Topological bound for the superfluid weight of a Bose-Einstein condensate

The non-trivial quantum geometry of the Bloch bands allows fermionic systems with quenched kinetic energy to carry a supercurrent. The relevance of quantum geometry in bosonic systems was pointed out only recently. On the kagome lattice, both the speed of sound and the quantum depletion of a Bose-Einstein condensate have a geometric origin. The finite-momentum condensate, however, breaks translational symmetry and hinders possible connections between the geometry and the topology of the bands. Here, we consider a specific model with narrow isolated bands with non-trivial fragile topology. We successfully formulate a lower bound for the superfluid weight of the Bose-Einstein condensate in terms of the topological indices of the non-interacting bands. This result parallels what is known in fermionic systems and allows the identification of promising models for Bose-Einstein condensates in the limit of weak interactions.