Fast Fourier-Chebyshev approach to real-space simulations of the Kubo formula

TL;DR

This paper introduces a fast, scalable numerical method combining Chebyshev expansions and divide-and-conquer techniques to accurately simulate transport properties in large disordered lattice systems using the Kubo formula.

Contribution

It presents a novel hybrid algorithm that enables efficient and accurate real-space simulations of large 2D systems for transport phenomena, surpassing previous computational limitations.

Findings

Successfully computed conductance for systems with over 10^7 sites.

Accurately resolved linear-response properties in disordered systems.

Demonstrated scalability and spectral accuracy of the method.

Abstract

The Kubo formula is a cornerstone in our understanding of near-equilibrium transport phenomena. While conceptually elegant, the application of Kubo's linear-response theory to interesting problems is hindered by the need for algorithms that are accurate and scalable to large lattice sizes beyond one spatial dimension. Here, we propose a general framework to numerically study large systems, which combines the spectral accuracy of Chebyshev expansions with the efficiency of divide-and-conquer methods. We use the hybrid algorithm to calculate the two-terminal conductance and the bulk conductivity tensor of 2D lattice models with over $10^7$ sites. By efficiently sampling the microscopic information contained in billions of Chebyshev moments, the algorithm is able to accurately resolve the linear-response properties of complex systems in the presence of quenched disorder. Our results lay the groundwork for future studies of transport phenomena in previously inaccessible regimes.