Quantum Geometric Origin of Superconducting Diode effect

The nonreciprocal critical supercurrent endows the superconducting diode effect (SDE) in noncentrosymmetric superconductors with time-reversal symmetry breaking. Recently, it was reported that the SDE has been observed in flat-band superconductors such as twisted trilayer graphene with zero magnetic field. In this Letter, we unveil the SDE in superconductors that host flat bands, where the supercurrent has a significant contribution from the wave-function quantum geometry. By establishing the phenomenological Ginzburg-Landau theory and addressing a mean-field numerical study, we show that besides the conventional band dispersion effect arising from the Fermi surface asymmetry, the microscopic origin of SDE from quantum geometry is the quantum metric dipole. Importantly, in the flat-band limit when the attractive interaction strength is much larger than the bandwidth, the supercurrent and diode effect contributed from quantum geometry becomes the leading order. The interaction-driven valley polarization which lifts the valley degeneracy is also necessary. Our theory applies to the flat-band superconductors especially the moir\'{e} materials in the recent experiments.