  In a flat band superconductor, the charge carriers' group velocity vF is
extremely slow, quenching their kinetic energy. The emergence of
superconductivity thus appears paradoxical, as conventional BCS theory implies
a vanishing coherence length, superfluid stiffness, and critical current. Here,
using twisted bilayer graphene (tBLG), we explore the profound effect of
vanishingly small vF in a Dirac superconducting flat band system Using
Schwinger-limited non-linear transport studies, we demonstrate an extremely
slow vF ~ 1000 m/s for filling fraction nu between -1/2 and -3/4 of the moire
superlattice. In the superconducting state, the same velocity limit constitutes
a new limiting mechanism for the critical current, analogous to a relativistic
superfluid. Importantly, our measurement of superfluid stiffness, which
controls the superconductor's electrodynamic response, shows that it is not
dominated by the kinetic energy, but instead by the interaction-driven
superconducting gap, consistent with recent theories on a quantum geometric
contribution. We find evidence for small pairs, characteristic of the BCS to
Bose-Einstein condensation (BEC) crossover, with an unprecedented ratio of the
superconducting transition temperature to the Fermi temperature exceeding
unity, and discuss how this arises for very strong coupling superconductivity
in ultra-flat Dirac bands.
