A hybrid approach to Fermi operator expansion

TL;DR

This paper introduces an improved hybrid method combining polynomial expansion and iterative inversion for efficiently computing the finite temperature density matrix, significantly reducing computational scaling for high-resolution ab-initio simulations.

Contribution

The authors develop a real-valued formalism using Chebyshev polynomials and fast summation, enhancing the efficiency and scalability of the previous hybrid approach.

Findings

Reduced algorithm scaling to the cubic root of the Hamiltonian spectrum width

Enhanced efficiency for high energy resolution ab-initio simulations

Significant improvements over the previous complex-valued formalism

Abstract

In a recent paper we have suggested that the finite temperature density matrix can be computed efficiently by a combination of polynomial expansion and iterative inversion techniques. We present here significant improvements over this scheme. The original complex-valued formalism is turned into a purely real one. In addition, we use Chebyshev polynomials expansion and fast summation techniques. This drastically reduces the scaling of the algorithm with the width of the Hamiltonian spectrum, which is now of the order of the cubic root of such parameter. This makes our method very competitive for applications to ab-initio simulations, when high energy resolution is required.