Flat-band Fulde-Ferrell-Larkin-Ovchinnikov State from Quantum Geometry Discrepancy

We propose a distinctive paradigm to realize the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) Cooper pairing states by creating an imbalance in the quantum geometry of paired electrons on an isolated flat band, which we term "Quantum Geometry Discrepancy (QGD)''. Based on a flat-band electronic Hamiltonian with continuously tunable quantum metrics for each spin species, we analytically investigate the QGD-induced FFLO instability near the superconducting critical temperature through the band-projection method. We found that a new quantum geometric quantity named "anomalous quantum distance'' can be used to capture QGD, and map out the FFLO instability on the flat band. To obtain the phase diagram of the BCS-FFLO transition driven by QGD, we perform numerical calculations using self-consistent mean-field theory, which aligns well with the analytical results. Additionally, we discuss the stability of the flat-band FFLO state when a finite band dispersion is turned on. We point out that QGD serves as a new protocol for stabilizing the flat-band FFLO state.