Obstructed Cooper pairs in flat band systems - weakly-coherent superfluids and exact spin liquids

Superconductivity in a partially filled flat band presents a vexing conceptual hurdle because the absence of a Fermi surface precludes a weak-coupling regime where one can extend insights from the Bardeen-Cooper-Schrieffer picture of a Fermi surface instability. We approach the strongly correlated problem of flat band superconductivity from the strong coupling limit of local attractive interactions on line-graph lattices, whose non-interacting bandstructures host exactly flat bands. In this limit, the pair kinetic energy which sets the superfluid stiffness is expected to scale inversely with the pair binding interaction. Here we demonstrate a striking counterexample. We show that when doped charges propagate on the line-graph of a lattice with strong pairing interaction, they bind into obstructed Cooper pairs whose motion is frustrated by destructive interference. As a result, the leading-order pair kinetic energy vanishes identically in the strong-coupling expansion, producing a flat bosonic band of compact localized pair states, zero superfluid stiffness at leading order, and an extensively degenerate many-body ground state manifold. At quarter filling, the frustrated pair dynamics maps onto a quantum dimer model with a $d$-wave resonating-valence-bond spin liquid ground state, which becomes exact at the analytically solvable Rokhsar-Kivelson point. The pairing Hamiltonian in this limit thus has a topologically ordered ground state with long-range entanglement and deconfined holon excitations. Interestingly, we find exact compact localized eigenstates and extensive degeneracies in the many-body eigenstates of this emergent dimer model. Our results establish a disorder-free mechanism for interaction-driven localization, in which strong pairing collapses the kinetic energy of Cooper pairs.