Abstract
Current theories of high-temperature superconductivity in flat-band systems predict a linear dependence of the transition temperature on the attractive interaction, T_c(U) = c|U|. However, neither the value of c nor the full nonlinear T_c(U) curve -- with a maximum at large |U| -- is known beyond mean-field and quantum geometry estimates. Using a controlled diagrammatic Monte Carlo technique, we trace the onset of superfluid response in the Lieb lattice with attractive Hubbard interaction. Focusing on the half-filled flat-band case, where the ordering mechanism differs fundamentally from both BCS and preformed Cooper pair scenarios, we find that the pairing response diverges linearly with decreasing temperature over a broad range of U, leading to a sharp crossover to long-range correlations at a characteristic temperature T_*, which provides a controlled upper bound on T_c. The highest T_* occurs when all three bands touch at a single momentum point, potentially corresponding to high T_c values.