Anderson localization: A disorder-induced quantum bound state

Abstract

Electrons at the Fermi energy may lose their ability to propagate to long distances in certain random media. We use Green functions and solve parquet equations for the non-local electron-hole vertex in high spatial dimensions to describe the vanishing of diffusion in Anderson localization. It is caused by forming a quantum bound state between the diffusing particle and the hole left behind. Divergence in a new time scale proportional to the electrical polarizability signals the Anderson localization transition. Consequently, the height of the peak of the dynamical conductivity at zero frequency, the static diffusion constant, is not pushed to zero at the localization transition but rather its width. Spatially localized quantum bound states in the localized phase cannot be described by the continuity and wave equations in the Hilbert space of Bloch waves.