Anderson Localization

The intuition — a wave that can't escape its own echoes

Picture an electron dropped into a perfect crystal. The atoms sit in a flawless periodic lattice, and Bloch's theorem guarantees the electron travels through it like a free particle dressed by the lattice — it spreads out and carries current forever. Now sprinkle in randomness: vacancies, impurities, dopants, a frozen-in jitter of the potential well depths. Each defect scatters the electron, sending little wavelets off in all directions.

Common sense says scattering just makes the electron diffuse more slowly, like a drunkard's random walk, and indeed at weak disorder that's exactly right. But the electron is a wave, and waves interfere. Among the countless scattered paths there are loops — paths that return to the starting point. For every such loop traversed clockwise, there is an identical loop traversed counter-clockwise (its time-reverse), and the two arrive back at the origin exactly in phase. Their amplitudes add coherently, so the probability of coming back is enhanced — more than the classical random walk predicts.

Crank the disorder up far enough and this return enhancement runs away. The wave's amplitude piles up around its origin and cancels everywhere else. The eigenstates stop being spread-out plane-wave-like things and become little exponentially-decaying bumps pinned to particular regions of the sample. Diffusion doesn't just slow down — it stops. The material is an insulator, built entirely out of interference, with no chemistry, no gap, and no inelastic loss. That is Anderson localization.

The Anderson model and the governing equation

P. W. Anderson's 1958 paper "Absence of Diffusion in Certain Random Lattices" stripped the problem to a tight-binding lattice. An electron lives on sites i of a grid, can hop to neighbors with amplitude t, and feels a site energy ε_i drawn at random:

H = Σ_i ε_i |i⟩⟨i|  −  t Σ_⟨i,j⟩ ( |i⟩⟨j| + |j⟩⟨i| )

The on-site energies ε_i are uniformly distributed in a box of width W: ε_i ∈ [−W/2, +W/2]. The single dimensionless control knob is the disorder ratio:

w = W / t      (disorder strength / hopping)

When w is small, hopping wins and the states are extended. When w is large, the random wells win, neighboring sites fall out of resonance, and the electron gets stuck. A localized eigenstate has the hallmark exponential envelope:

|ψ(r)|  ~  exp( −|r − r₀| / ξ )

where ξ is the localization length — the size of the trap. As disorder is reduced toward the transition, ξ diverges as ξ ∝ (w − w_c)^{−ν} with a critical exponent ν ≈ 1.57 for the 3D orthogonal universality class.

The most useful sanity check is the Ioffe–Regel criterion. Localization sets in when the electron can no longer complete even one wavelength between scattering events — when the mean free path ℓ shrinks to roughly the inverse Fermi wavevector:

k_F · ℓ  ≲  1        (Ioffe–Regel limit: scattering every wavelength)

Above this (k_F ℓ ≫ 1) you have a good metal where Drude theory works; at k_F ℓ ~ 1 the wave picture of free propagation collapses and you are in the localized regime.

Scaling theory — why dimension is destiny

The deepest result came in 1979 from Abrahams, Anderson, Licciardello, and Ramakrishnan — the "gang of four." They asked how the dimensionless conductance g of a sample of size L changes as you glue blocks together to make it bigger. Everything is captured by a single scaling function:

β(g) = d(ln g) / d(ln L)

If β > 0, conductance grows with size — a metal. If β < 0, conductance shrinks toward zero — an insulator. The limiting behaviors are fixed: at large g (good metal, Ohm's law) β → d − 2; at small g (deep insulator) β → ln g → −∞. Connecting these smoothly gives the verdict:

Dimension	β(g) at large g	Sign of β	Conclusion
1D	−1	Always negative	All states localized — any disorder insulates
2D (marginal)	0⁻ (logarithmic)	Always negative	All states localized; ξ can be astronomically large
3D	+1	Changes sign	True transition at a critical g_c — a genuine metal-insulator transition

The headline: in one and two dimensions there is no true metal at zero temperature — enough sample and enough patience and any wire or film localizes. Only in three dimensions does β cross zero, giving an unstable fixed point at g_c that is the mobility edge. This is why a clean copper film is conducting in practice (its 2D localization length is larger than the sample and the temperature dephases the loops long before they close), but it is not a metal in the strict, idealized sense.

Regimes — from weak localization to a hard insulator

Regime	Condition	Behavior	Signature
Good metal (Drude)	k_F ℓ ≫ 1	Diffusive, Ohm's law holds	Constant σ; ρ rises with T (phonons)
Weak localization	k_F ℓ > 1, large but finite L_φ	Small negative quantum correction to σ	Resistance dips in a magnetic field (negative magnetoresistance)
Critical / mobility edge	E_F = E_c, ξ → ∞	Multifractal states; σ → 0 as a power law	Scale-invariant, fractal wave functions
Localized insulator	k_F ℓ ≲ 1, ξ finite	No diffusion; states are exp. traps	σ(T→0) = 0; variable-range hopping at finite T

Notice that disorder doesn't flip a switch between metal and insulator. It walks you continuously through these regimes. Weak localization is the gentlest tax: even a clean metal pays a small interference penalty, measurable as a few-percent resistance change when you turn on a magnetic field that breaks the time-reversed-loop pairing.

Real numbers — where the transition actually lands

System	Quantity	Value at / near transition
3D Anderson model (box disorder, band center)	Critical disorder w_c = W_c/t	≈ 16.5
3D Anderson transition	Localization-length exponent ν	≈ 1.57
Doped Si:P (silicon doped with phosphorus)	Critical donor density n_c	≈ 3.7 × 10¹⁸ cm⁻³
Mott / Ioffe–Regel rule of thumb	n_c¹ᐟ³ · a_B	≈ 0.26 (a_B = effective Bohr radius)
Light in GaAs powder (Wiersma 1997)	k ℓ* at strongest scattering	~1 (near Ioffe–Regel limit)
Ultrasound in Al-bead network (Hu 2008)	First clean 3D classical-wave localization	Localized band ~2.4 MHz
Ultracold ⁸⁷Rb in laser speckle (2008)	Localization length ξ	~150–300 µm (directly imaged)

The phosphorus-doped silicon number is the classic experimental anchor: below about 3.7 × 10¹⁸ donors per cm³ the electrons are bound to localized impurity states and Si:P is an insulator at T = 0; cross that density and it abruptly conducts. The dimensionless combination n_c^{1/3} a_B ≈ 0.26 is the famous Mott criterion — localization gives way to metallicity when the wave functions of neighboring donors finally overlap.

Where it shows up

• Doped semiconductors. The metal-insulator transition in Si:P, Si:B and amorphous alloys is governed by the interplay of disorder (Anderson) and electron–electron interaction (Mott). It sets the doping you need before a semiconductor conducts.
• Light in random media. The bright coherent-backscattering cone reflected straight back from paint, paper and powdered semiconductors is weak localization of photons. Strong localization of light remains the holy grail of disordered photonics.
• Acoustics and seismology. Ultrasound localizes in random elastic networks; seismic coda waves and the long ringing of multiply-scattered sound share the same diffusion-then-trapping physics.
• Cold-atom quantum simulation. Releasing a Bose–Einstein condensate into a laser speckle potential lets you photograph an exponentially localized matter wave directly — disorder you can dial with a knob.
• Quantum Hall plateaus. The flat, quantized conductance plateaus of the quantum Hall effect exist precisely because disorder localizes most states, leaving only a thin sliver of extended states to carry current.
• Many-body localization. Adding interactions to a localized system can keep it from ever thermalizing — a quantum system that fails to reach equilibrium, of intense current interest for protecting quantum information.
• Random lasers. Multiple scattering plus gain in a disordered powder can produce lasing without a conventional cavity — the scattering does the feedback that mirrors usually provide.

Anderson insulator vs Mott insulator vs band insulator

Three different ways to forbid conduction, often confused:

Feature	Band insulator	Anderson insulator	Mott insulator
Cause	Periodic lattice symmetry	Disorder / randomness	Electron–electron repulsion
States at E_F?	None (gap)	Plenty, but localized	Split by interaction (correlation gap)
Needs disorder?	No	Yes (the whole point)	No (clean crystals do it)
Needs a gap?	Yes	No	Yes (interaction-driven)
Single particle picture?	Yes	Yes	No — fundamentally many-body
Knob to drive transition	Chemistry / pressure	Disorder strength W/t	U/t (repulsion / bandwidth)
Example	Diamond, NaCl	Heavily disordered Si:P near n_c, dirty metal films	NiO, undoped cuprates, V₂O₃

Real materials are messy mixtures: a doped Mott insulator near its transition is simultaneously disordered, so untangling Anderson localization from Mott physics — the "Anderson–Mott transition" — is one of the hard open problems of condensed matter.

Common misconceptions and edge cases

• "Disorder always increases resistance gradually." At weak disorder, yes. But localization is a phase transition, not a slope — in 3D there is a sharp threshold where conductivity drops to exactly zero at T = 0.
• "It's just classical trapping in deep wells." No. The electron has more than enough energy to escape any single well classically. Localization is purely a wave-interference effect; turn off the phase coherence and the trapping vanishes.
• "There's a gap, so no states." The opposite of a band insulator. The density of states stays finite at the Fermi energy; the states are simply immobile.
• "A real copper film is a true 2D metal." Strictly, scaling theory says all 2D states localize. Films conduct because their localization length exceeds the sample size and finite-temperature dephasing kills the interference long before loops close.
• "It needs electrons." Any coherent wave localizes — light, sound, microwaves, matter waves. The cleanest demonstrations use classical waves precisely because there are no pesky electron interactions.
• "Temperature doesn't matter — it's a ground-state property." Localization is the T = 0 statement. At finite T, inelastic phonon scattering dephases the loops over a length L_φ; once L_φ < ξ, conduction resumes via variable-range hopping, σ ∝ exp[−(T₀/T)^{1/4}] in 3D (Mott's law).