Geometric superfluid stiffness of Kekul\'e superconductivity in magic-angle twisted bilayer graphene

Abstract

Superconductivity in twisted graphene is probed by tunneling spectroscopy and superfluid stiffness, two observables that access the same order parameter from complementary perspectives. We show that a finite-momentum pair-density-wave (PDW) state, consistent with reported Kekul\'e signatures, reconciles substantial low-energy tunneling weight with an approximately $T^2$ suppression of the low-temperature superfluid stiffness. The PDW order produces a Bogoliubov Fermi surface and finite zero-bias conductance. The same gapless quasiparticles also enter the geometric superfluid response, yielding a low-temperature stiffness suppression that persists in the flat-band limit. We further predict that, under density or displacement-field tuning, enhanced residual zero-bias conductance should accompany reduced low-temperature stiffness, providing a direct experimental link between tunneling spectroscopy and phase rigidity in twisted graphene.