nLab Anderson localization

Idea

In solid state physics, Anderson localization is the phenomenon that quantum particles may localize in disordered media, meaning that instead of propagating freely they get stuck at certain positions. For electrons in a crystal this means that as the density of impurities becomes large, the crystal may turn from a conductor (where electrons move freely) to an insulator, then called an Anderson insulator.

Anderson localization is a quantum mechanical phenomenon in stark contrast to the situation in classical physics: For the motion of classical bouncing Billiard balls, the presence of “bumpers” (impurities) in their path will make their motion diffuse instead of localize. But for quanta described by wave mechanics the impurities tend to lead to destructive quantum interference of their wavefunction with itself, thus resulting in the localization phenomenon.

Related concepts

• Anderson localization has impeded the identification of Majorana zero modes in nanowires; see there.

References

The original article:

• Philip W. Anderson: Absence of Diffusion in Certain Random Lattices, Phys. Rev. 109 (1958) 1492 [doi:10.1103/PhysRev.109.1492]

and its modern develoment:

• E. Abrahams, Philip W. Anderson, D. C. Licciardello, Tiruppatur V. Ramakrishnan: Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions, Phys. Rev. Lett. 42 (1979) 673 [doi:10.1103/PhysRevLett.42.673]

Early experimental observation:

• Melvin Cutler, N. F. Mott: Observation of Anderson Localization in an Electron Gas, Phys. Rev. 181 (1969) 1336 [doi:10.1103/PhysRev.181.1336]

Review:

• Patrick A. Lee and Tiruppatur V. Ramakrishnan: Disordered electronic systems, Rev. Mod. Phys. 57 (1985) 287 [doi:10.1103/RevModPhys.57.287]

• B. Kramer, A. MacKinnon: Localization: theory and experiment, Rep. Prog. Phys. 56 (1993) 1469 [doi:10.1088/0034-4885/56/12/001]

• Ad Lagendijk, Bart van Tiggelen, Diederik S. Wiersma: Fifty years of Anderson localization, Physics Today (Aug 2009) [doi:10.1063/1.3206091]

• Eduardo C. Marino: Metals, Fermi Liquids, Mott and Anderson Insulators, chapter 14 in: Quantum Field Theory Approach to Condensed Matter Physics, Cambridge University Press (2017) [doi:10.1017/9781139696548]

See also:

• Wikipedia: Anderson localization

Mathematical discussion:

• Jürg Fröhlich, Thomas Spencer: A rigorous approach to Anderson localization, Physics Reports 103 1–4 (1984) 9-25 [doi:10.1016/0370-1573(84)90061-9]

• Günter Stolz: An Introduction to the Mathematics of Anderson Localization, Contemp. Math. 552 (2011) 71-108 [arXiv:1104.2317, doi:10.1090/conm/552]

Further discussion:

• Tobias Brandes, S. Kettemann (eds.): Anderson Localization and Its Ramifications – Disorder, Phase Coherence, and Electron Correlations, Lecture Notes in Physics 630, Springer (2003) [doi:10.1007/b13139]

• Marcel Filoche, Svitlana Mayboroda: Universal mechanism for Anderson and weak localization, PNAS 109 37 (2012) 14761-14766 [doi:10.1073/pnas.1120432109]