The ground state and transport properties of the Lieb lattice flat band in the presence of an attractive Hubbard interaction are considered. It is shown that the superfluid weight can be large even for an isolated and strictly flat band. Moreover the superfluid weight is proportional to the interaction strength and to the quantum metric, a band structure invariant obtained from the flat-band Bloch functions. These predictions are amenable to verification with ultracold gases and may explain the anomalous behaviour of the superfluid weight of high-Tc superconductors.
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