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arXiv:1512.06182 · Physical review. D/Physical review. D.

Entanglement entropy of a Maxwell field on the sphere

Horacio Casini, Marina Huerta

Published May 20, 2016• 43 citations
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Abstract

We compute the logarithmic coefficient of the entanglement entropy on a sphere for a Maxwell field in $d=3+1$ 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 $l=0$ 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 ${c}_{\mathrm{log}}^{M}$, the one for a $d=3+1$ massless scalar ${c}_{\mathrm{log}}^{S}$, and the logarithmic coefficient ${c}_{\mathrm{log}}^{{S}_{l=0}}$ for a $d=1+1$ scalar with a Dirichlet boundary condition at the origin. Using the accepted values for these coefficients ${c}_{\mathrm{log}}^{S}=\ensuremath{-}1/90$ and ${c}_{\mathrm{log}}^{{S}_{l=0}}=1/6$, 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 ${c}_{\mathrm{log}}^{M}$, ${c}_{\mathrm{log}}^{S}$ and ${c}_{\mathrm{log}}^{{S}_{l=0}}$, verifying the relation, as well as checked that they coincide with the corresponding logarithmic term in mutual information of two concentric spheres.

DOI
10.1103/physrevd.93.105031

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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 clogM=2(clogS\ensuremath−clogSl=0){c}_{\mathrm{log}}^{M}=2({c}_{\mathrm{log}}^{S}\ensuremath{-}{c}_{\mathrm{log}}^{{S}_{l=0}})clogM​=2(clogS​\ensuremath−clogSl=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 clogS=\ensuremath−1/90{c}_{\mathrm{log}}^{S}=\ensuremath{-}1/90clogS​=\ensuremath−1/90 and clogSl=0=1/6{c}_{\mathrm{log}}^{{S}_{l=0}}=1/6clogSl=0​​=1/6, we get clogM=\ensuremath−16/45{c}_{\mathrm{log}}^{M}=\ensuremath{-}16/45clogM​=\ensuremath−16/45, which coincides with Dowker's calculation, but does not match the coefficient \ensuremath−3145\ensuremath{-}\frac{31}{45}\ensuremath−4531​ 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.
We calculate numerically the logarithmic contribution to the entanglement entropy of a cylindrical region in three spatial dimensions for a Maxwell field. Our result does not agree with the analytical predictions concerning any conformal field theory in four dimensions according to which the coefficient is universal and proportional to the type c conformal anomaly. In cylindrical coordinates the problem decomposes into one dimensional ones along the radial coordinate for each angular momentum. The entanglement entropy of a Maxwell field is equivalent to the one of two identical decoupled scalars with an extra self interaction term.
We compute the entanglement entropy of a massless spin 2 field in a sphere in flat Minkowski space. We describe the theory with a linearized metric perturbation field ${h}_{\ensuremath{\mu}\ensuremath{\nu}}$ and decompose the field into independent modes using tensor spherical harmonics. We fix the gauge such that (a) the two dynamical modes for each angular momentum decouple and have the dynamics of scalar spherical modes, and (b) the gauge-fixed field degrees of freedom inside the sphere represent gauge invariant operators of the theory localized in the same region. In this way the entanglement entropy turns out to be equivalent to the one of a pair of free massless scalars where the contributions of the l=0l=0l=0 and l=1l=1l=1 modes have been subtracted. The result for the coefficient of the universal logarithmic term is $\ensuremath{-}61/45$ and coincides with that computed using a regularization based on the mutual information.
We study the universal logarithmic coefficient of the entanglement entropy in a sphere for free fermionic field theories in a d=4d=4d=4 Minkowski spacetime. As a warm-up, we revisit the free massless spin-1/21/21/2 field case by employing a dimensional reduction to the d=2d=2d=2 half-line and a subsequent numerical real-time computation on a lattice. Surprisingly, the area coefficient diverges for a radial discretization but is finite for a geometric regularization induced by the mutual information. The resultant universal logarithmic coefficient $\ensuremath{-}11/90$ is consistent with the literature. For the free massless spin-3/23/23/2 field, the Rarita-Schwinger field, we also perform a dimensional reduction to the half-line. The reduced Hamiltonian coincides with the spin-1/21/21/2 one, except for the omission of the lowest total angular momentum modes. This gives a universal logarithmic coefficient of $\ensuremath{-}71/90$. We discuss the physical interpretation of the universal logarithmic coefficient for free higher-spin field theories without a stress-energy tensor.
Abstract We consider the linearised graviton in 4d Minkowski space and decompose it into tensor spherical harmonics and fix the gauge. The Gauss law of gravity implies that certain radial components of the Riemann tensor of the graviton on the sphere labels the superselection sectors for the graviton. We show that among these 6 normal components of the Riemann tensor, 2 are related locally to the algebra of gauge-invariant operators in the sphere. From the two-point function of these components of the Riemann tensor on S2 we compute the logarithmic coefficient of the entanglement entropy of these superselection sectors across a spherical entangling surface. For sectors labelled by each of the two components of the Riemann tensor these coefficients are equal and their total contribution is given by −163 -\frac{16}{3} −316​ − 16 3 . We observe that this coefficient coincides with that extracted from the edge partition function of the massless spin-2 field on the 4-sphere when written in terms of its Harish-Chandra character. As a preliminary step, we also evaluate the logarithmic coefficient of the entanglement entropy from the superselection sectors labelled by the radial component of the electric field of the U(1) theory in even d dimensions. We show that this agrees with the corresponding coefficient of the edge Harish-Chandra character of the massless spin-1 field on Sd.
We study entanglement entropy (EE) for a Maxwell field in (2+1)(2+1)(2+1) dimensions. We do numerical calculations in two-dimensional lattices. This gives a concrete example of the general results of our recent work [1] on entropy for lattice gauge fields using an algebraic approach. To evaluate the entropies we extend the standard calculation methods for the entropy of Gaussian states in canonical commutation algebras to the more general case of algebras with center and arbitrary numerical commutators. We find that while the entropy depends on the details of the algebra choice, mutual information has a well defined continuum limit as predicted in [1]. We study several universal terms for the entropy of the Maxwell field and compare with the case of a massless scalar field. We find some interesting new phenomena: an ``evanescent'' logarithmically divergent term in the entropy with topological coefficient which does not have any correspondence with ultraviolet entanglement in the universal quantities, and a nonstandard way in which strong subadditivity is realized. Based on the results of our calculations we propose a generalization of strong subadditivity for the entropy on some algebras that are not in tensor product.