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Entanglement entropy of two dimensional systems and holography

Georgios Michalogiorgakis

Published Jun 18, 2008• 5 citations
OpenAlex

Abstract

In this note a new method for computing the entanglement entropy of a CFT holographically is explored. It consists of finding a bulk background with a boundary metric that has the conical singularities needed to compute the entanglement entropy in the usual QFT definition. An explicit calculation is presented for d=2.

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The purpose of these lecture notes is to give an introduction to entanglement entropy with an emphasis on holography. Apart from the standard definition and properties of entanglement entropy, we explicitly derive the relation between the coefficients of the conformal anomaly and the universal term in the entropy of a d-dimensional CFT. We also discuss the prescription for calculating the entanglement entropy holographically in theories dual to Einstein-Hilbert and Lovelock gravity and present several examples in detail.
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The relationship between Bekenstein-Hawking and entanglement entropy is one \nof the most intriguing problems of black hole and theoretical physics. Although both entropies have the same geometric characterization (they scale as an area), they have a different origin. The Bekenstein-Hawking entropy has it’s roots in the thermal correlations whereas the entanglement entropy describes the quantum correlations. In this thesis we address this problem from a holographic \nperspective, making large use of the AdS/CFT correspondence. \nWe first look at the BTZ black hole in which case the black hole horizon and \nthe conical singularity are related to each other. Using the modular transformations of the dual 2D CFT, we first obtain an expression of the holographicentanglement entropy for the Euclidean BTZ black hole, AdS3 vacua and conical singularity and then analyse the behaviour of the leading terms in the expansions of the holographic entanglement entropy for the BTZ black hole and the \nconical singularity. From these calculations, we have extracted the “signatures” through which entanglement entropy differentiates between the horizon and the conical singularity. When we deal with CFT at a finite temperature then it is well known that, entanglement entropy fails as a measure and hence it is replaced by entanglement negativity which is able to separate the thermal/classical \ncorrelations from the quantum correlations thereby capturing “distillable” entanglement. We address the problem of finding a suitable holographic prescription \nto calculate the entanglement negativity for two adjacent intervals when the CFT is at a finite temperature and in turn dual to a black hole in the bulk. We first propose a conjecture for the holographic entanglement negativity for two adjacent intervals in the \nAdS3/CFT2 setup and perform calculations in the \nvarious cases to support this conjecture. We further push forward the validity of this conjecture by calculating the holographic entanglement negativity for two \nadjacent subsystems when we have RN-AdS black holes in the bulk which in \nturn are dual to CFT with a conserved charge.
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.
With an aim towards understanding the time-dependence of entanglement entropy in generic quantum field theories, we propose a covariant generalization of the holographic entanglement entropy proposal of hep-th/0603001. Apart from providing several examples of possible covariant generalizations, we study a particular construction based on light-sheets, motivated in similar spirit to the covariant entropy bound underlying the holographic principle. In particular, we argue that the entanglement entropy associated with a specified region on the boundary in the context of the AdS/CFT correspondence is given by the area of a co-dimension two bulk surface with vanishing expansions of null geodesics. We demonstrate our construction with several examples to illustrate its reduction to the holographic entanglement entropy proposal in static spacetimes. We further show how this proposal may be used to understand the time evolution of entanglement entropy in a time varying QFT state dual to a collapsing black hole background. Finally, we use our proposal to argue that the Euclidean wormhole geometries with multiple boundaries should be regarded as states in a non-interacting but entangled set of QFTs, one associated to each boundary.
We derive the universal terms of entanglement entropy for 6d CFTs by applying the holographic and the field theoretical approaches, respectively. Our formulas are conformal invariant and agree with the results of [37, 38]. Remarkably, we find that the holographic and the field theoretical results match exactly for the C 2 and Ck 2 terms, where C and k denote the Weyl tensor and the extrinsic curvature, respectively. As for the k 4 terms, we meet the splitting problem of the conical metrics. The splitting problem in the bulk can be fixed by equations of motion. As for the splitting on the boundary, we assume the general forms and find that there indeed exists suitable splitting which can make the holographic and the field theoretical k 4 terms match. Since we have much more equations than the free parameters, the match for k 4 terms is non-trivial.