Anderson Localization

In condensed matter physics, Anderson localization (also known as strong localization) is the absence of diffusion of waves in a ''disordered'' medium. This phenomenon is named after the American physicist P. W. Anderson, who was the first to suggest that electron localization is possible in a lattice potential, provided that the degree of randomness (disorder) in the lattice is sufficiently large, as can be realized for example in a semiconductor with impurities or defects.

Anderson localization is a general wave phenomenon that applies to the transport of electromagnetic waves, acoustic waves, quantum waves, spin waves, etc. This phenomenon is to be distinguished from weak localization, which is the precursor effect of Anderson localization (see below), and from Mott localization, named after Sir Nevill Mott, where the transition from metallic to insulating behaviour is ''not'' due to disorder, but to a strong mutual Coulomb repulsion of electrons.

Introduction

In the original Anderson tight-binding model, the evolution of the wave function ''ψ'' on the ''d''-dimensional lattice Z^''d'' is given by the Schrödinger equation

:$i \hbar \frac = H \psi~,$

where the Hamiltonian ''H'' is given by

:$H \psi_j = E_j \psi_j + \sum_ V_ \psi_k~,$

with ''E''_''j'' random and independent, and potential ''V''(''r'') falling off faster than ''r''⁻³ at infinity. For example, one may take ''E''_''j'' uniformly distributed in minus;''W'',   +''W'' and

:$V(, r, ) = \begin 1, & , r, =, j-k, = 1 \\ 0, &\text \end$

Starting with ''ψ''₀ localised at the origin, one is interested in how fast the probability distribution $, \psi, ^2$ diffuses. Anderson's analysis shows the following:

* if ''d'' is 1 or 2 and ''W'' is arbitrary, or if ''d'' ≥ 3 and ''W''/ħ is sufficiently large, then the probability distribution remains localized:

::$\sum_ , \psi(t,n), ^2 , n, \leq C$

:uniformly in ''t''. This phenomenon is called Anderson localization.

* if ''d'' ≥ 3 and ''W''/ħ is small,

:$\sum_ , \psi(t,n), ^2 , n, \approx D \sqrt~,$

:where ''D'' is the diffusion constant.

Analysis

The phenomenon of Anderson localization, particularly that of weak localization, finds its origin in the wave interference between multiple-scattering paths. In the strong scattering limit, the severe interferences can completely halt the waves inside the disordered medium.

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 disorder-induced metal-insulator transition (MIT) exists for non-interacting electrons in three dimensions (3D) at zero magnetic field and in the absence of spin-orbit coupling. Much further work has subsequently supported these scaling arguments both analytically and numerically (Brandes ''et al.'', 2003; see Further Reading). 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 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 either decreasing system size for fixed disorder or increasing 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. Direct numerical solution of Maxwell equations to demonstrate Anderson localization of light has been implemented (Conti and Fratalocchi, 2008).

Recent work has shown that a non-interacting Anderson localized system can become many-body localized even in the presence of weak interactions. This result has been rigorously proven in 1D, while perturbative arguments exist even for two and three dimensions.

Experimental evidence

Two reports of Anderson localization of light in 3D random media exist up to date (Wiersma ''et al.'', 1997 and Storzer ''et al.'', 2006; see Further Reading), even though absorption complicates interpretation of experimental results (Scheffold ''et al.'', 1999). Anderson localization can also be observed in a perturbed periodic potential where the transverse localization of light is caused by random fluctuations on a photonic lattice. Experimental realizations of transverse localization were reported for a 2D lattice (Schwartz ''et al.'', 2007) and a 1D lattice (Lahini ''et al.'', 2006). Transverse Anderson localization of light has also been demonstrated in an optical fiber medium (Karbasi ''et al.'', 2012) and a biological medium (Choi ''et al.'', 2018), and has also been used to transport images through the fiber (Karbasi ''et al.'', 2014). It has also been observed by localization of a Bose–Einstein condensate in a 1D disordered optical potential (Billy ''et al.'', 2008; Roati ''et al.'', 2008). Anderson localization of elastic waves in a 3D disordered medium has been reported (Hu ''et al.'', 2008). The observation of the MIT has been reported in a 3D model with atomic matter waves (Chabé ''et al.'', 2008). The MIT, associated with the nonpropagative electron waves has been reported in a cm sized crystal (Ying ''et al.'', 2016). Random lasers can operate using this phenomenon.

Comparison with diffusion

Standard diffusion has no localization property, being in disagreement with quantum predictions. However, it turns out that it is based on approximation of the principle of maximum entropy, which says that the probability distribution which best represents the current state of knowledge is the one with largest entropy. This approximation is repaired in maximal entropy random walk, also repairing the disagreement: it turns out to lead to exactly the quantum ground state stationary probability distribution with its strong localization properties.Z. Burda, J. Duda, J. M. Luck, and B. Waclaw, ''Localization of the Maximal Entropy Random Walk''
Phys. Rev. Lett., 2009.J. Duda, ''Extended Maximal Entropy Random Walk''
PhD Thesis, 2012.

See also

*Aubry–André model

Notes

Further reading

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*{{ cite journal , last = Choi , first = Seung Ho , year = 2018 , title = Anderson light localization in biological nanostructures of native silk , journal = Nature Communications , volume = 9 , issue = 1, pages = 452 , doi = 10.1038/s41467-017-02500-5 , pmid = 29386508, pmc = 5792459 , bibcode = 2018NatCo...9..452C , display-authors=etal

External links

Fifty years of Anderson localization
Ad Lagendijk, Bart van Tiggelen, and Diederik S. Wiersma, Physics Today 62(8), 24 (2009).
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
Videos of multifractal electronic eigenstates at the MITAnderson localization of elastic wavesPopular scientific article on the first experimental observation of Anderson localization in matter waves

Mesoscopic physics
Condensed matter physics