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arXiv:1412.2568 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Renormalized entanglement entropy for BPS black branes

Arpan Bhattacharyya, S. Shajidul Haque, Álvaro Véliz-Osorio

Published Feb 19, 2015• 7 citations
arXivHTMLPDF

Abstract

We compute the renormalized entanglement entropy (REE) for BPS black solutions in $\mathcal{N}=2$, four-dimensional gauged supergravity. We find that this quantity decreases monotonically with the size of the entangling region until it reaches a critical point, then increases and approaches the entropy density of the brane. This behavior can be understood as a consequence of the renormalized entanglement entropy being driven by two competing factors, namely, entanglement and the mixedness of the black brane. In the UV, entanglement dominates, whereas in the IR, the mixedness takes over.

DOI
10.1103/physrevd.91.045026

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We compute the renormalized entanglement entropy (REE) for BPS black solutions in N=2\mathcal{N}=2N=2, four-dimensional gauged supergravity. We find that this quantity decreases monotonically with the size of the entangling region until it reaches a critical point, then increases and approaches the entropy density of the brane. This behavior can be understood as a consequence of the renormalized entanglement entropy being driven by two competing factors, namely, entanglement and the mixedness of the black brane. In the UV, entanglement dominates, whereas in the IR, the mixedness takes over.
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 study the entanglement entropy for open bosonic strings on multiple Dp -branes by using the covariant open string field theory. Choosing one of the spatial coordinates which are tangential to the hyperplane on which Dp -branes are located, we divide the hyperplane into two halves. By using the string wavefunction in the Fock space representation, we evaluate the entanglement entropy. The entanglement entropy is found to be proportional to the area of ( p − 1 ) -dimensional boundary of the bipartite hyperplanes and divergent in the ultraviolet (UV) region as well as in the infrared (IR) region. However, the leading divergences are mainly due to tachyon contributions to the entanglement entropy, which may be absent in supersymmetric string theories . Apart from the divergences thanks to tachyons, the entanglement entropy for open bosonic strings on Dp -branes is finite for 2 ≤ p ≤ d critical − 2 and logarithmically divergent for p = 1 , d critical − 1 .
Holographic entanglement entropy (EE) and its renormalized form in AdS solitons with gauge potential are computed across a range of dimensions. The renormalized EE is identified as a cutoff-independent universal component of EE. By taking into account Kaluza-Klein reduction and the constraints of the low-energy limit, the (d − 1)-dimensional renormalized EE is deduced from the odd-dimensional counterpart. The region associated with the shrinking circle, which is examined at large values of l is indicative of this relationship. Transitions of the minimal surface are from a disk-shaped to a cylindrical configuration as l increases. A quantum phase transition occurs at a critical subregion size, with the renormalized EE exhibiting non-monotonic behavior in the vicinity of this size. In all dimensions, massive modes become decoupled at the low energy limit, whereas degrees of freedom that include Wilson lines influence the behavior at smaller energy scales.
A recent proposal by Ryu and Takayanagi for a holographic interpretation of entanglement entropy in conformal field theories dual to supergravity on anti-de Sitter space is generalized to include entanglement entropy of black holes living on the boundary of anti-de Sitter space. The generalized proposal is verified in boundary dimensions d=2 and d=4 for both the uv-divergent and uv-finite terms. In dimension d=4 an expansion of entanglement entropy in terms of size L of the subsystem outside the black hole is considered. A new term in the entropy of dual strongly coupled conformal-field theory, which universally grows as L(2)lnL and is proportional to the value of the obstruction tensor at the black hole horizon, is predicted.
The renormalization of entanglement entropy of quantum field theories is investigated in the simplest setting with a λϕ4λϕ^4λϕ4 scalar field theory. The 3+1 dimensional spacetime is separated into two regions by an infinitely flat 2-dimensional interface. The entanglement entropy of the system across the interface has an elegant geometrical interpretation using the replica trick, which requires putting the field theory on a curved spacetime background. We demonstrate that the theory, and hence the entanglement entropy, is renormalizable at order λλλ once all the relevant operators up to dimension-4 are included in the action. This exercise has a one-to-one correspondence to entanglement entropy interpretation of the black hole entropy which suggests that our treatment is sensible. Our study suggests that entanglement entropy is renormalizable and is a physical quantity.
Black holes in general relativity carry an entropy whose value is given by the Bekenstein-Hawking formula, but whose statistical origin remains obscure. Such horizons also possess an entanglement entropy, which has a clear statistical meaning but no a priori relation to the dynamics of gravity. For free minimally-coupled scalar and spinor fields, these two quantities are intimately related: the entanglement entropy is the one-loop correction to the black hole entropy due to renormalization of Newton's constant. For gauge fields, the entanglement entropy and the one-loop correction to the black hole entropy differ. This dissertation addresses two issues concerning the entanglement entropy of gauge fields, and its relation black hole entropy. First, for abelian gauge fields Kabat identified a negative divergent contribution to the black hole entropy that is not part of the entanglement entropy, known as a ``contact term''. We show that the contact term can be attributed to an ambiguous expression for the gauge field's contribution to the Wald entropy. Moreover, in two-dimensional de Sitter space, the contact term arises from an incorrect treatment of zero modes and is therefore unphysical. In a manifestly gauge-invariant reduced phase space quantization of two-dimensional gauge theory, the gauge field contribution to the entropy is positive, finite, and equal to the entanglement entropy. This suggests that the contact term in more than two dimensions may also be unphysical. Second, we consider lattice gauge theory and point out that the Hilbert space corresponding to a region of space includes edge states that transform nontrivially under gauge transformations. By decomposing these edge states in irreducible representations of the gauge group, the entanglement entropy of an arbitrary state is shown to be a sum of a bulk entropy and a boundary entropy associated to the edge states. This entropy formula agrees with the two-dimensional results from the reduced phase space quantization. These results are applied to several examples, including the ground state in the strong coupling expansion of Kogut and Susskind, and the entropy of the edge states is found to be the dominant contribution to the entanglement entropy.