Non-local spin entanglement in a fermionic chain

TL;DR

This paper investigates the entanglement properties of two spins in a fermionic chain, revealing how external magnetic fields and spin-orbit coupling influence their quantum correlations and entropy at various distances.

Contribution

It provides a detailed analysis of the two-spin density matrix in a lattice fermionic system, highlighting the effects of magnetic fields and spin-orbit coupling on entanglement and entropy.

Findings

Maximally entangled X-state form at finite distance R

Von-Neumann entropy saturates at 2 ln 2 for large R in spin-degenerate systems

Spin-orbit coupling does not alter the asymptotic entropy results

Abstract

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 spinful Fermi sea, can be obtained from the two-electron density matrix following the framework prescribed in Phys. Rev. A 69, 054305 (2004). We note that the single spin density matrix (SSDM) obtained from this TSDM for generic spin-degenerate systems of free fermions is always pinned to the maximally mixed state $i.e.$ $(1/2) \ \mathbb{I}$, 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 $R\rightarrow 0$ limit while it saturates to an X-state with largest allowed value of von-Neumann entropy of $2 \ln2$ as $R \rightarrow \infty$ 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 $R\rightarrow \infty$ limit becomes a function of the chemical potential and it saturate to $2 \ln2$ 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 with previous works which were based on continuum models owing to physics which stem from the lattice model.