Computing spectral properties of topological insulators without artificial truncation or supercell approximation

TL;DR

This paper introduces novel computational methods to analyze the spectral properties of topological insulators directly in their infinite-dimensional form, avoiding artificial truncation or supercell approximations, thus enabling more accurate physical property calculations.

Contribution

The authors develop and demonstrate new techniques for computing spectral properties of topological insulators without artificial boundary conditions or supercell approximations.

Findings

Methods accurately compute spectra and eigenstates of infinite operators.

Techniques handle material defects and disorder effectively.

Numerical examples provided for the Haldane model.

Abstract

Topological insulators (TIs) are renowned for their remarkable electronic properties: quantised bulk Hall and edge conductivities, and robust edge wave-packet propagation, even in the presence of material defects and disorder. Computations of these physical properties generally rely on artificial periodicity (the supercell approximation), or unphysical boundary conditions (artificial truncation). In this work, we build on recently developed methods for computing spectral properties of infinite-dimensional operators. We apply these techniques to develop efficient and accurate computational tools for computing the physical properties of TIs. These tools completely avoid such artificial restrictions and allow one to probe the spectral properties of the infinite-dimensional operator directly, even in the presence of material defects and disorder. Our methods permit computation of spectra, approximate eigenstates, spectral measures, spectral projections, transport properties, and conductances. Numerical examples are given for the Haldane model, and the techniques can be extended similarly to other TIs in two and three dimensions.