In condensed matter physics, Anderson localization (also known as strong 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 to suggest that electron localization is possible in a lattice potential, provided that the degree of randomness (disorder) in the lattice is sufficiently large, as can be realized for example in a semiconductor with impurities or defects.

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

$${\displaystyle i\hbar {\frac {d\psi }{dt}}=H\psi ~,}$$

where the Hamiltonian H is given by

$${\displaystyle H\psi _{j}=E_{j}\psi _{j}+\sum _{k\neq j}V_{jk}\psi _{k}~,}$$

where ${\displaystyle j,k}$ are lattice locations. The self-energy ${\displaystyle E_{j}}$ is taken as random and independently distributed. The interaction potential ${\displaystyle V(r)=V(|j-k|)}$ is required to fall off faster than ${\displaystyle 1/r^{3}}$ in the ${\displaystyle r\to \infty }$ limit. For example, one may take ${\displaystyle E_{j}}$ uniformly distributed within a band of energies ${\displaystyle [-W,+W],}$ and

$${\displaystyle V(|r|)={\begin{cases}1,&|r|=|j-k|=1\\0,&{\text{otherwise.}}\end{cases}}}$$