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We present the fundamental model of a topological electromagnetic phase ofmatter: viscous Maxwell-Chern-Simons theory. Our model applies to a quantumHall fluids with viscosity. We solve both continuum and lattice regularizedsystems to demonstrate that this is the minimal (exactly solvable) gauge theorywith a nontrivial photonic Chern number ($C\neq 0$) for electromagnetic wavescoupled to a quantum Hall fluid. The interplay of symmetry and topology is alsocaptured by the spin-1 representations of a photonic skyrmion at high-symmetrypoints in the Brillouin zone. To rigorously analyze the topological physics, weintroduce the viscous Maxwell-Chern-Simons Lagrangian and derive the equationsof motion, as well as the boundary conditions, from the principle of leastaction. We discover topologically-protected chiral (unidirectional) edge stateswhich minimize the surface variation and correspond to massless photonicexcitations costing an infinitesimal amount of energy. Physically, ourpredicted electromagnetic phases are connected to a dynamical photonic mass inthe integer quantum Hall fluid. This arises from viscous (nonlocal) Hallconductivity and we identify the nonlocal Chern-Simons coupling with the Hallviscosity. The electromagnetic phase is topologically nontrivial $C\neq 0$ whenthe Hall viscosity inhibits the total bulk Hall response. Our work bridges thegap between electromagnetic and condensed matter topological physics while alsodemonstrating the central role of spin-1 quantization in nontrivial photonicphases.

Viscous Maxwell-Chern-Simons theory for topological electromagnetic phases of matter