Non-local spin entanglement in a fermionic chain

An effective two-spin density matrix (TSDM) for a pair of spin-1/2 degree of freedom, residing at a distance of R in a spinfull Fermi sea, can be obtained from the twoelectron density matrix following the framework prescribed in Oh and Kim (Phys Rev A 69:054305, 2004. https://doi.org/10.1103/PhysRevA.69.054305) . We note that the single-spin density matrix (SSDM) obtained from this TSDM for generic spindegenerate systems of free fermions is always pinned to the maximally mixed state, i.e. (1/2) II, independent of the distance R, while the TSDM confirms to the form for the set of maximally entangled mixed state (the so-called X-state) at finite R. The X-state reduces to a pure state (a singlet) in the R -> 0 limit, while it saturates to an X-state with the largest allowed value of von-Neumann entropy of 2 ln 2 as R ->infinity independent of the value of chemical potential. However, once an external magnetic field is applied to lift the spin-degeneracy, we find that the von-Neumann entropy of SSDM becomes a function of the distance R between the two spins. We also show that the von-Neumann entropy of TSDM in the R ->infinity limit becomes a function of the chemical potential and it saturates to 2 ln 2 only when the band in completely filled unlike the spin-degenerate case. Finally, we extend our study to include spin-orbit coupling and show that it does effect these asymptotic results. Our findings are in sharp contrast to previous works, which were based on continuum models owing to physics which stem from the lattice model.

Automated summary

This study investigates the properties of states in a spinful Fermi sea by using the density matrix between two spin-1/2 degrees of freedom. It is found that in the case of spin degeneracy, the density matrix is always in a maximally mixed state, while in the finite distance, it conforms to a set of maximally entangled mixed state. When an external magnetic field is applied to lift the spin degeneracy, the entropy of the density matrix becomes a function of the distance between the two spins. Additionally, the entropy of the density matrix becomes a function of the chemical potential in the limit of infinite distance and saturates to the maximum allowed value only when the band is completely filled.