Variational principle for an incompressible fluid with Hall viscosity and gapless surface modes

Hydrodynamics is a powerful tool to study strongly interacting systems, including for example the fractional quantum Hall effect (FQHE). The main features of the quantum Hall hydrodynamics are the incompressibility of the electron flow, due to the gap separation between the FQH ground state and the excited states, and the relation between the density of the fluid and its vorticity. Motivated by this example, we focus in this work on the variational and Hamiltonian formulations of incompressible two-dimensional fluid dynamics with free surface and nonvanishing Hall viscosity. We show that within the variational principle the Hall viscosity contribution corresponds to a purely geometric boundary term. This term modifies effective boundary conditions on the free surface. The modified boundary conditions have a interpretation describing an additional pressure at the free surface proportional to the angular velocity of the surface itself. These boundary conditions are believed to be universal since the proposed hydrodynamic action is fully determined by the symmetries of the system. The extension of this variational principle to compressible flows will be also considered in this talk.