Localization landscape for Dirac fermions

Physical review. B./Physical review. B, vol. 101

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\"odinger equation of spinless electrons. Here, we generalize the concept to the Dirac equation, which includes the effects of spin-orbit coupling and allows us to study quantum localization in graphene or in topological insulators and superconductors. The landscape function $u(\mathbit{r})$ is defined on a lattice as a solution of the differential equation $\stackrel{⎴}{H}u(\mathbit{r})=1$, where $\stackrel{⎴}{H}$ is the Ostrowski 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.

Authors 6

1. G. Lemut Aachen Instituut-Lorentz

   Leiden University

   Affiliation as printed

   Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands

2. M. J. Pacholski Aachen Instituut-Lorentz

   Leiden University

   Affiliation as printed

   Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands

3. O. Ovdat Aachen Instituut-Lorentz

   Leiden University

   Affiliation as printed

   Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands

4. Aurélien Grabsch Aachen Instituut-Lorentz

   Leiden University

   Affiliation as printed

   Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands

5. J. Tworzydło

   University of Warsaw

   Affiliation as printed

   Faculty of Physics, University of Warsaw, ulica Pasteura 5, 02-093 Warszawa, Poland

6. C. W. J. Beenakker Aachen Instituut-Lorentz

   Leiden University

   Affiliation as printed

   Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands