Entanglement negativity in free-fermion systems

We derive a general formula of the logarithmic negativity in free-fermion systems, using the overlap matrix to construct the partially transposed reduced density matrix $\rho_{A}^{T_{A_2}}$ of a subsystem $A = A_1 \bigcup A_2$. In particular, we consider the negativity between two adjacent or disjoint regions in three systems: a homogeneous one-dimensional chain, the dimerized Su-Schrieffer-Heeger model, and the integer Quantum Hall state. For the negativity of two adjacent intervals in a homogeneous one-dimensional gas, we find agreement with the conformal field theory [P. Calabrese {\it et al.} Phys. Rev. Lett. {\bf 109}, 130502 (2012)]. On the other hand, the negativity for the integer quantum Hall states satisfies the area law. Our method is applicable to the study of the negativity in any free-fermion systems.