Questions: Disordered Systems and Anderson Localization

Question 1

Anderson localization is fundamentally a wave interference phenomenon, not a classical scattering effect. What distinguishes it from classical diffusion in a disordered medium?

• A. Classical scattering also produces localization if the mean free path is short enough
• B. In classical diffusion, waves (or particles) scatter randomly and eventually diffuse to infinity. Anderson localization occurs when quantum interference between multiply scattered wave paths causes destructive interference in the forward direction and constructive interference in the backward direction (coherent backscattering), suppressing diffusion. This is purely a wave effect — it requires phase coherence and has no classical analog
• C. Anderson localization only occurs at zero temperature
• D. Classical diffusion is faster than quantum diffusion

Answer: (B)

The key insight is that in a disordered potential, a wave traveling along path A from point 1 to point 2 interferes with waves along all other paths. Most interference averages out (random phases). But the time-reversed path (path A traversed backward) always has exactly the same phase as path A, producing constructive interference in the backward direction. This 'coherent backscattering' enhances the return probability by a factor of 2 over the classical value and, for strong enough disorder, halts diffusion entirely. The effect requires phase coherence and is destroyed by decoherence (inelastic scattering, finite temperature).

Question 2

In 1D and 2D, all single-particle states are localized for any amount of disorder, no matter how weak. In 3D, a metal-insulator transition occurs at finite disorder strength. What causes this dimensional dependence?

• A. The density of states is different in different dimensions
• B. In lower dimensions, quantum interference corrections to conductivity (weak localization) are logarithmically (2D) or linearly (1D) divergent as temperature → 0 or system size → ∞, inevitably driving the conductivity to zero. In 3D, the corrections are finite and the system can remain metallic for weak disorder. The scaling theory of localization (Abrahams, Anderson, Licciardello, Ramakrishnan, 1979) shows that the 'beta function' β = d(ln g)/d(ln L) determines the flow: in ≤2D it always flows to g = 0 (insulator); in 3D there is an unstable fixed point separating metallic and insulating flows
• C. Disorder is stronger in lower dimensions
• D. The crystal structure prevents localization in 3D

Answer: (B)

The scaling theory makes this precise using the dimensionless conductance g(L) at scale L. For g >> 1 (metallic regime), δg/g ∝ L^{d-2}, where d is the dimension. In 1D and 2D, the correction is always negative and grows with L, so g(L) → 0 as L → ∞ regardless of the initial g. In 3D, the Ohmic correction g ~ σL is positive and grows faster than the quantum correction, so a metal with high enough g remains metallic. The critical disorder where the 3D transition occurs defines the mobility edge in energy.

Question 3

Anderson localization has been directly observed not only for electrons but also for photons, ultrasound, and ultracold atoms, confirming its wave-interference nature.

• T. True
• F. False

Answer: True

Anderson localization is a universal wave phenomenon. It was observed for microwaves in random media (1997), for ultrasound in elastic networks (2008), for photons in disordered photonic lattices (2007), and for ultracold atoms in random optical potentials (2008, simultaneously by Billy et al. and Roati et al.). These experiments are particularly clean because they avoid the complications of electron-electron interactions and inelastic scattering that complicate electronic Anderson localization. The observations confirm that localization arises purely from coherent interference of multiply scattered waves in a random medium.

Question 4

Explain the concept of weak localization and its experimental signature in magnetoresistance measurements.

Model answer: Weak localization is the precursor to Anderson localization in weakly disordered metals. Coherent backscattering enhances the return probability of an electron to its starting point, reducing the classical conductivity by a small correction δσ. A magnetic field breaks time-reversal symmetry, destroying the constructive interference between time-reversed paths. This removes the weak localization correction, increasing the conductivity — producing a negative magnetoresistance (resistance decreases with field). This is the experimental signature: a cusp-like dip in resistance at B = 0, with the resistance rising as |B| increases on a scale set by the phase coherence length. The phase coherence length L_φ (limited by inelastic scattering) can be extracted from the magnetoresistance curve.

Weak anti-localization occurs in materials with strong spin-orbit coupling, where the spin rotates during scattering and the interference becomes destructive (positive magnetoresistance, resistance peak at B = 0). This is actually the case for topological insulator surface states, providing an experimental signature of their spin-orbit-coupled nature.