Reflected entropy, symmetries and free fermions
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Springer
Received: April 13, 2020 Accepted: May 5, 2020 Published: May 22, 2020
Pablo Bueno and Horacio Casini Instituto Balseiro, Centro At´ omico Bariloche, 8400-S.C. de Bariloche, R´ıo Negro, Argentina
E-mail: [email protected], [email protected] Abstract: Exploiting the split property of quantum field theories (QFTs), a notion of von Neumann entropy associated to pairs of spatial subregions has been recently proposed both in the holographic context — where it has been argued to be related to the entanglement wedge cross section — and for general QFTs. We 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. This turns out to be given by the full-theory reflected entropy minus an entropy associated to the expectation value of the “twist” operator implementing the symmetry operation. Then we show that the reflected entropy for Gaussian fermion systems can be simply written in terms of correlation functions and we evaluate it numerically for two intervals in the case of a two-dimensional Dirac field as a function of the conformal cross-ratio. Finally, we explain how the aforementioned twist operators can be constructed and we compute the corresponding expectation value and reflected entropy numerically in the case of the Z2 bosonic subalgebra of the Dirac field. Keywords: Conformal Field Theory, Global Symmetries ArXiv ePrint: 2003.09546
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2020)103
JHEP05(2020)103
Reflected entropy, symmetries and free fermions
Contents 1 Introduction
1
2 Symmetries, twist operators, and type-I entropy
4 9 9 12 13 16 18
4 Twist operators 4.1 Type-I entropy for the bosonic subalgebra
21 23
5 Final comments
25
1
Introduction
In the context of quantum field theory (QFT), the entanglement entropy (EE) of spatial subregions is not a well-defined quantity. This is because as the cutoff is removed, more and more entanglement in ultraviolet modes across the surface is added up, leading to divergences. For the continuum model itself, the necessity of these divergences can be understood from a different perspective. Operator algebras attached to regions are typeIII von Neumann algebras. These are mathematical objects which (intrinsically) do not admit a well defined entropy — see e.g., [1, 2]. By the same reason, without a cutoff, a region and its complement cannot be associated with a tensor product decomposition of the Hilbert space. This tensor produt would give place to type-I factors — the algebras of operators acting on each of the Hilbert space factors in the tensor product — instead of type-III ones. Alternatively to the EE, there exist other statistical quantities that can be studied and which are finite in the continuum theory. A prototypical example is the mutual information I(A, B), which, as opposed to
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