Cooper pairing, flat-band superconductivity and quantum geometry in the pyrochlore-Hubbard model

We investigate the impacts of the quantum geometry of Bloch states, specifically through the band-resolved quantum-metric tensor, on Cooper pairing and flat-band superconductivity in a three-dimensional pyrochlore-Hubbard model. First we analyze the low-lying two-body spectrum exactly, and show that the pairing order parameter is uniform in this four-band lattice. This allowed us to establish direct relations between the superfluid weight of a multiband superconductor and ($i$) the effective mass of the lowest-lying two-body branch at zero temperature, ($ii$) the kinetic coefficient of the Ginzburg-Landau theory in proximity to the critical temperature, and ($iii$) the velocity of the low-energy Goldstone modes at zero temperature. Furthermore, we perform a comprehensive numerical analysis of the superfluid weight and Goldstone modes, exploring both their conventional and geometric components at zero temperature.