boundary terms used in [53]) [14] that have been recently proposed [15] 1 , a corresponding set of NLIEs was derived for a BSSG model [54]. We thus plan to analyze the IR limit of these NLIEs along the line of [53] and obtain the corresponding boundary S matrix.…”

“…In section 2, we briefly review the open spin-1 XXZ quantum spin chain, namely the Hamiltonian, the T -Q equations and the NLIEs of the model. The BSSG model and the set of NLIEs that describe the model are reviewed in section 3, which are reproduced from [54]. In section 4, we give the main results of the paper.…”