Superconductivity and geometric superfluid weight of a tunable flat band system

We study superconductivity and superfluid weight of the two-dimensional α-T₃ lattice with on-site asymmetries, where the degree of wavefunction delocalization is controlled by a parameter α and sublattice on-site energies, hosting an isolated quasi-flat band with an α-dependent bandwidth. Within a mean-field approximation of the attractive Hubbard model, we obtain the superconducting order parameters on the three inequivalent sublattices and show their strong dependence on α, interaction strength, and electron filling. At quasi-flat band filling, superconductivity develops for arbitrarily small interaction due to the diverging density of states, with the largest pairing amplitude localized on the more isolated sublattice. We calculate the superfluid weight from linear response theory and study its conventional and geometric contributions. While the conventional part proportional to band derivatives is suppressed in the quasi-flat band regime, the geometric contribution dominated by the quantum metric grows linearly for small interaction strength. We further demonstrate how tuning α enhances the quantum metric and thus the geometric superfluid weight, while increasing on-site asymmetries increases the conventional contribution by broadening the quasi-flat band. At finite temperature, we obtain the BKT transition temperature and demonstrate its strong dependence and enhancement with the α. Our results establish a flat band system, the α-T₃ lattice model, as a candidate for tunable quantum geometry and superfluid weight which motivates the exploration of tunable materials.