Finite-size corrections to the energy spectra of gapless one-dimensional systems in the presence of boundaries

TL;DR

This paper develops a finite-size scaling theory for one-dimensional quantum critical systems with boundaries, accounting for bulk and boundary perturbations, and verifies it with exact and numerical results.

Contribution

It provides a general expression for finite-size energy levels in boundary-perturbed systems, including a boundary-induced renormalization of system size.

Findings

Derived a universal formula for boundary-affected energy spectra

Confirmed the theory with exact solutions of the Ising chain

Validated results using DMRG on the three-state Potts chain

Abstract

We present the finite-size scaling theory of one-dimensional quantum critical systems in the presence of boundaries. While the finite-size spectrum in the conformal limit, namely of a conformal field theory with conformally invariant boundary conditions, is related to the dimensions of boundary operators by Cardy, the actual spectra of lattice models are affected by both bulk and boundary perturbations and contain non-universal boundary energies. We obtain a general expression of the finite-size energy levels in the presence of bulk and boundary perturbations. In particular, a generic boundary perturbation related to the energy-momentum tensor gives rise to a renormalization of the effective system size. We verify our field-theory formulation by comparing the results with the exact solution of the critical transverse-field Ising chain and with accurate numerical results on the critical three-state Potts chain obtained by Density-Matrix Renormalization Group.