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 $\ensuremath{\theta}$-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 $\ensuremath{\theta}=0$ 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.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 $\ensuremath{\theta}$-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 $\ensuremath{\theta}=0$ 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.

arXiv:2102.12193Physical review. B./Physical review. BOct 29, 20217 citations

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 $\ensuremath{\theta}$-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 $\ensuremath{\theta}=0$ 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.