Localization landscape for Dirac fermions

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 Schrödinger 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.

6 pages, 6 figures

References in corpus (15)

• Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect
• Anderson Transitions
• Substrate-induced bandgap in graphene on hexagonal boron nitride
• Kwant: a software package for quantum transport
• Majorana bound states without vortices in topological superconductors with electrostatic defects
• Localization landscape theory of disorder in semiconductors. III. Application to carrier transport and recombination in light emitting diodes
• Non-Hermitian disorder in two-dimensional optical lattices
• Localization landscape theory of disorder in semiconductors II: Urbach tails of disordered quantum well layers
• Effective confining potential of quantum states in disordered media
• Anderson transition in three-dimensional systems with non-Hermitian disorder
• Localization landscape theory of disorder in semiconductors I: Theory and modeling
• A road to reality with topological superconductors
• Finite difference method for transport properties of massless Dirac fermions
• Finite-difference method for transport of two-dimensional massless Dirac fermions in a ribbon geometry
• Many-Body Localization Landscape

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• A stochastic method to compute the $L^2$ localisation landscape
• Kernel polynomial method to Anderson transition in disordered $\beta$-graphyne
• Majorana-metal transition in a disordered superconductor: percolation in a landscape of topological domain walls
• Anderson localisation in two dimensions: insights from Localisation Landscape Theory, exact diagonalisation, and time-dependent simulations
• Second order accurate Dirichlet boundary conditions for linear nonlocal diffusion problems