Hydrodynamic stability of superfluids and the Landau criteria

The Landau criteria states that superfluidity can exist in nature because, up to some critical velocity, the superfluid may flow relative to a system at rest without creating dissipative excitations. This criteria governs the microscopics of superfluids but, like any other fluid, there also exists a hydrodynamic formulation describing the system in terms of symmetries and fluctuations around local thermodynamic equilibrium. In this formulation, microscopic excitations are repackaged into collective excitations of the system, namely sound and diffusive modes, that transport conserved densities and which are experimentally accessible. In this talk, I show how the Landau criteria appears in the hydrodynamic formulation of superfluidity. For superfluids that flow above some critical velocity, there exist unstable collective excitations that grow exponentially in time. Ultimately, this is connected to a thermodynamic instability that manifests as a divergence in the superfluid velocity's static susceptibility. In Helium-4, where a microscopic description exists, both microscopics and hydrodynamics agree that the critical velocity for the instability is given by the "roton minimum" of the dispersion relation for the massless Goldstone boson associated with the spontaneous symmetry breaking of a global U(1) symmetry.