  A basic diagnostic of entanglement in mixed quantum states is known as the
positive partial transpose (PT) criterion. Such criterion is based on the
observation that the spectrum of the partially transposed density matrix of an
entangled state contains negative eigenvalues, in turn, used to define an
entanglement measure called the logarithmic negativity. Despite the great
success of logarithmic negativity in characterizing bosonic many-body systems,
generalizing the operation of PT to fermionic systems remained a technical
challenge until recently when a more natural definition of PT for fermions that
accounts for the Fermi statistics has been put forward. In this paper, we study
the many-body spectrum of the reduced density matrix of two adjacent intervals
for one-dimensional free fermions after applying the fermionic PT. We show that
in general there is a freedom in the definition of such operation which leads
to two different definitions of PT: the resulting density matrix is Hermitian
in one case, while it becomes pseudo-Hermitian in the other case. Using the
path-integral formalism, we analytically compute the leading order term of the
moments in both cases and derive the distribution of the corresponding
eigenvalues over the complex plane. We further verify our analytical findings
by checking them against numerical lattice calculations.
