Finite size effects in the spin-1 XXZ and supersymmetric sine-Gordon models with Dirichlet boundary conditions

TL;DR

This paper derives nonlinear integral equations from the Bethe Ansatz solution of the open spin-1 XXZ chain to describe the boundary supersymmetric sine-Gordon model with Dirichlet conditions, confirming the boundary S matrix and analyzing finite size effects.

Contribution

It introduces a new set of NLIEs for the boundary supersymmetric sine-Gordon model derived from the spin-1 XXZ chain and establishes the relation between boundary parameters and the S matrix.

Findings

Boundary S matrix matches previous proposals.

Derived relation between UV boundary parameters and IR S matrix parameters.

Numerical solutions agree with analytical UV limit results.

Abstract

Starting from the Bethe Ansatz solution of the open integrable spin-1 XXZ quantum spin chain with diagonal boundary terms, we derive a set of nonlinear integral equations (NLIEs), which we propose to describe the boundary supersymmetric sine-Gordon model BSSG${}^{+}$ with Dirichlet boundary conditions on a finite interval. We compute the corresponding boundary $S$ matrix, and find that it coincides with the one proposed by Bajnok, Palla and Tak\'acs for the Dirichlet BSSG${}^{+}$ model. We derive a relation between the (UV) parameters in the boundary conditions and the (IR) parameters in the boundary $S$ matrix. By computing the boundary vacuum energy, we determine a previously unknown parameter in the scattering theory. We solve the NLIEs numerically for intermediate values of the interval length, and find agreement with our analytical result for the effective central charge in the UV limit and with boundary conformal perturbation theory.