  The pairwise entanglement, measured by concurrence and geometric phase in
$d=2,3$-dimensional free-fermion lattice systems have been studied for the
ground state in this paper. Their derivation with respect to the external
parameter show the singularity closed to the phase transition points, and can
be used to detect the phase transition in this model. Furthermore our studies
show for the free-fermion model that both concurrence and geometric phase shows
the intimate connection with the correlation functions. The possible relation
between concurrence and geometric phase has been also discussed.
