  We reconsider the moments of the reduced density matrix of two disjoint
intervals and of its partial transpose with respect to one interval for
critical free fermionic lattice models. It is known that these matrices are
sums of either two or four Gaussian matrices and hence their moments can be
reconstructed as computable sums of products of Gaussian operators. We find
that, in the scaling limit, each term in these sums is in one-to-one
correspondence with the partition function of the corresponding conformal field
theory on the underlying Riemann surface with a given spin structure. The
analytical findings have been checked against numerical results for the Ising
chain and for the XX spin chain at the critical point.
