Entanglement entropy in (3 + 1)-d free U(1) gauge theory

Abstract

We consider the entanglement entropy for a free U(1) theory in 3+1 dimensions in the extended Hilbert space definition. By taking the continuum limit carefully we obtain a replica trick path integral which calculates this entanglement entropy. The path integral is gauge invariant, with a gauge fixing delta function accompanied by a Faddeev -Popov determinant. For a spherical region it follows that the result for the logarithmic term in the entanglement, which is universal, is given by the a anomaly coefficient. We also consider the extractable part of the entanglement, which corresponds to the number of Bell pairs which can be obtained from entanglement distillation or dilution. For a spherical region we show that the coefficient of the logarithmic term for the extractable part is different from the extended Hilbert space result. We argue that the two results will differ in general, and this difference is accounted for by a massless scalar living on the boundary of the region of interest.

Entanglement entropy and superselection sectors. Part I. Global symmetries

“…In relation to the order parameter, notice first that the two states appearing in the relative entropy, namely ω 12 and ω 12 • E 12 , are invariant under the action of the global symmetry group. Second, we have that E(F 1 ⊗ F 2 ) = (O (12) ) , where the commutants are taken in O. Therefore, because of (3.7), we conclude that the order parameter can be computed equivalently as:…”

“…First we have a general property valid for any state φ and conditional expectation E : A → B that preserves the trace tr A (x) = tr A (E(x)) in matrix algebras. 12 This is not the general case for conditional expectations but the conditional expectation (3.5) preserves the trace in any subalgebra since it is an average over automorphisms. In this case we have…”

“…In this sense, local algebras may contain a center related to ambiguities on the choice of algebra at the boundary of the region in a lattice model. This type of ambiguities has attracted attention especially in relation with gauge models (for the discussion around this topic see for example [11][12][13][14][15][16][17][18][19]). However, this kind of local ambiguities does not survive the continuum limit, leaving the mutual information as a well-defined quantity [10].…”

Logarithmic coefficient of the entanglement entropy of a Maxwell field

“…where G E and G B are the radial electric and magnetic correlator kernels on the surface. The calculation of this type of contributions was done [22] in the context of the electric center contribution to the entropy. 9 The result for the universal piece is 1 3 log(R/ ).…”

“…As discussed in the previous section, with respect to the calculation in[22], we have a difference in an…”