Breakdown of Diffusion on the Edge

We show that dirty Quantum Hall systems exhibit large hydrodynamic fluctuations at their edge that lead to anomalously damped charge excitations in the Kardar-Parisi-Zhang universality class $\omega \simeq ck - i \mathcal D k^{3/2} $. The dissipative optical conductivity of the edge is singular at low frequencies $\sigma(\omega) \sim 1/\omega^{1/3}$. These results are direct consequences of the charge continuity relation, the chiral anomaly, and thermalization on the edge -- in particular translation invariance is not assumed. Diffusion of heat similarly breaks down, with a universality class that depends on whether the bulk thermal Hall conductivity vanishes. We further establish the theory of fluctuating hydrodynamics for surface chiral metals, where charge fluctuations give logarithmic corrections to transport.