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

Abstract

We consider the infrared (IR) limit of the nonlinear integral equations (NLIEs) for the boundary supersymmetric sine-Gordon (BSSG) model, previously obtained from the NLIEs for the inhomogeneous open spin-1 XXZ quantum spin chain with general integrable boundary terms, for values of the boundary parameters which satisfy a certain constraint. In particular, we compute the boundary S matrix and determine the "lattice -IR" relation for the BSSG parameters. 1 email: rmurgan@svsu.edu 1 The lattice boundary parameters satisfy a pair of real constraint. 2 Note that in [50], two BSSG models are discussed: BSSG − and BSSG + . Our results are related to the latter model by Bajnok et al.

The solution of an open XXZ chain with arbitrary spin revisited

“…One application of such a solution of the open XXZ chain with arbitrary spin is, the s = 1 case enables one to investigate the boundary version of the supersymmetric sine-Gordon model [21][22][23]. In particular, the Bethe ansatz solutions of the open spin-1 XXZ chain have been used to derive the nonlinear integral equations for the supersymmetric sine-Gordon model [24,25].…”

GeneralizedT–Qrelations and the open spin-sXXZ chain with nondiagonal boundary terms

“…Perhaps one could carry out a more thorough treatment and analysis of the functional equation to yield the exact form of the Q i (u) functions, thus avoiding the need for an ansatz such as (3.14). Another interesting problem is to see the relation of the s = 1 case to the supersymmetric sine-Gordon (SSG) model, along the lines of [58,59], but now for the spin-1 chain with nondiagonal boundary terms described by the generalized T -Q relations instead of the conventional T -Q relation. One could also try to generalize the solutions presented in [48] for the spin-1/2 case, where all six boundary parameters are completely arbitrary, to any spin s, and analyze the s = 1 case for this general solution in relation to the SSG model.…”

“…We remark that a more general form of T -Q relations were found in [48] involving multiple Q(u) functions. Moreover, the relationship of the s = 1 case to the supersymmetric sine-Gordon (SSG) model [49]- [54], especially the boundary SSG model [55]- [59], has inspired us to consider the problem. We stress that these results hold for cases with at most two arbitrary boundary parameters at roots of unity, namely when the bulk anisotropy parameter has values η = iπ/2, iπ/4, .…”

Generalized T-Q relations and the open spin-s XXZ chain with nondiagonal boundary terms

“…Perhaps one could carry out a more thorough treatment and analysis of the functional equation to yield the exact form of the Q i (u) functions, thus avoiding the need for an ansatz such as (3.14). Another interesting problem is to see the relation of s = 1 case to the supersymmetric sine-Gordon (SSG) model, along the lines of [58] and [59], but now for spin-1 chain with nondiagonal boundary terms described by the generalized T − Q relations instead of the conventional T − Q relation. One could also try to generalize the solutions presented in [48] for the spin-1/2 case, where all six boundary parameters are completely arbitrary, to any spin s, and analyze the s = 1 case for this general solution in relation to the SSG model.…”

“…We remark that a more general form of T −Q relations were found in [48] involving multiple Q(u) functions. Moreover, the relation of s = 1 case to the supersymmetric sine-Gordon (SSG) model [49]- [54], especially the boundary SSG model [55]- [59], has inspired us to consider the problem. We stress that these results hold for cases with at most two arbitrary boundary parameters at roots of unity, namely when the bulk anisotropy parameter has vales η = iπ 2 , iπ 4 , .…”