Reflected entropy, symmetries and free fermions

Entanglement entropy (EE) is an intrinsically divergent quantity in quantum field theory (QFT). Recently, a well-defined notion of von Neumann entropy associated to pairs of spatial subregions has been proposed both in the holographic context — where it has been argued to be related to the entanglement wedge cross section — and for general QFTs. I will show that, similarly to the EE, the reflected entropy for Gaussian fermion systems can be simply obtained in terms of correlation functions and I will explicitly evaluate it numerically for two intervals in the case of a two-dimensional Dirac field. Then, I will argue that the definition of this “reflected entropy” can be canonically generalized in a way which is particularly suitable for orbifold theories — those obtained by restricting the full algebra of operators to those which are neutral under a global symmetry group. I will evaluate explicitly this new "type-I entropy” for the $\mathbb{Z}_2$ bosonic subalgebra of the Dirac field. A key role in the construction is played by type-I von Neumann algebras, which differ from the usual type-III algebras associated to spatial regions in QFT. Using the free-fermions case, I will perform very explicit comparisons between both types of algebras.