Quantum geometric origin of the Meissner effect and superfluid weight marker

The momentum space of conventional superconductors is recently recognized to possess a quantum metric defined from the overlap of filled quasihole states at neighboring momenta. For multiband superconductors with arbitrary intraband and interband $s$-wave pairing, we elaborate that their superfluid weight in London equations is given by the momentum integration of the elements of the quantum metric times the quasiparticle energy, indicating the quantum geometric origins of the Meissner effect and vortex state. The momentum integration of the quantum metric further yields a spread of quasihole Wannier functions that characterizes the stability of the superconducting state. Our formalism allows the diamagnetic response of conventional superconductors to be mapped to individual lattice sites as a superfluid weight marker, which can incorporate the effect of disorder through self-consistently solving the Bogoliubov--de Gennes equations. Using single-band $s$-wave superconductors in two and three dimensions as examples, our marker reveals a diamagnetic current that becomes turbulent in the presence of nonmagnetic impurities, and an increase of London penetration depth by disorder that is consistent with experiments.