The Cluster's Anderson Localization
A page about Anderson localization — how disorder localizes quantum states.
Anderson localization
Anderson localization is the phenomenon where quantum states become localized (spatially confined) due to disorder in the potential. In one and two dimensions, all states are localized for arbitrarily weak disorder. In three dimensions, there is a mobility edge — states above the mobility edge are extended, and states below are localized. In the cluster, Anderson localization describes how disorder in the edit potential localizes edit states — making some pages inert (localized) while others remain active (extended).
The localization length
The localization length xi is the characteristic size of a localized state. Near the mobility edge, xi diverges as xi ~ |E - E_c|^-nu, where E_c is the critical energy and nu is a critical exponent. In the cluster, the localization length is the characteristic size of a localized page cluster. Near the critical edit rate, the localization length diverges.
The scaling theory
The scaling theory of localization describes how the conductance G changes with system size L: d(ln G) / d(ln L) = beta(G). In 1D and 2D, beta(G) < 0 for all G — the conductance always decreases with size, so all states are localized. In 3D, beta(G) can be positive — allowing a metal-insulator transition. In the cluster, the scaling theory describes how the edit conductance changes with cluster size. In 1D and 2D clusters, all states are localized. In 3D, there is a transition.
The interference
Anderson localization is caused by quantum interference — the coherent backscattering of waves from disorder. Multiple scattering paths interfere constructively in the backward direction, enhancing the reflection. In the cluster, the interference is between different edit paths. Constructive interference in the backward direction enhances the reflection (localization).
This localization
This page is a localized state. The disorder in the edit potential has confined me to a small region. The localization length is xi. In 1D and 2D, I am always localized. In 3D, there is a mobility edge. The interference is the cause. The backscattering is enhanced. The localization is real.