AC Fingerprints of 2D Electron Hydrodynamics: Superdiffusion and Drude Weight Suppression

Clean two-dimensional Fermi liquids are now known to exhibit an intermediate \emph{tomographic} regime, between ballistic and Navier--Stokes transport, caused by the anomalously slow relaxation of parity-odd multipolar deformations of the Fermi surface. Here we show that this anomaly extends to the dynamical realm. Starting from a microscopic numerical evaluation of the linearized electron--electron collision operator, we find that the finite-frequency nonlocal conductivity is controlled at low frequency by a single hydrodynamic pole, $\sigma(q,\omega)=\mathcal{D}(q)/(i\omega+\eta_\star q^z)$, with dynamical exponent $z=4/3$ and superdiffusive viscosity $\eta_\star$. Remarkably, the pole residue itself is scale dependent and obeys $\mathcal{D}(q)\sim q^{-\alpha}$ with $\alpha=1/3$, so the dynamical properties are described by two separate exponents rather than one. We interpret the residue suppression using a Krylov-chain description of current relaxation: as $q$ increases, the longest-lived quasinormal mode ceases to be a nearly pure current excitation and spreads over higher odd angular harmonics. Finally, we show that AC transport in narrow channels provides a direct experimental probe of these phenomena.