  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.
