Locobatic theorem for disordered media and validity of linear response

Wojciech De Roeck, Alexander Elgart, Martin Fraas

Published 2022-03-08Version 1

Spectral localization is intrinsically unstable under perturbation. As a result, the adiabatic theorem of quantum mechanics cannot generally hold for localized eigenstates. However, it turns out that a remnant of the adiabatic theorem, which we name the "locobatic theorem", survives: The physical evolution of a typical eigenstate $\psi$ for a random system remains close, with high probability, to the spectral flow for $\psi$ associated with a restriction of the full Hamiltonian to a region where $\psi$ is supported. We make the above statement precise for a class of Hamiltonians describing a particle in a disordered background. Our argument relies on finding a local structure that remains stable under the small perturbation of a random system. An application of this work is the justification of the linear response formula for the Hall conductivity of a two-dimensional system with the Fermi energy lying in a mobility gap. Additional results are concerned with eigenvector hybridization in a one-dimensional Anderson model and the construction of a Wannier basis for underlying spectral projections.