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We present exact results that give insight into how interactions lead to transport and superconductivity in a flat band where the electrons have no kinetic energy. We obtain bounds for the optical spectral weight for flat-band superconductors that lead to upper bounds for the superfluid stiffness and the two-dimensional (2D) Formula: see text We focus on on-site attraction Formula: see text on the Lieb lattice with trivial flat bands and on the π-flux model with topological flat bands. For trivial flat bands, the low-energy optical spectral weight Formula: see text with Formula: see text , where n is the flat-band density and Ω is the Marzari-Vanderbilt spread of the Wannier functions (WFs). We also obtain a lower bound involving the quantum metric. For topological flat bands, with an obstruction to localized WFs respecting all symmetries, we again obtain an upper bound for Formula: see text linear in Formula: see text We discuss the insights obtained from our bounds by comparing them with mean-field and quantum Monte Carlo results.
Verma et al. (2021) studied this question.