Anderson localization

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

${\displaystyle i\hbar {\frac {d\psi }{dt}}=H\psi ~,}$

where the Hamiltonian H is given by[2]

${\displaystyle H\psi _{j}=E_{j}\psi _{j}+\sum _{k\neq j}V_{jk}\psi _{k}~,}$

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 [−W,   +W], and

${\displaystyle V(|r|)={\begin{cases}1,&|r|=|j-k|=1\\0,&{\text{otherwise.}}\end{cases}}}$

Starting with ψ₀ localised at the origin, one is interested in how fast the probability distribution ${\displaystyle |\psi |^{2}}$ 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:

${\displaystyle \sum _{n\in \mathbb {Z} ^{d}}|\psi (t,n)|^{2}|n|\leq C}$

uniformly in t. This phenomenon is called Anderson localization.

• if d ≥ 3 and W/ħ is small,

${\displaystyle \sum _{n\in \mathbb {Z} ^{d}}|\psi (t,n)|^{2}|n|\approx D{\sqrt {t}}~,}$

where D is the diffusion constant.

Example of a multifractal electronic eigenstate at the Anderson localization transition in a system with 1367631 atoms.

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.[3] 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 or only an apparent MIT.[4] 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.

Anderson localization can 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).

In 3D, observations are more rare. 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.

The existence of Anderson localization for light in 3D was debated for years (Skipetrov et al., 2016) and remains unresolved today. Reports of Anderson localization of light in 3D random media were complicated by the competing/masking effects of absorption (Wiersma et al., 1997; Storzer et al., 2006; Scheffold et al., 1999; see Further Reading) and/or fluorescence (Sperling et al., 2016). Recent experiments (Naraghi et al., 2016; Cobus et al., 2023) support theoretical predictions that the vector nature of light prohibits the transition to Anderson localization (John, 1992; Skipetrov et al., 2019).

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.[5][6]

• Aubry–André model

• Brandes, T. & Kettemann, S. (2003). The Anderson Transition and its Ramifications --- Localisation, Quantum Interference, and Interactions. Lecture Notes in Physics. Berlin: Springer Verlag. ISBN 978-3-642-07398-4.

• Wiersma, Diederik S.; et al. (1997). "Localization of light in a disordered medium". Nature. 390 (6661): 671–673. Bibcode:1997Natur.390..671W. doi:10.1038/37757. S2CID 46723942.
• Störzer, Martin; et al. (2006). "Observation of the critical regime near Anderson localization of light". Phys. Rev. Lett. 96 (6): 063904. arXiv:cond-mat/0511284. Bibcode:2006PhRvL..96f3904S. doi:10.1103/PhysRevLett.96.063904. PMID 16605998. S2CID 12180478.
• Scheffold, Frank; et al. (1999). "Localization or classical diffusion of light?". Nature. 398 (6724): 206–207. Bibcode:1999Natur.398..206S. doi:10.1038/18347. S2CID 4347650.
• Schwartz, T.; et al. (2007). "Transport and Anderson Localization in disordered two-dimensional Photonic Lattices". Nature. 446 (7131): 52–55. Bibcode:2007Natur.446...52S. doi:10.1038/nature05623. PMID 17330037. S2CID 4429992.
• Lahini, Y.; et al. (2008). "Anderson localization and nonlinearity in one dimensional disordered photonic lattices". Physical Review Letters. 100 (1): 013906. arXiv:0704.3788. Bibcode:2008PhRvL.100a3906L. doi:10.1103/PhysRevLett.100.013906. PMID 18232768. S2CID 6376064.
• Karbasi, S.; et al. (2012). "Observation of transverse Anderson localization in an optical fiber". Optics Letters. 37 (12): 2304–6. Bibcode:2012OptL...37.2304K. doi:10.1364/OL.37.002304. PMID 22739889.
• Karbasi, S.; et al. (2014). "Image transport through a disordered optical fibre mediated by transverse Anderson localization". Nature Communications. 5: 3362. arXiv:1307.4160. Bibcode:2014NatCo...5.3362K. doi:10.1038/ncomms4362. PMID 24566557. S2CID 205323503.
• Billy, Juliette; et al. (2008). "Direct observation of Anderson localization of matter waves in a controlled disorder". Nature. 453 (7197): 891–894. arXiv:0804.1621. Bibcode:2008Natur.453..891B. doi:10.1038/nature07000. PMID 18548065. S2CID 4427739.
• Roati, Giacomo; et al. (2008). "Anderson localization of a non-interacting Bose-Einstein condensate". Nature. 453 (7197): 895–898. arXiv:0804.2609. Bibcode:2008Natur.453..895R. doi:10.1038/nature07071. PMID 18548066. S2CID 4388940.
• Ludlam, J. J.; et al. (2005). "Universal features of localized eigenstates in disordered systems". Journal of Physics: Condensed Matter. 17 (30): L321–L327. Bibcode:2005JPCM...17L.321L. doi:10.1088/0953-8984/17/30/L01. S2CID 17243205.
• Conti, C; A. Fratalocchi (2008). "Dynamic light diffusion, three-dimensional Anderson localization and lasing in inverted opals". Nature Physics. 4 (10): 794–798. arXiv:0802.3775. Bibcode:2008NatPh...4..794C. doi:10.1038/nphys1035. S2CID 119115156.
• Hu, Hefei; et al. (2008). "Localization of ultrasound in a three-dimensional elastic network". Nature Physics. 4 (12): 945–948. arXiv:0805.1502. Bibcode:2008NatPh...4..945H. doi:10.1038/nphys1101. S2CID 119097566.
• Chabé, J.; et al. (2008). "Experimental Observation of the Anderson Metal-Insulator Transition with Atomic Matter Waves". Phys. Rev. Lett. 101 (25): 255702. arXiv:0709.4320. Bibcode:2008PhRvL.101y5702C. doi:10.1103/PhysRevLett.101.255702. PMID 19113725. S2CID 773761.
• Ying, Tianping; et al. (2016). "Anderson localization of electrons in single crystals: Li_xFe₇Se₈". Science Advances. 2 (2): e1501283. Bibcode:2016SciA....2E1283Y. doi:10.1126/sciadv.1501283. PMC 4788481. PMID 26989781.
• Choi, Seung Ho; et al. (2018). "Anderson light localization in biological nanostructures of native silk". Nature Communications. 9 (1): 452. Bibcode:2018NatCo...9..452C. doi:10.1038/s41467-017-02500-5. PMC 5792459. PMID 29386508.
• Skipetrov, Sergey; et al. (2016). "Red light for Anderson localization". New Journal of Physics. 18 (2): 021001. arXiv:1601.07848. Bibcode:2016NJPh...18b1001S. doi:10.1088/1367-2630/18/2/021001. S2CID 118497908.

• 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 MIT
• Anderson localization of elastic waves
• Popular scientific article on the first experimental observation of Anderson localization in matter waves