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arXiv:1412.2473

Boundary effects on the lattice/continuum correspondence: the spin-1/2 XXZ chain and the sine-Gordon model

Chihiro Matsui

Published Dec 8, 2014• 0 citations
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Abstract

We derived the corresponding boundary condition on Fermi fields to the spin-1/2 Heisenberg chain with boundary magnetic fields. In order to obtain the correct boundary condition from the variation of the action at the edges, we carefully treat the oscillating terms which emerge as a result of the chiral decomposition of fermions and do not contribute to the bulk Lagrangian. The obtained result is checked by compared with the exact result derived from the Bethe ansatz, by considering the mode expansion of fermions on the light-cone coordinates. We also give the spin-wave interpretation to the emergence of boundary bound states.

DOI
10.48550/arxiv.1412.2473

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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 SSS 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 SSS 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.