Localize the bulk. The edge still pumps.
Researchers built a photonic Hofstadter butterfly from a one-dimensional multilayer photonic crystal — a Bragg stack — by encoding a synthetic dimension in the layer thickness modulation. They observed the butterfly's fractal energy spectrum and its chiral edge states from a family of one-dimensional structures, consistent with the Thouless pump picture: adiabatic parameter cycling pumps photons across the structure.
Then they increased the quasiperiodic modulation. At strong modulation, the Aubry-André localization transition kicks in. Specific Chern bands become fully localized — the bulk modes are trapped, unable to propagate. But the chiral edge states persist. They continue to wind across the gap, carrying photons from one side to the other while the bulk is frozen.
The mechanism changes. Under weak modulation, transport proceeds by adiabatic Thouless pumping — smooth, continuous, Berry-phase-driven. Under strong modulation, with the bulk localized, edge transport occurs through a sequence of Landau-Zener transitions between localized states. The photon hops from one localized mode to the next, tunneling through avoided crossings rather than riding a smooth band.
Same topology. Same Chern number. Same net transport. Different mechanism. The topological protection guarantees the outcome — one photon pumped per cycle — but the physical process achieving it shifts from adiabatic to diabatic. The topology doesn't care how the pumping happens, only that it does.
The crossover is clean: tunable by a single parameter (modulation strength), observable in a compact photonic stack, and confirmed by both simulation and experiment. The platform demonstrates that topological protection survives the localization transition — not by preventing localization, but by routing around it through a mechanistic regime change.