Superfluid Hydrodynamics of multi-component Bosonic systems

The superfluid hydrodynamic equations of a bosonic gas at zero temperature are often derived from the Gross-Pitaevskii equation (GPE), which is valid in the dilute region when the Bose-EinsteinCondensation dominates and the effects beyond the mean-field are negligible. In this paper, we generally show that the zero-temperature superfluid hydrodynamic equation of a multi-component bosonic system with short-ranged interactions can be obtained in the path-integral formalism under the local equilibrium assumption, which is even valid beyond the dilute region. Our approach can be extended to systems with long-range interactions such as dipole-dipole interactions by treating the Hartree energy properly. The Andreev-Bashkin effect of a multi-component superfluid can be described in this formalism. The implication of our results on quantum droplets is discussed.