NLIE and finite size effects of the spin-1/2 XXZ and sine-Gordon models with two boundaries revisited

TL;DR

This paper derives a nonlinear integral equation for the sine-Gordon model with boundaries from the XXZ spin chain, computes boundary and Casimir energies, and confirms the effective central charge matches theoretical predictions.

Contribution

It introduces a new NLIE for the sine-Gordon model with boundaries derived from the XXZ chain and establishes a relation between lattice and continuum boundary parameters.

Findings

Derived NLIE from T-Q equation for open XXZ chain.

Computed boundary and Casimir energies for the sine-Gordon model.

Numerical results for the effective central charge agree with theoretical UV limit.

Abstract

Starting from the T-Q equation of an open integrable spin-1/2 XXZ quantum spin chain with nondiagonal boundary terms, we derive a nonlinear integral equation (NLIE) of the sine-Gordon model on a finite interval. We compute the boundary energy and the Casimir energy for the sine-Gordon model with both left and right boundaries. A relation between the boundary parameters of the continuum model and the lattice model is given. We also present numerical results for the effective central charge of an open spin-1/2 XXZ quantum spin chain which find agreement with our analytical result for the central charge of the sine-Gordon model in the ultraviolet (UV) limit.