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.
A definition for the entanglement entropy in a gauge theory was given recently in arXiv:1501.02593. Working on a spatial lattice, it involves embedding the physical state in an extended Hilbert space obtained by taking the tensor product of the Hilbert space of states on each link of the lattice. This extended Hilbert space admits a tensor product decomposition by definition and allows a density matrix and entanglement entropy for the set of links of interest to be defined. Here, we continue the study of this extended Hilbert space definition with particular emphasis on the case of Non-Abelian gauge theories. We extend the electric centre definition of Casini, Huerta and Rosabal to the Non-Abelian case and find that it differs in an important term. We also find that the entanglement entropy does not agree with the maximum number of Bell pairs that can be extracted by the processes of entanglement distillation or dilution, and give protocols which achieve the maximum bound. Finally, we compute the topological entanglement entropy which follows from the extended Hilbert space definition and show that it correctly reproduces the total quantum dimension in a class of Toric code models based on Non-Abelian discrete groups.
A bstract Despite the seeming simplicity of the theory, calculating (and even defining) entanglement entropy for the Maxwell theory of a U(1) gauge field in (3+1) dimensions has been the subject of controversy. It is generally accepted that the ground state entanglement entropy for a region of linear size L behaves as an area law with a subleading logarithm, S = αL 2 − γ log L . While the logarithmic coefficient γ is believed to be universal, there has been disagreement about its precise value. After carefully accounting for subtle boundary corrections, multiple analyses in the high energy literature have converged on an answer related to the conformal trace anomaly, which is only sensitive to the local curvature of the partition. In contrast, a condensed matter treatment of the problem yielded a topological contribution which is not captured by the conformal field theory calculation. In this perspective piece, we review aspects of the various calculations and discuss the resolution of the discrepancy, emphasizing the important role played by charged states (the “extended Hilbert space”) in defining entanglement for a gauge theory. While the trace anomaly result is sufficient for a strictly pure gauge field, coupling the gauge field to dynamical charges of mass m gives a topological contribution to γ which survives even in the m → ∞ limit. For many situations, the topological contribution from dynamical charges is physically meaningful and should be taken into account. We also comment on other common issues of entanglement in gauge theories, such as entanglement distillation, algebraic definitions of entanglement, and gauge-fixing procedures.
We propose a definition for the entanglement entropy of a gauge theory on a spatial lattice. Our definition applies to any subset of links in the lattice, and is valid for both Abelian and Non-Abelian gauge theories. For
ZN and U(1) theories, without matter, our definition agrees with a particular case of the definition given by Casini, Huerta and Rosabal. We also argue that in general, both for Abelian and Non-Abelian theories, our definition agrees with the entanglement entropy calculated using a definition of the replica trick. Our definition, however, does not agree with some standard ways to measure entanglement, like the number of Bell pairs which can be produced by entanglement distillation.
We study the entanglement entropy (EE) for pure gauge theories in 1[Formula: see text]+[Formula: see text]1 dimensions with the lattice regularization. Using the definition of the EE for lattice gauge theories proposed in a previous paper,1 we calculate the EE for arbitrary pure as well as mixed states in terms of eigenstates of the transfer matrix in (1[Formula: see text]+[Formula: see text]1)-dimensional lattice gauge theory. We find that the EE of an arbitrary pure state does not depend on the lattice spacing, thus giving the EE in the continuum limit, and show that the EE for an arbitrary pure state is independent of the real (Minkowski) time evolution. We also explicitly demonstrate the dependence of EE on the gauge fixing at the boundaries between two subspaces, which was pointed out for general cases in the paper. In addition, we calculate the EE at zero as well as finite temperature by the replica method, and show that our result in the continuum limit corresponds to the result obtained before in the continuum theory, with a specific value of the counterterm, which is otherwise arbitrary in the continuum calculation. We confirm the gauge dependence of the EE also for the replica method.
In three dimensions, the pure Maxwell theory with a compact
U(1) gauge group is dual to a free compact scalar, and it flows from the Maxwell theory with a noncompact gauge group in the ultraviolet to a noncompact free massless scalar theory in the infrared. We compute the vacuum disk entanglement entropy all along this flow and show that the renormalized entropy
F(r) decreases monotonically with the radius
r as predicted by the F-theorem, interpolating between a logarithmic growth for small
r (matching the behavior of the
S3 free energy) and a constant at large
r (equal to the free energy of the conformal scalar). The calculation is carried out by the replica trick, employing the scalar formulation of the theory. The R\'enyi entropies for
n>1 are given by a sum over winding sectors, leading to a Riemann-Siegel theta function. The extrapolation to
n=1, to obtain the von Neumann entropy, is done by analytic continuation in the large- and small-
r limits and by a numerical extrapolation method at intermediate values. We also compute the leading contribution to the renormalized entanglement entropy of the compact free scalar in higher dimensions. Finally, we point out some interesting features of the reduced density matrix for the compact scalar, and its relation to that for the noncompact theory.
We extend the entanglement entropy calculation performed in the seminal paper by Srednicki (Phys Rev Lett 71:666, 1993) for free real massive scalar field theories in
1+1 1 + 1 ,
2+1 2 + 1 and
3+1 3 + 1 dimensions. We show that the inverse of the scalar field mass can be used as an expansion parameter for a perturbative calculation of the entanglement entropy. We perform the calculation for the ground state of the system and for a spherical entangling surface at third order in the inverse mass expansion. The calculated entanglement entropy contains a leading area law term, as well as subleading terms that depend on the regularization scheme, as expected. Universal terms are non-perturbative effects in this approach. Interestingly, this perturbative expansion can be used to approximate the coefficient of the area law term, even in the case of a massless scalar field in
2+1 2 + 1 and
3+1 3 + 1 dimensions. The presented method provides the spectrum of the reduced density matrix as an intermediate result, which is an important advantage in comparison to the replica trick approach. Our perturbative expansion underlines the relation between the area law and the locality of the underlying field theory.