  In this paper, we calculate the entanglement negativity in free-fermion
systems by use of the overlap matrices. For a tripartite system, if the ground
state can be factored into triples of modes, we show that the partially
transposed reduced density matrix can be factorized and the entanglement
negativity has a simple form. However, the factorability of the ground state in
a tripartite system does not hold in general. In this situation, the partially
transposed reduced density matrix can be expressed in terms of the Kronecker
product of matrices. We explicitly compute the entanglement negativity for the
Su-Schrieffer-Heeger model, the integer Quantum Hall state, and a homogeneous
one-dimensional chain. We find that the entanglement negativity for the integer
quantum Hall states shows an area law behavior. For the entanglement negativity
of two adjacent intervals in a homogeneous one-dimensional gas, we find
agreement with the conformal field theory. Our method provides a numerically
feasible way to study the entanglement negativity in free-fermion systems.
