Superpotentials, flat bands and the role of Quantum Geometry for the superfluid stiffness

We investigate the stability and enhancement of superconductivity in a system subject to a 2D periodic superpotential. This externally imposed modulation leads to a reconstructed multi-band structure with flat bands emerging in certain energy regions. When the chemical potential is tuned into one of these flat bands, the associated increase in the density of states can strongly enhance the superconducting critical temperature Tc compared to the homogeneous case while maintaining the superfluid stiffness Ds at a sizeable value. We also clarify the role of the quantum geometric effect for flat bands in such topologically trivial systems. While the overall paramagnetic interband contribution to the stiffness is negative we identify a class of certain interband scattering processes which enhance Ds and scale with the interaction, playing therefore the same role as quantum geometric effects in conventional flat band models. The enhancement of superconductivity is found to be robust against weak to moderate disorder, suggesting that the system is not overly sensitive to imperfections which is an important prerequisite for experimental realizations. Our results demonstrate that engineered superpotentials offer a compelling alternative to twist-based moire systems, providing a pathway toward tailoring superconductivity via lithographic design.