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 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.^[1]

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 isolating 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

$i \hbar \dot{\psi}_t = H \psi_t~,$

where the Hamiltonian H is given by

$(H \phi)(j) = E_j \phi(j) + \sum_{k \neq j} V(|k-j|) \phi(k)~,$

with E_j random and independent, and interaction V(r) falling of as r^-2 at infinity. For example, one may take E_j uniformly distributed in [−W,   +W], and

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

Starting with ψ₀ localised at the origin, one is interested in how fast the probability distribution |ψ_t|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:

$\sum_{n \in \mathbb{Z}^d} |\psi_t(n)|^2 |n| \leq C$

uniformly in t. This phenomenon is called Anderson localization.

• if d ≥ 3 and W/ħ is small,

$\sum_{n \in \mathbb{Z}^d} |\psi_t(n)|^2 |n| \approx D \sqrt{t}~,$

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.^[2] 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 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 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)

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). 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). Random lasers can operate using this phenomenon.

Notes

1. ^ Anderson, P. W. (1958). "Absence of Diffusion in Certain Random Lattices". Phys. Rev. 109 (5): 1492–1505. Bibcode 1958PhRv..109.1492A. doi:10.1103/PhysRev.109.1492. 
2. ^ Abrahams, E.; Anderson, P.W.; Licciardello, D.C.; Ramakrishnan, T.V. (1979). "Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions". Phys. Rev. Lett. 42 (10): 673–676. Bibcode 1979PhRvL..42..673A. doi:10.1103/PhysRevLett.42.673. http://link.aps.org/doi/10.1103/PhysRevLett.42.673. 

Further reading

• Brandes, T. & Kettemann, S. (2003). The Anderson Transition and its Ramifications --- Localisation, Quantum Interference, and Interactions. Berlin: Springer Verlag 

• 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. 

• 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. 

• Scheffold, Frank; et al. (1999). "Localization or classical diffusion of light?". Nature 398 (6724): 206–207. Bibcode 1999Natur.398..206S. doi:10.1038/18347. 

• 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. 

• Lahini, Y.; et al. (2006). "Direct Observation of Anderson Localized Modes and the Effect of Nonlinearity". Photonic Metamaterials: From Random to Periodic (META), Grand Bahama Island, The Bahamas, June 5, 2006, Postdeadline Papers. http://www.opticsinfobase.org/abstract.cfm?URI=META-2006-ThC4. 

• Billy, Juliette; et al. (2008). "Direct observation of Anderson localization of matter waves in a controlled disorder". Nature 453 (7197): 891–894. Bibcode 2008Natur.453..891B. doi:10.1038/nature07000. PMID 18548065. 

• Roati, Giacomo; et al. (2008). "Anderson localization of a non-interacting Bose-Einstein condensate". Nature 453 (7197): 895–898. Bibcode 2008Natur.453..895R. doi:10.1038/nature07071. PMID 18548066. 

• 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. 

• Conti, C; A. Fratalocchi (2008). "Dynamic light diffusion, three-dimensional Anderson localization and lasing in inverted opals". Nature Physics 4 (10): 794–798. Bibcode 2008NatPh...4..794C. doi:10.1038/nphys1035. 

• Hu, Hefei; et al. (2008). "Localization of ultrasound in a three-dimensional elastic network". Nature Physics 4 (12): 945. Bibcode 2008NatPh...4..945H. doi:10.1038/nphys1101. 

• Chabé, J.; et al. (2008). "Experimental Observation of the Anderson Metal-Insulator Transition with Atomic Matter Waves". Phys. Rev. Lett. 101 (25): 255702. Bibcode 2008PhRvL.101y5702C. doi:10.1103/PhysRevLett.101.255702. PMID 19113725. 

External links

• Fifty years of Anderson localization Physics Today, August 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
• Anderson localization of elastic waves
• Popular scientific article on the first experimental observation of Anderson localization in matter waves