Obstructed Atomic Insulators and Superfluids of Fermions Coupled to $mathbb{Z}_2$ Gauge Fields*

We study spin-$frac{1}{2}$ fermions coupled to $mathbb{Z}_2$ gauge fields on the square lattice and show how a spatial modulation of the fermion hopping amplitude allows for the realization of various obstructed atomic insulators that host higher-order band topology. Up on including the effect of quantum dynamics of the gauge fields within a simplified model, we find a rich phase diagram of this system with a number of superfluid phases (which host distinct types of topological defects) arising from the attractive interactions meditated by the gauge fields. The evolution of the superfluid obtained by the destabilization of the obstructed atomic insulators from the Bardeen-Cooper-Schrieffer (BCS) type to a Bose-Einstein condensate (BEC) of tightly bound pairs occurs via the realization of these different superfluid phases separated by first-order transitions.