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

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 infinitedimensional 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.

SpecSolve: Spectral methods for spectral measures

“…The SpecSolve framework can also compute spectral projections E ( [𝑎, 𝑏]) associated with the projection-valued measure by omitting the inner product step in Algorithm 1 and applying endpoint corrections [23]. Figure 2 displays a scalar spectral measure and spectral projection for the partial integro-differential operator…”