Localization landscape for Dirac fermions

TL;DR

This paper extends the localization landscape theory to Dirac fermions, enabling prediction of wave function localization in systems like graphene and topological insulators without solving eigenvalue problems.

Contribution

It generalizes the localization landscape concept to the Dirac equation, incorporating spin-orbit effects and establishing a mapping between Hermitian and non-Hermitian Anderson models.

Findings

Landscape function predicts localization sites in Dirac systems.

Comparison matrix defines equivalence classes for localization.

Mapping between Hermitian and non-Hermitian models established.

Abstract

In the theory of Anderson localization, a landscape function predicts where wave functions localize in a disordered medium, without requiring the solution of an eigenvalue problem. It is known how to construct the localization landscape for the scalar wave equation in a random potential, or equivalently for the Schr\"{o}dinger equation of spinless electrons. Here we generalize the concept to the Dirac equation, which includes the effects of spin-orbit coupling and allows to study quantum localization in graphene or in topological insulators and superconductors. The landscape function $u(r)$ is defined on a lattice as a solution of the differential equation $\overline{{H}}u(r)=1$, where $\overline{{H}}$ is the Ostrowsky comparison matrix of the Dirac Hamiltonian. Random Hamiltonians with the same (positive definite) comparison matrix have localized states at the same positions, defining an equivalence class for Anderson localization. This provides for a mapping between the Hermitian and non-Hermitian Anderson model.