Bootstrapping Flat-band Superconductors: Rigorous Lower Bounds on Superfluid Stiffness

The superfluid stiffness fundamentally constrains the transition temperature of superconductors, especially in the strongly coupled regime. However, accurately determining this inherently quantum many-body property in microscopic models remains a significant challenge. In this work, we show how the quantum many-body bootstrap framework, specifically the reduced density matrix (RDM) bootstrap, can be leveraged to obtain rigorous lower bounds on the superfluid stiffness in frustration-free models with a superconducting ground state. We numerically apply the method to two classes of frustration-free models for flat-band superconductivity: spinful s-wave superconductors with uniform pairing and spinless chiral p-wave superconductors. In the first class, which also refers to the so-called quantum geometric nesting model, we uncover a general relation between the stiffness and the pair mass. Going beyond the familiar Hubbard case within this class, we find how additional interactions, notably simple intra-unit-cell magnetic couplings, can enhance the superfluid stiffness. Furthermore, the RDM bootstrap unexpectedly reveals that the trion-type correlations are essential for bounding the stiffness, offering new insights into the structure of these models. In the second class, although no exact results can be obtained, the bootstrap, as a general method, can still certify a non-zero stiffness extrapolable to the thermodynamic limit, validating the existence of chiral superconductivity in certain systems. A straightforward generalization of the method can yield bounds on susceptibilities that complement variational approaches. Our findings underscore the immense potential of the quantum many-body bootstrap as a powerful tool to derive rigorous bounds on physical quantities beyond energy.