Anderson localization

In condensed matter physics, Anderson localization is the absence of diffusion of waves in a disordered medium. This phenomenon is named after the American physicist P. W. Anderson, who was the first one to suggest the possibility of electron localization inside a semiconductor, provided that the degree of randomness of the impurities or defects is sufficiently large.

Anderson localization is a general wave phenomenon that applies to the transport of electromagnetic waves, acoustic waves, quantum waves, spin waves, etc. This phenomenon is to be distinguished from weak localization, which is the precursor effect of Anderson localization (see below), and from Mott localization, named after Sir Nevill Mott, where the transition from metallic to insulating behaviour is not due to disorder, but to a strong mutual Coulomb repulsion of electrons.

In the original Anderson tight-binding model, the evolution of the wave function ψ on the d-dimensional lattice Z^d is given by the Schrödinger equation

where the Hamiltonian H is given by

with E_j random and independent, and interaction V(r) falling off as r⁻² at infinity. For example, one may take E_j uniformly distributed in [−W,   +W], and

Starting with ψ₀ localised at the origin, one is interested in how fast the probability distribution ${\displaystyle |\psi |^{2}}$ diffuses. Anderson's analysis shows the following:

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