Abstract: Band structure theory and the BCS theory of superconductivity are two cornerstones of modern condensed matter physics. They have been used to explain many properties of crystalline solids and have found important practical applications. It is believed that the interplay between the atomic lattice and the attractive force between electrons, whose origin is still a matter of debate, is at the root of the phenomenon of high-Tc superconductivity. In weakly-coupled superconductors the effect of the lattice amounts to a simple renormalization of the electron mass and of the density of states. On the contrary, in high-Tc superconductors the coherence length is of the order of the lattice spacing and new phenomena may occur. An extreme example in this sense are 'flat bands', namely bands where the electron effective mass diverges. In this talk I will present our ongoing work on the problem of superconductivity and superfluidity in flat band systems with special emphasis on the transport properties. Naively one can expect that in a flat band, where the charge carriers are very heavy, transport is absent or at least strongly suppressed. However, we have recently shown that in the flat band limit the superfluid weight Ds is not controlled by the effective mass but rather by a geometric invariant of the band, the quantum metric, which in a sense measures the overlap between neighbouring lattice wave functions. The quantum metric is intimately related to a topological invariant, the Chern number, and as a consequence we obtain the inequality Ds ? |C| between superfluid weight and Chern number C. We show that this geometric effect is important in a number of lattice models of current interest for material science and ultracold gases.
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