Localization landscape for Dirac fermions

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 Schrodinger 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(r) is defined on a lattice as a solution of the differential equation (sic)u(r) = 1, where (sic) 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.