Functional approach to superfluid stiffness: Role of quantum geometry in unconventional superconductivity

Nontrivial quantum geometry of electronic bands has been argued to facilitate superconductivity even for the case of flat dispersions where the conventional contribution to the superfluid weight is suppressed by the large effective mass. However, most previous work focused on the case of conventional superconductivity while many contemporary superconducting quantum materials are expected to host unconventional pairing. Here, we derive a generalized expression for the superfluid weight employing mean-field BCS theory for systems with time-reversal symmetry in the normal state and arbitrary unconventional superconducting order with zero-momentum intraband pairing. Our derivation reveals the necessity of incorporating functional derivatives of the grand potential with respect to the superconducting gap function. Through perturbative analysis in the isolated narrow-bands limit, we demonstrate that this contribution arises from quantum geometrical effects, specifically due to a nontrivial Wilczek-Zee connection. Utilizing the newly obtained expressions for the superfluid weight, we apply our framework to an extended Kane-Mele model, contrasting conventional $s$-wave superconductivity with chiral $d$-wave superconductivity.