Anderson localization

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Anderson localization, also known as strong localization, refers to the absence of diffusion of waves in a random medium. This phenomenon is named after the American physicist P. W. Anderson, who is 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 wave, acoustic wave, quantum wave and spin wave, etc. This phenomenon is to be distinguished from weak localization, which is the precursor effect of Anderson localization. This phenomenon finds its origin in the wave interference between multiple-scattering paths. In strong scattering limit, the severe interferences can completely halt the waves inside the random medium.

Localized states can easily be observed inside bandgaps upon structural disorders in periodic structures.

For non-interacting electrons, a highly successful approach was put forward in 1979 by Abrahams et al. This scaling hypothesis of localization suggests that a metal-insulator transition (MIT) exists for non-interacting electrons in three dimensions (3D) at zero magnetic field B and in the absence of spin-orbit coupling. Much further work has subsequently supported these scaling arguments both analytically and numerically. In 1D and 2D, the same hypothesis shows that there are no extended states and thus no MIT. However, since 2 is the lower critical dimension of the localization problem, the 2D case is in a sense close to 3D: states are only marginally localized for weak disorder and a small magnetic field or spin-orbit coupling can lead to the existence of extended states and thus an MIT. Consequently, the localization lengths of a 2D system with potential disorder can be quite large so that in numerical approaches one can always find a localization-delocalization transition when decreasing either system size for fixed disorder or disorder for fixed system size.

Most numerical approaches to the localization problem use the standard tight-binding Anderson Hamiltonian with onsite potential disorder. Characteristics of the electronic eigenstates are then investigated by studies of participation numbers obtained by exact diagonalization, multifractal properties, level statistics and many others. Especially fruitful is the transfer-matrix method (TMM) which allows a direct computation of the localization lengths and further validates the scaling hypothesis by a numerical proof of the existence of a one-parameter scaling function.