Quantitative test of geometric superfluid stiffness in moire superconductors

For an isolated flat band with pairing gap $\Delta$, mean-field theory gives a geometric superfluid weight $D_s^{\mathrm{geom}}$ proportional to $\Delta \sqrt{\nu_f (1 - \nu_f)}$ times the Brillouin-zone integral of the quantum metric $g(k)$ ($\nu_f$ the band filling fraction), bounded below by band topology. Compare the measured $D_s(T \to 0)$ in magic-angle twisted bilayer and trilayer graphene with $D_s^{\mathrm{geom}}$ computed from interaction-renormalized bands and the measured $\Delta$. An answer is the ratio $D_s^{\mathrm{meas}} / D_s^{\mathrm{geom}}$ with uncertainty at several fillings.

Same-device measurement of $\Delta$ (tunnelling) and $D_{s}$ (kinetic inductance) compared with a calculation of the quantum metric of the interacting bands.