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arXiv:1310.4886 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Disk entanglement entropy for a Maxwell field

César A. Agón, Matthew Headrick, Daniel Louis Jafferis, Skyler Kasko

Published Jan 28, 2014• 56 citations
arXivHTMLPDF

Abstract

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 ${S}^{3}$ 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.

DOI
10.1103/physrevd.89.025018

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In three dimensions, the pure Maxwell theory with a compact U(1)U(1)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)F(r)F(r) decreases monotonically with the radius rrr as predicted by the F-theorem, interpolating between a logarithmic growth for small rrr (matching the behavior of the S3{S}^{3}S3 free energy) and a constant at large rrr (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>1n>1n>1 are given by a sum over winding sectors, leading to a Riemann-Siegel theta function. The extrapolation to n=1n=1n=1, to obtain the von Neumann entropy, is done by analytic continuation in the large- and small-rrr 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.
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We develop a framework of calculating entanglement entropy for non-conformal field theories with the use of the dilaton effective action. To illustrate it, we locate a theory on a cylinder R×S2 \mathbb{R}\times {\mathbb{S}}^2 R×S2 and compute entanglement entropy of a cap-like region perturbatively with respect to the mass for a free massive scalar field. A renormalized entanglement entropy (REE) is proposed to regularize the ultraviolet divergence on the cylinder. We find that the REE decreases monotonically both in the small and large mass regions as the mass increases. We confirm all of these behaviors by the numerical calculations, which further shows the monotonic decrease of the REE in the entire renormalization group flow.
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.
We investigate the connection between entanglement entropy and the notion of irreversibility in the renormalization-group flow in the context of a simple theory for which an explicit calculation from first principles is feasible. The change of the entanglement entropy for a spherical entangling surface as its radius grows from zero to infinity corresponds to the flow from the UV to the IR. Through analytical and numerical means, we compute the entanglement entropy for a free massive scalar theory, making use of the method of correlation functions. We deduce a c -function in 1 + 1 dimensions and an a -function in 3 + 1 dimensions. Both functions are monotonic and vary continuously between one and zero, which are the expected values for this theory.
We compute the logarithmic coefficient of the entanglement entropy on a sphere for a Maxwell field in d=3+1d=3+1d=3+1 dimensions. In spherical coordinates the problem decomposes into one-dimensional ones along the radial coordinate for each angular momentum. We show that the entanglement entropy of a Maxwell field is equivalent to one of two identical massless scalars from which the mode of l=0l=0l=0 has been removed. This shows the relation ${c}_{\mathrm{log}}^{M}=2({c}_{\mathrm{log}}^{S}\ensuremath{-}{c}_{\mathrm{log}}^{{S}_{l=0}})$ between the logarithmic coefficient in the entropy for a Maxwell field clogM{c}_{\mathrm{log}}^{M}clogM​, the one for a d=3+1d=3+1d=3+1 massless scalar clogS{c}_{\mathrm{log}}^{S}clogS​, and the logarithmic coefficient clogSl=0{c}_{\mathrm{log}}^{{S}_{l=0}}clogSl=0​​ for a d=1+1d=1+1d=1+1 scalar with a Dirichlet boundary condition at the origin. Using the accepted values for these coefficients ${c}_{\mathrm{log}}^{S}=\ensuremath{-}1/90$ and clogSl=0=1/6{c}_{\mathrm{log}}^{{S}_{l=0}}=1/6clogSl=0​​=1/6, we get ${c}_{\mathrm{log}}^{M}=\ensuremath{-}16/45$, which coincides with Dowker's calculation, but does not match the coefficient $\ensuremath{-}\frac{31}{45}$ in the trace anomaly for a Maxwell field. We have numerically evaluated these three numbers clogM{c}_{\mathrm{log}}^{M}clogM​, clogS{c}_{\mathrm{log}}^{S}clogS​ and clogSl=0{c}_{\mathrm{log}}^{{S}_{l=0}}clogSl=0​​, verifying the relation, as well as checked that they coincide with the corresponding logarithmic term in mutual information of two concentric spheres.
The entanglement entropy in one-dimensional critical systems with boundaries has been associated with the noninteger ground-state degeneracy. This quantity, being a characteristic of boundary fixed points, decreases under renormalization group flow, as predicted by the g theorem. Here, using conformal field theory methods, we exactly calculate the entanglement entropy in the boundary Ising universality class. Our expression can be separated into the well-known bulk term and a boundary entanglement term, displaying a universal flow between two boundary conditions, in accordance with the g theorem. These results are obtained within the replica trick approach, where we show that the associated twist field, a central object generating the geometry of an nnn-sheeted Riemann surface, can be bosonized, giving simple analytic access to multiple quantities of interest. We argue that our result applies to other models falling into the same universality class. This includes the vicinity of the quantum critical point of the two-channel Kondo model, allowing one to track in real space the presence of a region containing one-half of a qubit with entropy 12log(2)\frac{1}{2}log(2)21​log(2), associated with a free local Majorana fermion.