Decay properties of spectral projectors with applications to electronic structure

TL;DR

This paper provides a rigorous theoretical analysis of the exponential decay of spectral projectors in large sparse Hermitian matrices, supporting linear scaling methods in electronic structure calculations for non-metallic systems.

Contribution

It offers a mathematical proof of decay properties of spectral projectors, extending to metallic systems at positive temperature, with implications for quantum chemistry and physics.

Findings

Proves exponential decay of density matrices in gapped systems

Supports linear scaling methods in electronic structure calculations

Discusses decay properties in metallic systems at positive temperature

Abstract

Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the exponential off-diagonal decay ("nearsightedness") for the density matrix of gapped systems at zero electronic temperature in both orthogonal and non-orthogonal representations, thus providing a firm theoretical basis for the possibility of linear scaling methods in electronic structure calculations for non-metallic systems. We further discuss the case of density matrices for metallic systems at positive electronic temperature. A few other possible applications are also discussed.