Polaron spectra and Orthogonality Catastrophe for correlated flat bands

The spectrum of a single immobile impurity embedded in a Fermi sea, which is related to the Anderson Orthogonality Catastrophe (OC), can be computed exactly and is characterized by a divergent power law. We generalize this canonical problem by replacing the impurity with a single particle moving in a flat band with non-trivial Bloch band geometry. We show that, when lattice effects are sufficiently weak or the Fermi momentum is small enough, there is a universal correction to the power law exponent which is proportional to the Fermi energy times the quantum metric of the heavy band. We illustrate this numerically for the Lieb lattice and draw connections to polaron-type ultracold gas experiments.