Origin of topological order in a Cooper-pair insulator

We present a microscopic derivation of the Cooper-pair insulator (CPI), a topologically ordered counterpart of the s-wave superconductor. For this, we study a generalized model of electrons with attractive interactions with the unitary renormalization group method. The effective Hamiltonian for the CPI corresponds to a gapped, insulating state of quantum matter arising solely from interparticle interactions (and without any need for disorder or coupling to an external magnetic field). The CPI ground-state manifold displays signatures of topological order, including a fourfold degeneracy. The CPI effective Hamiltonian can be written in terms of Wilson loops and contains a topological theta-term known to be equivalent to the Chern-Simons term in two spatial dimensions. We show that the long-ranged many-particle entanglement content of the CPI ground state is driven by interhelicity two-particle scattering processes. The state with theta = 0 possesses the largest bipartite entanglement entropy (EE) and scales logarithmically with subsystem size (L). Passage from the CPI to the s-wave BCS superconducting ground state at T = 0 under RG is demonstrated through the replacement of the long-ranged entanglement of the former by the short-ranged entanglement of the latter. A study of the renormalization of the entanglement in k space shows that the CPI state possesses a hierarchy of scales of entanglement, and that this hierarchy collapses in the BCS state. Our work offers clear evidence for the microscopic origins of topological order in a prototypical system and lays the foundation for similar investigations in other systems of correlated electrons.