Finite size effects in the XXZ and sine-Gordon models with two boundaries

TL;DR

This paper calculates boundary and Casimir energies in the XXZ spin chain and sine-Gordon models with boundaries, providing a comprehensive analysis for various boundary conditions and coupling constants.

Contribution

It introduces a Bethe Ansatz-based method to analyze finite size effects in boundary quantum models, including a nonlinear integral equation for the sine-Gordon ground state.

Findings

Explicit formulas for boundary and Casimir energies.

Valid for general bulk coupling constants and boundary interactions.

Results encompass both diagonal and nondiagonal boundary cases.

Abstract

We compute the boundary energy and the Casimir energy for both the spin-1/2 XXZ quantum spin chain and (by means of the light-cone lattice construction) the massive sine-Gordon model with both left and right boundaries. We also derive a nonlinear integral equation for the ground state of the sine-Gordon model on a finite interval. These results, which are based on a recently-proposed Bethe Ansatz solution, are for general values of the bulk coupling constant, and for both diagonal and nondiagonal boundary interactions. However, the boundary parameters are restricted to obey one complex (two real) constraints.