Quantum Entanglement in Second-quantized Condensed Matter Systems

The entanglement between occupation-numbers of different single particle basis states depends on coupling between different single particle basis states in the second-quantized Hamiltonian. Thus in principle, interaction is not necessary for occupation-number entanglement to appear. However, in order to characterize quantum correlation caused by interaction, we use the eigenstates of the single-particle Hamiltonian as the single particle basis upon which the occupation-number entanglement is defined. Using the proper single particle basis, we discuss occupation-number entanglement in important eigenstates, especially ground states, of systems of many identical particles. The discussions on Fermi systems start with Fermi gas, Hatree-Fock approximation, and the electron-hole entanglement in excitations. The entanglement in a quantum Hall state is quantified as -fln f-(1-f)ln(1-f), where f is the proper fractional part of the filling factor. For BCS superconductivity, the entanglement is a function of the relative momentum wavefunction of the Cooper pair, and is thus directly related to the superconducting energy gap. For a spinless Bose system, entanglement does not appear in the Hatree-Gross-Pitaevskii approximation, but becomes important in the Bogoliubov theory.