  We unveil the microscopic origin of the topologically ordered counterpart of
the s-wave superconductor in this work. For this, we employ the recently
developed unitary renormalisation group (URG) method on a generalised model of
2D electrons attractive interactions. The effective Hamiltonian obtained at the
stable low-energy fixed point of the RG flow corresponds to a gapped,
insulating state of quantum matter we call the Cooper pair insulator (CPI). We
show that the CPI ground state manifold displays several signatures of
topological order, including a four-fold degeneracy when placed on the torus.
Spectral flow arguments reveal the emergent gauge-theoretic structure of the
effective Hamiltonian, as it can be written entirely in terms of non-local
Wilson loops. It also contains a topological $\theta$-term whose coefficient is
quantised, in keeping with the requirement of invariance of the ground state
under large gauge transformations. We find that the long-ranged many-particle
entanglement content of the CPI ground state is driven by inter-helicity
two-particle scattering processes. Analysis of the passage from CPI to BCS
superconducting ground state shows the RG flow promotes fluctuations in the
number of condensed Cooper pairs and lowers those in the conjugate global
phase. Consequently, the distinct signatures of long-ranged entanglement in the
CPI are replaced by the well-known short-ranged entanglement of the BCS state.
Finally, we study the renormalisation of the entanglement in $k$-space for both
the CPI and BCS ground states. The topologically ordered CPI state is shown to
possess an emergent 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 this prototypical system, and lays
the foundation for similar investigations in other systems of correlated
electrons.
