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 to describe the vanishing of diffusion in Anderson localization. It is caused by a diverging timescale responsible for forming a quantum bound state between the diffusing particle and the hole left behind. Divergence in this new timescale, proportional to the electrical polarizability, signals the Anderson localization transition. The critical behavior with this scale becomes symmetric on both sides of the transition. Consequently, the height of the peak of the dynamic conductivity in the metallic phase, the static diffusion constant, is not pushed to zero at the localization transition, but rather its width. Spatially localized quantum bound states cannot be described by the standard continuity and diffusion equations in the Hilbert space of Bloch waves with real wave vectors.