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arXiv:2307.00057 · Physical review. D/Physical review. D.

Entanglement entropy of a Rarita-Schwinger field in a sphere

Valentín Benedetti, Lucas Daguerre

Published Oct 11, 2023• 3 citations
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Abstract

We study the universal logarithmic coefficient of the entanglement entropy in a sphere for free fermionic field theories in a $d=4$ Minkowski spacetime. As a warm-up, we revisit the free massless spin-$1/2$ field case by employing a dimensional reduction to the $d=2$ 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-$3/2$ field, the Rarita-Schwinger field, we also perform a dimensional reduction to the half-line. The reduced Hamiltonian coincides with the spin-$1/2$ 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.

DOI
10.1103/physrevd.108.086015

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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\ensuremath{-}11/90\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\ensuremath{-}71/90\ensuremath−71/90. We discuss the physical interpretation of the universal logarithmic coefficient for free higher-spin field theories without a stress-energy tensor.
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 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.
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.
The entanglement entropy of an arbitrary spacetime region A in a three-dimensional conformal field theory (CFT) contains a constant universal coefficient, F(A). For general theories, the value of F(A) is minimized when A is a round disk, F_{0}, and in that case it coincides with the Euclidean free energy on the sphere. We conjecture that, for general CFTs, the quantity F(A)/F_{0} is bounded above by the free scalar field result and below by the Maxwell field one. We provide strong evidence in favor of this claim and argue that an analogous conjecture in the four-dimensional case is equivalent to the Hofman-Maldacena bounds. In three dimensions, our conjecture gives rise to similar bounds on the quotients of various constants characterizing the CFT. In particular, it implies that the quotient of the stress-tensor two-point function coefficient and the sphere free energy satisfies C_{T}/F_{0}≤3/(4π^{2}log2-6ζ[3])≃0.14887 for general CFTs. We verify the validity of this bound for free scalars and fermions, general O(N) and Gross-Neveu models, holographic theories, N=2 Wess-Zumino models and general ABJM theories.
We consider the universal logarithmic divergent term in the entanglement entropy of gauge fields in the Minkowski vacuum with an entangling sphere. Employing the mapping in arXiv:1102.0440, we analyze the corresponding thermal entropy on open Einstein universe and on the static patch of de Sitter. Using the heat kernel of the vector Laplacian we resolve a discrepancy between the free field calculation and the expected Euler conformal anomaly. The resolution suggests a modification of the well known formulas for the vacuum expectation value of the spin-1 energy-momentum tensor on conformally flat space-times.
We calculate the universal contribution to the $\ensuremath{\alpha}$-R\'enyi entropy from a cubic trihedral corner in the boundary of the entangling region in 3+13+13+1 dimensions for a massless free scalar. The universal number ${v}_{\ensuremath{\alpha}}$ is manifest as the coefficient of a scaling term that is logarithmic in the size of the entangling region. Our numerical calculations find that this universal coefficient has both larger magnitude and the opposite sign to that induced by a smooth spherical entangling boundary in 3+13+13+1 dimensions, for which there is a well-known subleading logarithmic scaling. Despite these differences, up to the uncertainty of our finite-size lattice calculations, the functional dependence of the trihedral coefficient ${v}_{\ensuremath{\alpha}}$ on the R\'enyi index $\ensuremath{\alpha}$ is indistinguishable from that for a sphere, which is known analytically for a massless free scalar. We comment on the possible source of this $\ensuremath{\alpha}$ dependence arising from the general structure of (3+1)(3+1)(3+1)-dimensional conformal field theories, and suggest calculations past the free scalar which could further illuminate the general structure of the trihedral divergence in the R\'enyi entropy.