Numerically Exact Study of Flat-Band Superconductivity

Current theories of high-temperature superconductivity in flat-band systems predict a linear dependence of the transition temperature on the attractive interaction, $T_c(U)=c\mathrm{\mid }U\mathrm{\mid }$. However, neither the value of $c$ nor the full nonlinear $T_c(U)$ curve – with a maximum at large $\mathrm{\mid }U\mathrm{\mid }$ – 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 $T_{\ast }$, which provides a controlled upper bound on $T_c$. The highest $T_{\ast }$ occurs when all three bands touch at a single momentum point, potentially corresponding to high $T_c$ values.