In a Galilean superfluid, the depletion of superfluid density with rising temperature can be attributed to thermally excited noninteracting phonons. For systems without Galilean symmetry, it has been shown [Berger et al., Phys. Rev. Lett. 136, 166001 (2026)] that “phonon wind” is no longer responsible for the depletion of superfluid density. In this work, we develop the theory of superfluid density at low temperature ($T$) and provide detailed derivations of all results announced by Berger et al. [Phys. Rev. Lett., submitted]. Using Popov's hydrodynamic action, we show that the theory of low-temperature depletion in a $d$-dimensional quantum superfluid maps onto the problem of finite-size ($L$) corrections in a $(d+1)$-dimensional anisotropic (pseudo-)classical-field system with $U$(1)-symmetric complex-valued action. In addition to generalizing Landau's (canonical) formula, we develop the grand canonical theory, which in a broader context reveals a universal scaling, $T^{d+1}$ and $1/L^{d+1}$, for finite-$T$ and finite-$L$ effects of many thermodynamic quantities. We validate our theory with numeric simulations of interacting lattice bosons and the $J$-current model.

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• Superfluidity
• Superfluids
• Effective field theory
• Monte Carlo methods