We study pairwise quantum entanglement in systems of fermions itinerant in a lattice from a second-quantized perspective. Entanglement in the grand-canonical ensemble is studied, both for energy eigenstates and for the thermal state. Relations between entanglement and superconducting correlations are discussed in a BCS-like model and for η-pair superconductivity.
We study pairwise quantum entanglement in systems of fermions itinerant in a lattice from a second-quantized perspective. Entanglement in the grand-canonical ensemble is studied, both for energy eigenstates and for the thermal state. Relations between entanglement and superconducting correlations are discussed in a BCS-like model and for η-pair superconductivity.
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 this so-called proper single particle basis, if there is no interaction, the many-particle second-quantized Hamiltonian is diagonalized and thus cannot generate entanglement, while its eigenstates can always be chosen to be non-entangled. If there is interaction, entanglement in the proper single particle basis arises in energy eigenstates and can be dynamically generated. Using the proper single particle basis, we discuss occupation-number entanglement in important eigenstates, especially ground states, of systems of many identical particles, in exploring insights the notion of entanglement sheds on many-particle physics. The discussions on Fermi systems start with Fermi gas, the Hartree–Fock approximation and the electron–hole entanglement in excitations. In the ground state of a Fermi liquid, in terms of the Landau quasiparticles, entanglement becomes negligible. The entanglement in a quantum Hall state is quantified as − f ln 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 g k , and is thus directly related to the superconducting energy gap, and vanishes if and only if superconductivity vanishes. For a spinless Bose system, entanglement does not appear in the Hartree–Gross–Pitaevskii approximation, but becomes important in the Bogoliubov theory, as a characterization of two-particle correlation caused by the weak interaction. In these examples, the interaction-induced entanglement as calculated is directly related to the macroscopic physical properties.
The notion of ``paired'' fermions is central to important condensed-matter phenomena such as superconductivity and superfluidity. While the concept is widely used and its physical meaning is clear, there exists no systematic and mathematical theory of pairing that would allow us to unambiguously characterize and systematically detect paired states. We propose a definition of pairing and develop methods for its detection and quantification applicable to current experimental setups. Pairing is shown to be a quantum correlation different from entanglement, giving further understanding in the structure of highly correlated quantum systems. In addition, we will show the resource character of paired states for precision metrology, proving that the BCS states allow phase measurements at the Heisenberg limit.
The Hubbard model describes interacting itinerant fermion systems and is the simplest model capable of describing the essential physics of strongly correlated electron systems. In this model, both spin correlations and charge correlations are important for the entanglement, thus the magnetic susceptibility is not an appropriate thermal entanglement witness. The specific heat reflects spin correlations and charge correlations. We obtain a relation that permits the specific heat to be used directly as a thermal entanglement witness. We calculate, away from half filling, the entanglement critical temperature TE, below which entanglement is detected, for small linear clusters and the hypercubic lattice in the limit of infinite dimensions by using the exact numerical diagonalization method and the dynamical mean-field theory, respectively. We find, in both cases, that TE increases with increasing strength of the on-site Coulomb repulsion U.
We examine distinct measures of fermionic entanglement in the exact ground state of a finite superconducting system. It is first shown that global measures such as the one-body entanglement entropy, which represents the minimum relative entropy between the exact ground state and the set of fermionic Gaussian states, exhibit a close correlation with the BCS gap, saturating in the strong superconducting regime. The same behavior is displayed by the bipartite entanglement between the set of all single-particle states k of positive quasimomenta and their time-reversed partners k. In contrast, the entanglement associated with the reduced density matrix of four single-particle modes $k,\overline{k}, {k}^{\ensuremath{'}},{\overline{k}}^{\ensuremath{'}}$, which can be measured through a properly defined fermionic concurrence, exhibits a different behavior, showing a peak in the vicinity of the superconducting transition for states $k,{k}^{\ensuremath{'}}$ close to the Fermi level and becoming small in the strong coupling regime. In the latter, such reduced state exhibits, instead, a finite mutual information and quantum discord. While the first measures can be correctly estimated with the BCS approximation, the previous four-level concurrence lies strictly beyond the latter, requiring at least a particle-number projected BCS treatment for its description. Formal properties of all previous entanglement measures are as well discussed.
We investigate the behavior of entanglement between a single fermionic level and a fermionic bath in three distinct thermodynamic regimes. First, in thermal equilibrium, we analyze the dependence of entanglement on the considered statistical ensemble: for the grand canonical state, it is generated only for a sufficiently strong system-bath coupling, whereas it is present for arbitrarily weak couplings for the canonical state with a fixed particle number. The threshold coupling strength at which entanglement appears is shown to strongly depend on the bath bandwidth. Second, we consider the relaxation to equilibrium. In this case a transient entanglement in a certain time interval can be observed even in the weak-coupling regime, when the reduced dynamics and thermodynamics of the system can be well described by an effectively classical and Markovian master equation for the state populations. At strong-coupling strengths, entanglement is preserved for long times and converges to its equilibrium value. Finally, in voltage-driven junctions, a steady-state entanglement is generated for arbitrarily weak system-bath couplings at a certain threshold voltage. It is enhanced in the strong-coupling regime, and it is reduced by either the particle-hole or the tunnel coupling asymmetry.