An efficient and accurate decomposition of the Fermi operator

TL;DR

This paper introduces a hybrid polynomial and iterative method for efficiently computing the Fermi function of Hamiltonians, achieving linear scaling and high accuracy, especially for metallic systems, without relying on orbital localization.

Contribution

The paper presents a novel hybrid approach combining polynomial expansion and Newton-like iterations for Fermi operator decomposition, improving scalability and robustness over previous methods.

Findings

Achieves linear scaling with system size

Demonstrates high accuracy on metallic LiAl alloy

Overcomes limitations of previous Fermi expansion techniques

Abstract

We present a method to compute the Fermi function of the Hamiltonian for a system of independent fermions, based on an exact decomposition of the grand-canonical potential. This scheme does not rely on the localization of the orbitals and is insensitive to ill-conditioned Hamiltonians. It lends itself naturally to linear scaling, as soon as the sparsity of the system's density matrix is exploited. By using a combination of polynomial expansion and Newton-like iterative techniques, an arbitrarily large number of terms can be employed in the expansion, overcoming some of the difficulties encountered in previous papers. Moreover, this hybrid approach allows us to obtain a very favorable scaling of the computational cost with increasing inverse temperature, which makes the method competitive with other Fermi operator expansion techniques. After performing an in-depth theoretical analysis of computational cost and accuracy, we test our approach on the DFT Hamiltonian for the metallic phase of the LiAl alloy.