Efficient Linear Scaling Approach for Computing the Kubo Hall Conductivity

TL;DR

This paper introduces an efficient order-N computational method for calculating the Kubo Hall conductivity in large, disordered 2D systems under realistic magnetic fields, enabling new insights into quantum transport phenomena.

Contribution

The paper presents a novel wavepacket propagation and continued fraction expansion approach that scales linearly with system size, allowing analysis of systems with tens of millions of orbitals.

Findings

Validated method against brute-force diagonalization in disordered graphene

Enabled analysis of large-scale 2D materials under realistic magnetic fields

Facilitated deeper understanding of microscopic structures from experimental data

Abstract

We report an order-N approach to compute the Kubo Hall conductivity for disorderd two-dimensional systems reaching tens of millions of orbitals, and realistic values of the applied external magnetic fields (as low as a few Tesla). A time-evolution scheme is employed to evaluate the Hall conductivity $\sigma_{xy}$ using a wavepacket propagation method and a continued fraction expansion for the computation of diagonal and off-diagonal matrix elements of the Green functions. The validity of the method is demonstrated by comparison of results with brute-force diagonalization of the Kubo formula, using (disordered) graphene as system of study. This approach to mesoscopic system sizes is opening an unprecedented perspective for so-called reverse engineering in which the available experimental transport data are used to get a deeper understanding of the microscopic structure of the samples. Besides, this will not only allow addressing subtle issues in terms of resistance standardization of large scale materials (such as wafer scale polycrystalline graphene), but will also enable the discovery of new quantum transport phenomena in complex two-dimensional materials, out of reach with classical methods.