Derivation of Kubo's formula for disordered systems at zero temperature

This work justifies the linear response formula for the Hall conductance of a two-dimensional disordered system. The proof rests on controlling the dynamics associated with a random time-dependent Hamiltonian. The principal challenge is related to the fact that spectral and dynamical localization are intrinsically unstable under perturbation, and the exact spectral flow - the tool used previously to control the dynamics in this context - does not exist. We resolve this problem by proving a local adiabatic theorem: With high probability, the physical evolution of a localized eigenstate $\psi$ associated with a random system remains close to the spectral flow for a restriction of the instantaneous Hamiltonian to a region $R$ where the bulk of $\psi$ is supported. Allowing $R$ to grow at most logarithmically in time ensures that the deviation of the physical evolution from this spectral flow is small. To substantiate our claim on the failure of the global spectral flow in disordered systems, we prove eigenvector hybridization in a one-dimensional Anderson model at all scales.