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arXiv:1601.06418 · Physical review. D/Physical review. D.

Anomalies, entropy, and boundaries

Dmitri V. Fursaev, Sergey N. Solodukhin

Published Apr 13, 2016• 52 citations
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Abstract

A relation between the conformal anomaly and the logarithmic term in the entanglement entropy is known to exist for CFTs in even dimensions. In odd dimensions, the local anomaly and the logarithmic term in the entropy are absent. As was observed recently, there exists a nontrivial integrated anomaly if an odd-dimensional spacetime has boundaries. We show that, similarly, there exists a logarithmic term in the entanglement entropy when the entangling surface crosses the boundary of spacetime. The relation of the entanglement entropy to the integrated conformal anomaly is elaborated for three-dimensional theories. Distributional properties of intrinsic and extrinsic geometries of the boundary in the presence of conical singularities in the bulk are established. This allows one to find contributions to the entropy that depend on the relative angle between the boundary and the entangling surface.

DOI
10.1103/physrevd.93.084021

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A relation between the conformal anomaly and the logarithmic term in the entanglement entropy is known to exist for CFTs in even dimensions. In odd dimensions, the local anomaly and the logarithmic term in the entropy are absent. As was observed recently, there exists a nontrivial integrated anomaly if an odd-dimensional spacetime has boundaries. We show that, similarly, there exists a logarithmic term in the entanglement entropy when the entangling surface crosses the boundary of spacetime. The relation of the entanglement entropy to the integrated conformal anomaly is elaborated for three-dimensional theories. Distributional properties of intrinsic and extrinsic geometries of the boundary in the presence of conical singularities in the bulk are established. This allows one to find contributions to the entropy that depend on the relative angle between the boundary and the entangling surface.
Entanglement entropy in even-dimensional conformal field theories (CFTs) contains well-known universal terms arising from the conformal anomaly. R\'enyi entropies are natural generalizations of the entanglement entropy that are much less understood. Above two spacetime dimensions, the universal terms in the R\'enyi entropies are unknown for general entangling geometries. We conjecture a new structure in the dependence of the four-dimensional R\'enyi entropies on the intrinsic and extrinsic geometry of the entangling surface. We provide evidence for this conjecture by direct numerical computations in the free scalar and fermion field theories. The computation involves relating the four-dimensional free massless R\'enyi entropies across cylindrical entangling surfaces to corresponding three-dimensional massive R\'enyi entropies across circular entangling surfaces. Our numerical technique also allows us to directly probe other interesting aspects of three-dimensional R\'enyi entropy, including the massless renormalized R\'enyi entropy and calculable contributions to the perimeter law.
We study the entanglement entropy of theories that are derived from relevant perturbation of given CFTs for regions with a singular boundary by using the AdS/CFT correspondence. In the smooth case, it is well known that a relevant deformation of the boundary theory by the relevant operator with scaling dimension Δ=d+22Δ=\frac{d+2}{2}Δ=2d+2​ generates a logarithmic universal term to the entanglement entropy. As the smooth case, when the boundary CFT deformed by a relevant operator, we find that the entanglement entropy of singular surface also contains a new logarithmic term which is due to relevant perturbation of the conformal field theory, and depends on the scaling dimension of relevant operator. We also find for extended singular surfaces, cn×Rmc_{n}\times R^{m}cn​×Rm, as well as logarithmic term, the new universal double logarithmic terms may appear depending on the scaling dimension of relevant operator and spacetime dimensions. These new terms are due to relevant perturbation of the boundary theory.
A bstract We study the structure of divergences and universal terms of the entanglement and Rényi entropies for singular regions. First, we show that for (3 + 1)-dimensional free conformal field theories (CFTs), entangling regions emanating from vertices give rise to a universal contribution Snuniv=−18πfb(n)∫γk2log⁡2(R/δ) {S}_n^{\mathrm{univ}}=-\frac{1}{8\pi }{f}_b(n){\int}_{\gamma }{k}^2{\log}^2\left(R/\delta \right) Snuniv​=−8π1​fb​(n)∫γ​k2log2(R/δ) S n univ = − 1 8 π f b n ∫ γ k 2 log 2 R / δ , where γ is the curve formed by the intersection of the entangling surface with a unit sphere centered at the vertex, and k the trace of its extrinsic curvature. While for circular and elliptic cones this term reproduces the general-CFT result, it vanishes for polyhedral corners. For those, we argue that the universal contribution, which is logarithmic, is not controlled by a local integral, but rather it depends on details of the CFT in a complicated way. We also study the angle dependence for the entanglement entropy of wedge singularities in 3+1 dimensions. This is done for general CFTs in the smooth limit, and using free and holographic CFTs at generic angles. In the latter case, we show that the wedge contribution is not proportional to the entanglement entropy of a corner region in the (2 + 1)-dimensional holographic CFT. Finally, we show that the mutual information of two regions that touch at a point is not necessarily divergent, as long as the contact is through a sufficiently sharp corner. Similarly, we provide examples of singular entangling regions which do not modify the structure of divergences of the entanglement entropy compared with smooth surfaces.
We study whether the relations between the Weyl anomaly, entanglement entropy (EE), and thermal entropy of a two-dimensional (2D) conformal field theory (CFT) extend to 2D boundaries of 3D CFTs, or 2D defects of D≥3 CFTs. The Weyl anomaly of a 2D boundary or defect defines two or three central charges, respectively. One of these, b, obeys a c theorem, as in 2D CFT. For a 2D defect, we show that another, d_{2}, interpreted as the defect's "conformal dimension," must be non-negative if the averaged null energy condition holds in the presence of the defect. We show that the EE of a sphere centered on a planar defect has a logarithmic contribution from the defect fixed by b and d_{2}. Using this and known holographic results, we compute b and d_{2} for 1/2-Bogomol'nyi-Prasad-Sommerfield surface operators in the maximally supersymmetric (SUSY) 4D and 6D CFTs. The results are consistent with b's c theorem. Via free field and holographic examples we show that no universal "Cardy formula" relates the central charges to thermal entropy.
In the vacuum state of a CFT, the entanglement entropy of singular surfaces contains a logarithmic universal term which is only due to the singularity of the entangling surface. We consider the relevant perturbation of a three dimensional CFT for singular entangling surface. We observe that in addition to the universal term due to the entangling surface, there is a new logarithmic term which corresponds to a relevant perturbation of the conformal field theory with a coefficient depending on the scaling dimension of the relevant operator. We also find a new power law divergence in the holographic entanglement entropy. In addition, we study the effect of a relevant perturbation in the Gauss-Bonnet gravity for a singular entangling surface. Again a logarithmic term shows up. This new term is proportional to both the dimension of the relevant operator and the Gauss-Bonnet coupling. We also introduce the renormalized entanglement entropy for a kink region which in the UV limit reduces to a universal positive finite term.
Entanglement entropy in holographic conformal field theories with boundaries of generic spacetime signature cannot be described entirely within the standard holographic duality to a real bulk spacetime. We show, both in static and time-dependent setups, that an exact match with the conformal field theory predictions in two boundary dimensions entails an extension of the dual spacetime to complex coordinates. As a consequence, relevant bulk objects such as Ryu-Takayanagi surfaces and branes are in general complex, yet anchored on a real asymptotic boundary preserving the physical meaning of field-theoretical quantities. We also provide a third numerical check of the holographic prediction with a two-dimensional system of free Gaussian fermions exhibiting the same asymptotic regimes across different entanglement phases.