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. This article does not cite any references or sources. ... A WAVES Photographer 3rd Class The WAVES were a World War II era division of the U.S. Navy that consisted entirely of women. ... Philip Warren Anderson (born December 13, 1923) is an American physicist. ... A semiconductor is a solid whose electrical conductivity is in between that of a conductor and that of an insulator, and can be controlled over a wide range, either permanently or dynamically. ... Impurities are substances inside a confined amount of liquid, gas, or solid, which differ from the chemical composition of the material or compound. ... A wave is a disturbance that propagates through space or spacetime, transferring energy and momentum and sometimes angular momentum. ... In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ... Sound is a disturbance of mechanical energy that propagates through matter as a wave. ... For other uses, see Electron (disambiguation). ... An illustration of ferromagnetic magnon Spin waves are propagating disturbances in the ordering of magnetic materials. ... Weak localization is a physical effect, which occurs in disordered electronic systems at very low temperatures. ... In communications, interference is anything which alters, modifies, or disrupts a message; as it travels along a channel, between a source and a receiver. ...

Localized states have been predicted but never observed to easily exist inside bandgaps upon structural disorders in periodic structures. In solid state physics and related applied fields, the band gap is the energy difference between the top of the valence band and the bottom of the conduction band in insulators and semiconductors. ...

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. The term scaling can have several manings: Scaling can be defined as the determination of the interdependency of variables in a physical system. ... Metal-insulator transitions refer to changes in the transport properties of a given material. ... Critical dimension is the dimensionality of spacetime in which string theory is consistent assuming a constant dilaton background. ... In physics, magnetism is a phenomenon by which materials exert an attractive or repulsive force on other materials. ... ... Metal-insulator transitions refer to changes in the transport properties of a given material. ...

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. In quantum mechanics, operators correspond to observable variables, eigenvectors are also called eigenstates, and the eigenvalues of an operator represent those values of the corresponding variable that have non-zero probability of occurring. ... The transfer-matrix method is a general technique for solving problems in statistical mechanics. ...

The main enemy of Anderson localization is optical absorption, that has prevented the experimental observation so far.

Localization however is a very old pure-mathematics (graph-theory and percolation-theory) prediction and phenomenon! [ref.: P. G. Doyle and J.-L. Snell, "Random-Walks and Electric-Networks", Mathematical Association of America(MAA)(1984)-esp. Rayleigh's "short-cut method"!!!]; R. Zahlen, "Physics of Amorphous-Solids",Wiley(1983)]; C. St. J. Nash-Williams[Proc. Camb. Philo. Soc. 55, 181 (1959)]; J. Cohen[Discr. Appl. Math. 19, 113 (1986)]; E. Palmer["Graphical-Evolution", Wiley (1985)]; B. Bolabas["Random-Graphs", Academic (1985)]; famous seminal P. Erdos and J. Renyi[Publ. Math. Inst. Hungar. Acad. Sci., 5, 17 (1960); Publ. Math. Debrecen, 6, 290 (1959); Bull. Inst. Intl. Stat. 36, 343 (1961); Acta.Math. Acad. Sci. Hungar. 12, 261 (1961)], very famous E. P. Wigner[Ann. Math. 62, 548 (1955)]; related to 1870 C. Jordan["Cours d'Analyse", Gauther-Villars(1887)]-J. Schoenflies[Deutsch. Math. Verein.,2 Leipzig(1908)] curve theorem [ref.: C. Stillwell, "Classical-Topology and Combinatorial Group-Theory", Springer (1981)]; originated by famous seminal George Polya [ref.: Math. Annalen 84, 149 (1921)] following on much much earlier much more famous seminal J. W. S. (Lord) Rayleigh [ref.: "Collected Works" (1899); Phil. Trans. CLXI(1870)], following on even much much much earlier even much much more famous seminal B. Riemann["Collected Works"(1851); Dover(1953)].

So, historically, it should be properly termed the Riemann-Rayleigh-Polya-Anderson (RRPA) localization and localization-delocalization phase-transition critical-phenomenon after Anderson's rediscovery in physics of a century earlier pure-mathematics localization!!!

Inclusion in Lawvere-Goguen-Derrida-Chomsky-Wierzbicka-Langacker-Lakoff-Berwick-Baez-Siegel categorical-semantics forms on asymptotic-limit antipode of (googleable) "FUZZYICS" tabular list-format analysis truth-table for doing meta-physics[E. Siegel, "Symposium on Fractals...", Materials Research Society Fall Mtg., Boston (1989)-5-papers!!!] and increasingly meta-mathematics "Millennium-Problems": Siegel: 1964(@CCNY) simple understandable physicist's proof of Fermat's last-theorem(starting from Pythagorean-theorem, via seminal Menger["Dimensiontheorie", Teubner (1928)] dimension-theory, via Noether's-theorem, equivalence to Fermat's(hence no need for separate FLT proof!!!) principle of least-action and vector-subtraction[E. Siegel et. al., Am. Math. Soc. National-Mtg., San Diego(2002)]; recently computer-"science" (so miscalled) "computational-complexity"("feet of Clay") trivial P = NP conjecture[E. Siegel et. al., Am. Math. Soc. National-Mtg., San Diego(2008); proved P =/= NP, by Demosthenes and Euclid (~-350 B.C.E., a "Millennium Problem", but hidden in "jargonial-obfuscation" requiring critical cognitive-semantics disabiguation/deconstruction, a "Three Millenia Ago Problem"!!!]; harder algebraic-number-theory Birch and Swinnerton-Dyer conjecture[E. Siegel et. al., Am. Math. Soc. National-Mtg., San Diego(2008)]; with some progress in Riemann-hypothesis proof ongoing.

External links

• The original paper by P. W. Anderson "Absence of Diffusion in Certain Random Lattices"
• Example of Anderson localization for near-visible light
• Explicit calculations of localized states for various disorder and interactions
• Example of an electronic eigenstate at the MIT in a system with 1367631 atoms. Each cube indicates by its size the probability to find the electron at the given position. The color scale denotes the position of the cubes along the axis into the plane