  Logarithmic negativity is a widely used entanglement measure in quantum
information theories, which can also be efficiently computed in quantum
many-body systems by replica trick or by relating to correlation matrices. In
this paper, we demonstrate that in free-fermion systems with conserved charge,
R\'enyi and logarithmic negativity can be expanded by connected charge
correlators, analogous to the case for entanglement entropy in the context of
full counting statistics (FCS). We confirm the rapid convergence of this
expansion in random all-connected Hamiltonian through numerical verification,
especially for systems with only local hopping. We find that the replica trick
that get logarithmic negativity from the limit of R\'enyi negativity is valid
in this method only for translational invariant systems. Using this expansion,
we analyze the scaling behavior of negativity in extensive free-fermion
systems. In particular, in 1+1 dimensional free-fermion systems, we observe
that the scaling behavior of negativity from our expansion is consistent with
known results from the method with Toeplitz matrix. These findings provide
insights into the entanglement properties of free-fermion systems, and
demonstrate the efficacy of the expansion approach in studying entanglement
measures.
