Viscosity of Quantum Fluids: Kubo Formulas, Hall Viscosity, and Relation with Conductivity

The analogue of friction in fluid dynamics is provided by viscosity, which causes dissipation of the energy of the flow. However, if time reversal symmetry is broken, the viscosity tensor may have non-dissipative components, termed “Hall viscosity”, similarly to the non-dissipative Hall conductivity. The Hall viscosity was shown recently to be topologically-protected in rotationally-invariant systems, and to be equal to half the particle density times the orbital angular momentum per particle. Its observation can therefore be of interest in elucidating the nature of the more exotic quantum Hall filling fractions and related systems (e.g., p+ip superfluids), including the possibility of non-abelian statistics and its use for topological quantum computation. However, no concrete measurement scheme has hitherto been proposed.

With this motivation in mind we developed a microscopic linear response theory of viscosity (dissipative as well as Hall), based on the equivalence between a strain rate and time-dependence of the spatial metric of the system. We applied this formalism to rederive and extend previous results on the Hall viscosity. Furthermore, we have established a general relation between the viscosity tensor and the wave-vector dependent conductivity tensor for Galilean-invariant quantum fluids. This relation enables one to extract the Hall viscosity, as well as other viscosity coefficients (shear and bulk) when relevant, from electromagnetic response measurements.