Variational discrete variable representation for excitons on a lattice

TL;DR

This paper introduces a scalable numerical basis for lattice excitons that reduces computational effort and preserves the variational principle, enabling detailed study of exciton properties beyond continuum models.

Contribution

It presents a novel variational discrete variable representation that efficiently computes exciton states on a lattice, bridging the gap to the continuum limit and improving over standard methods.

Findings

Reduced computational scaling in three dimensions

Ability to compute binding energies and spectra accurately

Application to excitons in cuprous oxide revealing non-hydrogenic behavior

Abstract

We construct numerical basis function sets on a lattice, whose spatial extension is scalable from single lattice sites to the continuum limit. They allow us to compute small and large bound states with comparable, moderate effort. Adopting concepts of discrete variable representations, a diagonal form of the potential term is achieved through a unitary transformation to Gaussian quadrature points. Thereby the computational effort in three dimensions scales as the fourth instead of the sixth power of the number of basis functions along each axis, such that it is reduced by two orders of magnitude in realistic examples. As an improvement over standard discrete variable representations, our construction preserves the variational principle. It allows for the calculation of binding energies, wave functions, and excitation spectra. We use this technique to study central-cell corrections for excitons beyond the continuum approximation. A discussion of the mass and spectrum of the yellow exciton series in the cuprous oxide, which does not follow the hydrogenic Rydberg series of Mott-Wannier excitons, is given on the basis of a simple lattice model.