Density dependence of the superconducting transition temperature

Determine whether the superconducting transition temperature T_c as a function of the flat-band fermion density n_f in the attractive standard Lieb-lattice Hubbard model with U=-2t develops a non-monotonic peak or rapidly saturates when connecting the dilute-pair regime to the half-filled flat band.

Background

The paper establishes the dilute-density behavior of T_c through a Berezinskii–Kosterlitz–Thouless transition of mobile bound pairs, with the slope determined by the pair effective mass. At flat-band fermion density n_f=1, however, the preformed-pair description is no longer applicable, and T_c is bounded by the crossover scale T_* for the onset of long-range superconducting response obtained in earlier work.

The unresolved issue is how T_c connects these two regimes. The authors identify two possible scenarios: a non-monotonic maximum at intermediate density or rapid saturation. Resolving the density dependence would determine how broadly the high critical temperatures persist away from the dilute limit and near half-filling.