Boundary layer theory for quantum Hall hydrodynamics

In this talk, we consider non-dissipative two-dimensional broken parity fluids with odd viscosity and magnetic field with a free surface boundary condition. We show that for non-dissipative fluids, incompressibility must be violated at free surfaces to have a consistent hydrodynamic description of the dynamical boundary. Violation to incompressibility sources vorticity within the boundary layer which is an exclusive manifestation of odd viscosity. The thickness of the boundary layer is controlled by the velocity of sound. A non-dissipative boundary layer mechanism allows us to write the full variational principle for both bulk and boundary. In the limit of large sound velocity, we recover Quantum Hall like fluids where the bulk is incompressible but the thin boundary layer manifests compression modes. We discuss the applicability of our results to (fractional) quantum Hall fluids with a free surface.