Entanglement spectrum and number fluctuations in the spin-partitioned BCS ground state

We study entanglement between the spin components of the BCS ground state by calculating the full entanglement spectrum and the corresponding von Neumann entanglement entropy. The entanglement spectrum is effectively modeled by a generalized Gibbs ensemble (GGE) of noninteracting electrons, which may be approximated by a canonical ensemble at the BCS critical temperature. We further demonstrate that the entanglement entropy is jointly proportional to the pairing energy and to the number of electrons about the Fermi surface (an area law). Furthermore, the entanglement entropy is also proportional to the number fluctuations of either spin component in the BCS state.

Figure 1

Entanglement spectrum of the spin-partitioned BCS ground state. Discontinuities at $\pm {\epsilon }_{\mathrm{D}}$ have been smoothed out.

Figure 2

The entanglement spectrum $spec{\rho }^{\uparrow }$ (solid line) is comparable to the orbital occupancies of an effective thermal theory of noninteracting fermions at reciprocal temperature ${\beta }_{\mathrm{e}}^0\approx 1.76/\Delta$ (dashed line). For the latter $p\left(\xi \right)=1-{(1+e^{{\beta }_{\mathrm{e}}^0\xi })}^{-1}$ and ${(1+e^{{\beta }_{\mathrm{e}}^0\xi })}^{-1}$.

Figure 3

Density of states weighted entanglement entropy of energy orbitals. The density of states used here $g\left(\xi \right)={(\xi +\mu )}^{1/2}$ corresponds to a three-dimensional dispersion relation quadratic in momentum. The total spin-EE is dominated by the contributions of orbitals in the vicinity of the Fermi energy.