Anderson localization

Anderson localization originates from the wave nature of particles and the interference of multiple scattering paths in a disordered potential. In his 1958 paper, Anderson considered a tight-binding model with random on-site energies and showed that for sufficiently strong disorder, the eigenstates become exponentially localized, leading to the absence of diffusion and electrical conduction.¹ The key parameter is the ratio of the disorder strength to the hopping amplitude; when this ratio exceeds a critical value, the wavefunction decays as exp(-|r-r0|/ξ), where ξ is the localization length.

The phenomenon is intimately connected to the scaling theory of localization, which predicts that in one and two dimensions all states are localized for any infinitesimal disorder, whereas in three dimensions there exists a mobility edge separating localized and extended states.² This scaling behavior was established by Abrahams, Anderson, Licciardello, and Ramakrishnan in 1979 using renormalization group arguments.

Although originally proposed for electrons, Anderson localization has been observed in many classical wave systems. In optics, localization of light was demonstrated in disordered photonic lattices and in suspensions of dielectric spheres.³ In acoustics, ultrasonic waves in disordered aluminum beads show localized modes. Ultracold atoms in optical potentials have provided a highly controllable platform to study localization, including the observation of the exponential localization of matter waves in a quasi-periodic lattice.⁴

For electrons, direct observation is complicated by interactions and decoherence, but experiments on disordered nanowires and thin films have shown signatures of localization, such as the exponential increase of resistance with length. The transition from classical diffusion to localization has been studied in detail in microwave and ultrasound experiments, confirming the predicted critical behavior.

Anderson localization is not limited to static disorder; it also occurs in time-dependent or dynamic disorder, leading to phenomena such as dynamical localization in kicked systems. In addition, the concept has been extended to many-body systems, where interactions can either destroy or enhance localization, giving rise to many-body localization (MBL).⁵ MBL is a topic of intense research because it challenges the conventional understanding of thermalization in isolated quantum systems.

Another niche aspect is the role of topology: in certain disordered systems, topological invariants can protect delocalized states even in the presence of strong disorder, leading to the quantum spin Hall effect and other topological phases. Furthermore, Anderson localization has been proposed as a mechanism for the absence of diffusion in certain biological systems, such as energy transport in photosynthetic complexes, although this remains debated.

Anderson localization has practical implications for the design of materials and devices. In electronics, it sets a fundamental limit on the miniaturization of transistors, as disorder in nanoscale devices can lead to localization and increased resistance. In photonics, localization can be used to confine light in disordered media, enabling random lasers and enhanced light-matter interactions.⁶ In acoustics, it can be exploited for vibration damping and sound insulation.

The concept also influences other fields, such as seismology, where localization of seismic waves in heterogeneous media can affect the propagation of earthquake waves. Moreover, the mathematical framework of Anderson localization has been applied to the study of quantum chaos and the metal-insulator transition, providing a deep connection between disorder and quantum coherence.

Mobility edge

The energy threshold separating localized and extended states in a disordered system.

Localization length

The characteristic length over which the wavefunction decays exponentially in a localized state.

Many-body localization

A phenomenon in which an isolated interacting quantum system fails to thermalize due to disorder.