Optical $N$-invariant of graphene's viscous Hall fluid

Over the past three decades, graphene has become the prototypical platform for discovering unique phases of topological matter. Both the Chern insulator $C\in\mathbb{Z}$ and the quantum spin Hall insulator $\nu\in\mathbb{Z}_2$ were first predicted in graphene, which led to a veritable explosion of research in topological materials. Here, we introduce a new topological classification of two-dimensional matter -- the optical $N$-phases $N\in\mathbb{Z}$. The $C$ and $\nu$ phases are related to charge and spin transport respectively, whereas the $N$-phases are connected to polarization transport. In all three cases, transportation of charge/spin/polarization quanta is forbidden in the bulk but permitted on the edge. One fundamental difference is that the $N$-invariant is defined for dynamical electromagnetic waves over all Matsubara frequencies and wavevectors. We show this topological quantum number is captured solely by the spatiotemporal dispersion of the susceptibility tensor $\chi(\omega,\mathbf{q})$. We also prove $N\neq 0$ is nontrivial in graphene's viscous Hall fluid with the underlying physical mechanism being Hall viscosity $\eta_H$. In the nontrivial phase, we discover a deep sub-wavelength phenomenon reminiscent of the Meissner effect: at a particularly large photon momentum $q=D_H^{-1}$ defined by the Hall diffusion length $D_H$, the magnetic field is completely expelled from the viscous Hall fluid. We propose a new probe of topological matter, evanescent magneto-optic Kerr effect (e-MOKE) spectroscopy, to unravel this novel optical $N$-invariant and verify the magnetic field expulsion. Our work indicates that graphene with Hall viscosity is the first candidate material for a topological electromagnetic phase of matter.