---
title: "Anderson localization: disorder could stop diffusion"
description: "Why can a quantum wave remain localized even when the lattice still has connected paths?"
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language: "en"
page_type: "story"
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  zh-CN: "https://sandboxphysics.com/zh/stories/condensed-matter/anderson-localization/"
last_modified: "2026-10-23"
---

[Magnetism, disorder and topological response](https://sandboxphysics.com/stories/condensed-matter/#magnetism)

Materials · Measurements · Explanation

# Anderson localization: disorder could stop diffusion

Why can a quantum wave remain localized even when the lattice still has connected paths?

Sandbox Physics · 2026-10-23

Nobel contribution: 1977-1

1. Randomize site energies
2. Allow coherent hopping
3. Track a wave or transmission
4. Compare size and disorder dependence

Classical connectivity and quantum interference are different mechanisms. A finite transmission drop alone does not prove an infinite-system transition.

## The 1958 model kept hopping but randomized energies

Anderson’s 1958 paper studied a lattice with site energies drawn from a random distribution. Neighboring sites remained coupled, but sufficiently strong randomness could prevent diffusion of a coherent quantum excitation.

The result challenged the assumption that disorder merely shortens a mean free path. A connected graph does not guarantee extended quantum eigenstates.

Sources: [3] [1]

## Interference distinguishes localization from a blocked path

Scattered amplitudes add with phase. Repeated coherent scattering can suppress long-range propagation without removing the geometric route. Turning off phase coherence or adding interactions changes the problem, even if the same random landscape remains.

Sources: [1] [3]

## Size and dimension matter to the inference

A finite sample can show low transmission through a barrier, a resonance mismatch or localization. Varying sample size and disorder distinguishes these possibilities. The existence and character of a mobility edge depend on dimension and symmetry, so one chain does not represent every material.

Sources: [1] [2]

## The existing Lab keeps the single-particle question

The localization Lab compares spatial confinement and transport in its specified coherent model. It is reused here rather than rebuilt. The 1977 contribution also includes local-moment physics; interaction-driven Mott insulation is discussed separately because it requires a different mechanism.

Sources: [1] [4]

Continue with the physics

## Try the related models

### [Disorder and wave transport](https://sandboxphysics.com/labs/anderson-localization-transport/)

Change disorder and compare transmission and spatial localization.

The single-particle model omits electron interactions and cannot produce a Mott transition.

## Sources and reading notes

1. [Anderson: Local moments and localized states (1977)](https://www.nobelprize.org/uploads/2018/06/anderson-lecture-1.pdf)

   The local-moment and disorder-localization problems are related contributions with different Hamiltonians.

2. [Mott: Electrons in glass (1977)](https://www.nobelprize.org/uploads/2018/06/mott-lecture.pdf)

   Mobility edges, amorphous semiconductors and hopping; do not merge these with interaction-driven Mott localization.

3. [Anderson: Absence of diffusion in certain random lattices (1958)](https://doi.org/10.1103/PhysRev.109.1492)

   Original random-site-energy model and its transport criterion.

4. [Nobel Prize in Physics 1977: official award](https://www.nobelprize.org/prizes/physics/1977/summary/)

   Official attribution and award year. The observation or theoretical paper may be much earlier.

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