Disordered Systems and Anderson Localization

Core Idea

Anderson localization is the absence of diffusion of waves in a disordered medium. In a crystal with random potential disorder, sufficiently strong disorder causes all electronic wavefunctions to become exponentially localized: |psi(r)| ~ exp(-|r - r_0|/xi_loc), where xi_loc is the localization length. In 1D and 2D, all states are localized for any amount of disorder. In 3D, a mobility edge separates localized states (in the band tails) from extended states (in the band center), and the metal-insulator transition (Anderson transition) occurs when the Fermi level crosses the mobility edge. Anderson localization is a wave interference phenomenon — it applies to light, sound, and matter waves, not just electrons.

Explainer

Bloch's theorem tells us that electrons in a perfect crystal propagate freely as Bloch waves. But real materials always contain disorder: impurities, vacancies, grain boundaries, lattice distortions. Philip Anderson showed in 1958 that sufficiently strong disorder causes a qualitative change in the nature of electronic states: they become exponentially localized in space, with wavefunctions decaying as |psi| ~ exp(-|r|/xi_loc). A localized electron cannot propagate to infinity and does not contribute to DC transport. If all states at the Fermi level are localized, the material is an Anderson insulator.

The mechanism is quantum interference among multiply scattered wave paths. In a disordered potential, an electron follows many scattering paths from point A to point B, and their amplitudes add coherently. For most paths, the phases are random and average out. But there is a special class of paths: for every path from A back to A, the time-reversed path has exactly the same phase (by time-reversal symmetry). This coherent backscattering doubles the return probability compared to the classical expectation, suppressing diffusion. When this effect is strong enough — in strongly disordered systems or low dimensions — diffusion halts entirely and all states become localized.

The scaling theory of localization (1979) provides the dimensional classification. The key quantity is the dimensionless conductance g(L) of a sample of size L. In 1D and 2D, quantum corrections always drive g(L) to zero as L increases, meaning all states are localized for any disorder strength. In 3D, a metallic regime (g increasing with L) can persist for weak disorder, and the Anderson metal-insulator transition occurs at a critical disorder strength. At the transition, the localization length diverges: xi_loc ~ |W - W_c|^{-nu}, with a universal critical exponent nu ~ 1.57 in 3D.

The practical manifestation of weak disorder in metals is weak localization — a small quantum correction to the classical (Drude) conductivity. Weak localization reduces the conductivity at zero magnetic field but is destroyed by an applied field (which breaks time-reversal symmetry and removes coherent backscattering). The resulting negative magnetoresistance — resistance dipping at B = 0 — is one of the most commonly measured quantum transport signatures in mesoscopic physics. In materials with strong spin-orbit coupling, the sign flips (weak anti-localization, positive magnetoresistance), providing a direct probe of spin-orbit physics. Anderson localization extends far beyond electrons: it has been observed for light, sound, and cold atoms, confirming its universal wave-interference origin and establishing it as one of the most fundamental phenomena in wave physics.