Hall viscosity, orbital spin, and geometry: Paired superfluids and quantum Hall systems

The Hall viscosity, a nondissipative transport coefficient analogous to Hall conductivity, is considered for quantum fluids in gapped or topological phases. The relation of the Hall viscosity to the mean orbital spin per particle (s) over bar (discovered in previous work) is elucidated with the help of examples and of the geometry of shear transformations and rotations. For noninteracting particles in a magnetic field, there are several ways to derive the result (even at nonzero temperature), including standard linear response theory. Arguments for the quantization, and the robustness of (s) over bar to small changes in the Hamiltonian that preserve rotational invariance, are given. Numerical calculations of adiabatic transport are performed to check the predictions for quantum Hall systems, with excellent agreement for trial states. The coefficient of k(4) in the static structure factor is also considered and shown to be exactly related to the orbital spin and robust to perturbations in rotation invariant systems.