The idea that the finite effective mass of Cooper pairs in a flat band is the combined result of interactions and the geometric properties of the localized wave functions of the flat band, encoded in the quantum metric, has been put forward in S. Peotta and P. Törmä, Nature Communications 6, 8944 (2015). In FLATOPS the results of this first work have been applied to a number of lattice models with flat bands relevant both for experiments in ultracold gases and for solid state materials. An important example is the Lieb lattice. The superfluid weight for the Lieb lattice is shown in Fig 1. For fillings in between one and two the flat band is partially filled. Precisely for a partially filled flat band the superfluid weight reaches its maximum. This is a striking example how flat bands can enhance the superfluid weight, which in turn controls the critical temperature. The Lieb lattice and other models studied in this project have been recently realized with ultracold gases in optical lattices, where the results of FLATOPS are likely to find a first immediate application. The results shown in Fig. 1 are obtained using mean-field theory, which is the simplest approximation that can be used to describe the superconducting state. However, in this project mean-field results have been systematically checked using beyond mean-field methods. An example is shown in Fig. 2, where the superfluid weight of the Creutz ladder, another lattice model with flat bands, obtained from mean-field is compared to the essentially exact result calculated using Density Matrix Renormalization Group, which is an important beyond mean-field method used for one-dimensional systems. An important goal of FLATOPS was to further explore the relation between superfluidity and topological properties of the band structure. Indeed the superfluid weight of a flat band is bounded from below by the Chern number, a topological invariant of the band structure as shown in S. Peotta and P. Törmä, Nature Communications 6, 8944 (2015). It is a natural question to ask whether similar bounds can be found also in the case of other topological invariants that are currently known. Within FLATOPS it has been shown that also the winding number, an important topological invariant for one-dimensional systems, provides a bound from below to the superfluid weight, in the same way as the Chern number.
