---
tag: MXEZM0QJ1K
kind: concept
name: Anderson localization
msc: [82B44, 47B80, 81Q10]
sources:
  - url: https://en.wikipedia.org/wiki/Anderson_localization
same_as:
  wikidata: Q491409
  wikipedia: Anderson localization
---

## Summary

Anderson localization is the absence of wave transport in a disordered medium.
Mathematically, a random [[Schrödinger operator|89AA6XJY5K]] has, in some energy range, only eigenvalues, with eigenfunctions that decay exponentially away from a centre.
It was proposed by the physicist P. W. Anderson in 1958 for electrons in a random lattice.
It is proved for the one-dimensional Anderson model at all energies and in every dimension at large disorder.

## Explanations

### L1 human

Imagine a wave moving through a crystal whose atoms have random energies.
Each atom scatters the wave a little, and the scattered pieces interfere.
In a strongly disordered medium the interference is destructive on average, and the wave stays trapped near where it started.
For an electron this means the material does not conduct: the states are localized.
Mathematically, localized states are eigenvectors of the energy operator that are concentrated near one point and very small far away.