Correlations of the current density in many-body Landau level states

Motivated by recent advances in quantum gas microscopy, we investigate correlation functions of the current density in many-body Landau level states, such as the Laughlin state of the fractional quantum Hall effect. For states fully in the lowest Landau level, or in a vortexable Aharonov-Casher band, we present an exact relationship which shows that all correlation functions involving the current density are directly related to correlation functions of the number density. We calculate perturbative corrections to this relationship arising from Landau level mixing due to interparticle interactions and show that this provides a method by which to extract the system's interaction energy. Finally, we comment on the applicability of our results to lattice systems.