Hall Viscosity Revealed via Density Response

Quantum Hall systems are recently shown to possess a quantity sensitive to the spatial geometry and topology of the system, dubbed the Hall viscosity $\eta_H$. Despite the extensive theoretical discussions on its properties, the question of how to measure $\eta_H$ still poses a challenge. In this paper, we present a general relation between Hall viscosity and susceptibility for systems with Galilean invariance. Thus, it allows for determination of $\eta_H$ through density response signatures. The relations are verified in the integer quantum Hall example, and is further illustrated in an effective hydrodynamic analysis. Since the derivation is based on Kubo formulae and assumes no more than conservation laws and translational symmetry, the results are applicable to a wide range of systems.