Ab-initio superfluid weight and superconducting penetration depth

Machine learning and high-throughput screening approaches to superconductor discovery require physically meaningful descriptors that capture essential physics while remaining computationally tractable. The superfluid weight is an ideal descriptor as it is a prerequisite for superconductivity, determines the magnetic penetration depth and the Berezinskii-Kosterlitz-Thouless transition temperature in two-dimensional materials, may limit the critical temperature in unconventional superconductors through phase coherence, and reveals quantum geometric contributions to supercurrent transport. We develop a computationally efficient framework for calculating the zero-temperature, mean-field superfluid weight for uniform pairing from density functional theory band structures and Bloch wavefunctions. We separately evaluate the conventional contribution from band curvature and the geometric contribution from quantum geometry. To validate the method, we calculate London penetration depths for a few conventional superconductors (Al, Pb, Nb, MgB$_2$, LuRu$_3$B$_2$ and YRu$_3$B$_2$) and find good agreement with experiment after accounting for nonlocal corrections, strong-coupling effects, and sample quality. The conventional contribution dominates by orders of magnitude in these wide-band materials, as expected. This framework provides a foundation for large-scale screening of superconducting candidates and exploring quantum geometric effects in unconventional superconductors.