Quantum geometry and the superfluid weight of a Bose-Einstein condensate

Nontrivial quantum geometry enables superconductivity even in perfectly flat bands, where the single-particle effective mass is infinite. Recent studies have pointed out that quantum geometry also plays a role in Bose-Einstein condensates. Of particular importance is the quantum metric at the condensation momentum, which determines quantities such as the speed of sound in a flat band condensate. Here, we study the superfluid weight of a Bose-Einstein condensate within multiband Bogoliubov theory, and show that its geometric part contains contributions related to both the quantum metric at the condensation momentum and the quantum metric integrated over the Brillouin zone. We formulate conditions for the stability of a flat band condensate in terms of geometric properties. We find that a nontrivial integrated quantum metric, which relates to topological properties, can hinder the formation of a stable condensate in flat bands, in contrast to fermionic superconductors. A nontrivial quantum metric at the condensation momentum, on the other hand, is advantageous for a stable condensate.