Superfluid stiffness within Eliashberg theory: the role of vertex corrections

It has long been known that the superfluid stiffness of a Galilean invariant system at $T=0$ must be $D_s = n/m$ with $n$ the total electron density and $m$ the bare electron mass.

However, outside of BCS theory superconductivity generally arises in systems which break Galilean invariance, either due to an underlying lattice structure, or due to to retardation effects in the electron-electron interactions.

Motivated by this interplay, we consider the superfluid stiffness within Eliashberg theory with a focus on the corrections to the supercurrent vertex and their role in reproducing the Galilean invariant result.

We consider systems with interactions peaked at small momentum transfer and systems with nearly momentum-independent interaction. In the first case, we show that superfluid stiffness is close to $n/m$ because the current vertex approximately satisfies the same Ward identity as the density vertex and nearly cancels out mass renormalization. In the second case, we show that vertex corrections are irrelevant, and stiffness can be substantially reduced by mass renormalization.