Quantum entanglement in second-quantized condensed matter systems

Abstract

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 singleparticle 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, then the manyparticle 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, Hatree-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 Hatree-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.

Entanglement in many-body systems

“…This has been exemplified in (Vedral, 2003) (see also Shi, 2004) for the case where the scattering strength is independent of the momentum transfer q:…”

“…This line has been followed in (Shi, 2004; for analyzing a connection to BCS superconductivity and also to the phenomenon of η-pairing, a possible scenario for high T c superconductivity (see also Fan et al, 2004;Vedral, 2004a,b). Such states appear as eigenstates of the Hubbard model with off diagonal long range order (see III.B).…”

“…The concurrence of the two qubits represented by the modes k and −k has been found to be a monotonically increasing function of the order parameter; it drops to zero significantly before the critical temperature is reached, though. For electrons with spin, a connection between the BCS order parameter and the local von Neumann entropy in the particle number projected BCS ground state has been proposed by Shi, 2004 (see also (Gedik, 2002)). …”

Signatures of metal-insulator and topological phase transitions in the entanglement of one-dimensional disordered fermions

“…For example, the spatial entanglement of quantum states reveals the non-local properties of many-body wave functions 8,9 which naturally suggests its usefulness for identifying localized and delocalized states. Additionally, entanglement is particularly useful in capturing universal properties that cannot be obtained through the expectation value of local operators 10,11 .…”

Relation between two measures of entanglement in spin-1∕2and spinless fermion quantum chain systems

“…With these recognitions, it is found that modes, rather than particle labels, are universal concepts to properly describe the quantum entanglement of indistinguishable * Electronic address: songtc@nankai.edu.cn particle systems. This idea is used to characterize the ground-state entanglement of the BCS model [24] and superconductivity [23,29], and also it is investigated to derive the interesting relation between entanglement and quantum state transfer [30]. Some related works were also proposed by different authors [25,26,27,28] recently.…”