Hyper-sparsity of the density matrix in a wavelet representation

TL;DR

This paper demonstrates that using wavelet basis sets reveals a new hyper-sparsity in the density matrix, exploiting localization in both real and Fourier space, enabling more efficient large-scale quantum calculations.

Contribution

It introduces the concept of hyper-sparsity in the density matrix, combining real and Fourier space localization to improve O(N) methods for large quantum systems.

Findings

Density matrix exhibits hyper-sparsity in wavelet basis

Applicable to both insulating and metallic systems

Enables compact representation of large systems

Abstract

O(N) methods are based on the decay properties of the density matrix in real space, an effect sometimes refered to as near-sightedness. We show, that in addition to this near-sightedness in real space there is also a near-sightedness in Fourier space. Using a basis set with good localization properties in both real and Fourier space such as wavelets, one can exploit both localization properties to obtain a density matrix which exhibits additional sparseness properties compared to the scenario where one has a basis set with real space localization only. We will call this additional sparsity hyper-sparsity. Taking advantage of this hyper-sparsity, it is possible to represent very large quantum mechanical systems in a highly compact way. This can be done both for insulating and metallic systems and for arbitrarily accurate basis sets. We expect that hyper-sparsity will pave the way for O(N) calculations of large systems requiring many basis functions per atom, such as Density Functional calculations.