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

TL;DR

This paper derives and verifies boundary conditions for fermionic fields in the spin-1/2 Heisenberg chain with boundary magnetic fields, connecting lattice models to continuum field theories and exploring boundary bound states.

Contribution

It provides a detailed derivation of boundary conditions for Fermi fields in the Heisenberg chain and confirms them through Bethe ansatz comparisons, linking lattice and continuum descriptions.

Findings

Correct boundary conditions derived for Fermi fields.

Verification through Bethe ansatz confirms the results.

Insight into boundary bound states via spin-wave interpretation.

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.