We review the concept of superfluidity and, based on real and thought experiments, we use the formalism of second quantization to derive expressions that allow the calculation of the superfluid density for general Hamiltonians with path-integral methods. It is well known that the superfluid density can be related to the response of the free energy to a boundary phase twist, or to the fluctuations of the winding number. However, we show that this is true only for a particular class of Hamiltonians. In order to treat other classes, we derive general expressions of the superfluid density that are valid for various Hamiltonians. While the winding number is undefined when the number of particles is not conserved, our general expressions allow us to calculate the superfluid density in all cases. We also provide expressions of the superfluid densities associated to the individual components of multispecies Hamiltonians, which remain valid when interspecies conversions occur. The cases of continuous and discrete spaces are discussed, and we emphasize common mistakes that occur when considering lattices with nonorthonormal primitive vectors.

• Received 25 April 2014
• Revised 16 September 2014

DOI:https://doi.org/10.1103/PhysRevB.90.134503