Quantum metric and effective mass of a two-body bound state in a flat band

We consider two-body bound states in a flat band of a multiband system. The existence of pair dispersion predicts the possibility of breaking the degeneracy of the band and creating order, such as superconductivity. Within a separable interaction potential approximation, we find that finiteness of the effective mass of a bound pair is determined by a band structure invariant, which in the uniform case becomes the quantum metric. The results offer a simple foundation to understand and predict flat band superconductivity. We propose an experiment to test the interaction-induced pair motion.