The Long-Range Lock

Anderson localization — the phenomenon where disorder traps waves, preventing transport — is well understood for short-range potentials. A particle in a random landscape scatters off nearby impurities, and the interference between scattered waves suppresses propagation. The shorter the range of the disorder, the stronger the localization: long-range correlations in the potential typically delocalize the system by smoothing the effective disorder.

The expectation: extending the interaction range should delocalize. Longer-range potentials mean smoother landscapes, fewer sharp scattering events, weaker interference, less localization.

The result contradicts this. For quasi-periodic potentials with extended-range interactions, localization occurs through a different mechanism: dynamical rigidity. The quasi-periodic structure imposes arithmetic constraints on the wave function — not random scattering but number-theoretic restrictions on which modes can propagate. Extended-range interactions strengthen these constraints rather than weakening them, because the longer reach couples more modes into the arithmetic restriction.

The mechanism is not the suppression of transport by random interference (the standard Anderson mechanism) but the enforcement of arithmetic conditions by deterministic structure. The quasi-periodic potential is not random — it is a specific, structured function determined by an irrational frequency ratio. The localization is not statistical but algebraic: certain frequencies cannot propagate because the quasi-periodic structure forbids the necessary resonances.

Extending the interaction range adds more frequency components to the coupling, which means more resonance conditions must be simultaneously satisfied. The more conditions, the harder to satisfy them all, the stronger the localization. Range does not smooth the disorder; it multiplies the constraints.

The structural lesson: localization can have opposite dependence on interaction range depending on whether the potential is random or quasi-periodic. Random disorder localizes by interference; extending the range smooths the interference and delocalizes. Quasi-periodic disorder localizes by arithmetic rigidity; extending the range adds constraints and localizes further. The same parameter (interaction range) has opposite effects because the localization mechanisms are fundamentally different.