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In this work we consider the superfluid stiffness of a genericallynon-Galilean invariant interacting system and investigate under what conditionsthe stiffness may nonetheless approach the Galilean-invariant value $n/m$.Within Eliashberg theory we find that the renormalized stiffness isapproximately given by $n/m$ in the case when the $l=0$ and $l=1$ components ofthe effective Fermi-surface projected interaction are approximately equal overa range of frequencies. This holds, in particular, when the interaction ispeaked at zero momentum transfer. We examine this result through threecomplementary lenses: the $\delta (\omega)$ term in the conductivity, the phasedependence of the Luttinger-Ward free energy, and the coupling of the amplitudeand phase sectors in the Hubbard-Stratonovich collective mode action. Fromthese considerations we show that the value of the stiffness is determined bythe strength of renormalization of the current vertex and that the latter canbe interpreted as the shift of the self-consistent solution due to flow of thecondensate, or alternatively as coupling of the phase mode to $l=1$fluctuations of the order parameter. We highlight that even though thesuperfluid stiffness in some non-Galilean systems approaches the Galileanvalue, this is not enforced by symmetry, and in general the stiffness may bestrongly suppressed from its BCS value. As a corollary we obtain the genericform of the phase action within Eliashberg theory and charge and spin Wardidentities for a superconductor with frequency dependent gap function.

Superfluid stiffness within Eliashberg theory: the role of vertex corrections