Using the conformal equivalence of translational KMS states on chiral theories with dilational KMS states obtained from restricting the vacuum state to an interval (the chiral inversion of the Unruh-effect) it was shown in a previous publications that the diverging volume (length) factor of the thermodynamic limit corresponds to the logarithmic increase in the attenuation length of the localization-caused vacuum polarization cloud near the causal boundary. This is not a coincidence but rather a structural consequence of the fact that both operator algebras are of the same unique von Neumann type which is completely different from that met in quantum mechanical algebras. Together with the technique of holographic projection this leads to the universal area proportionality. The main aim in this paper is to present a derivation which is more in the spirit of recent work on entanglement entropy in condensed matter physics, especially to that of the replica trick as used by Cardy and collaborators. The essential new ingredient is the use of the split property which already has shown its constructive power in securing the existence of models of factorizing theories.
Using the conformal equivalence of translational KMS states on chiral theories with dilational KMS states obtained from restricting the vacuum state to an interval (the chiral inversion of the Unruh-effect) it was shown in a previous publications that the diverging volume (length) factor of the thermodynamic limit corresponds to the logarithmic increase in the attenuation length of the localization-caused vacuum polarization cloud near the causal boundary. This is not a coincidence but rather a structural consequence of the fact that both operator algebras are of the same unique von Neumann type which is completely different from that met in quantum mechanical algebras. Together with the technique of holographic projection this leads to the universal area proportionality. The main aim in this paper is to present a derivation which is more in the spirit of recent work on entanglement entropy in condensed matter physics, especially to that of the replica trick as used by Cardy and collaborators. The essential new ingredient is the use of the split property which already has shown its constructive power in securing the existence of models of factorizing theories.
The previously developed algebraic lightfront holography is used in conjunction with the tensor splitting of the chiral theory on the causal horizon. In this way a universal area law for the entanglement entropy of the vacuum relative to the split (tensor factorized) vacuum is obtained. The universality of the area law is a result of the kinematical structure of the properly defined lightfront degrees of freedom. We consider this entropy associated with causal horizon of the wedge algebra in Minkowski spacetime as an analog of the quantum Bekenstein black hole entropy similar to the way in which the Unruh temperature for the wedge algebra may be viewed as an analog in Minkowski spacetime of the Hawking thermal behavior. My more recent preprint hep-th/0202085 presents other aspects of the same problem.
Abstract Given a QFT net A A of local von Neumann algebras A(O) A ( O ) , we consider the von Neumann entropy SA(O,O) S A ( O , O ~ ) of the restriction of the vacuum state to the canonical intermediate type I factor for the inclusion of von Neumann algebras A(O)⊂A(O) A ( O ) ⊂ A ( O ~ ) (split property). This canonical entanglement entropy SA(O,O) S A ( O , O ~ ) is finite for the chiral conformal net on the circle generated by finitely many free Fermions (here double cones are intervals). The finiteness property is derived by an explicit formula of entropy and an observation that the operators in the definition are closely related to Hankel operators. In this paper we give further analysis of this entropy using a variety of techniques that have been developed in different context, and in particular we show that there is an upper bound given by a positive constant multiply by ∣lnη∣ | ln η | , where η η is the cross ratio of the underlying system, when η→0 η → 0 .
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.
The algebraic approach to QFT, which for several decades has enriched QFT with structural theorems, has recently shown its utility in various constructions of actual interest. In these lecture notes I explain how AQFT (in particular the modular theory of operator algebras) implies paradigmatic conceptual and mathematical changes while fully preserving the physical principles which underly QFT. As an illustration of actual interest I use holography on null-surfaces and the ensuing area law for entropy of localized matter in the vacuum state.
After a short presentation of KMS states and modular theory as the unifying description of thermalizing systems we propose the absence of transverse vacuum fluctuations in the holographic projections as the principle reason for an area behavior (the transverse area) of localization entropy as opposed to the volume dependence of ordinary heat bath entropy. Thermalization through causal localization is not a property of QM but results from the omnipresent vacuum polarization in QFT and does not require a Gibbs type ensemble avaraging (coupling to a heat bath).
We study holographic entanglement entropy in 3-dimensional AdS gravity with conformal boundary conditions which fix the conformal class of the boundary metric and its extrinsic curvature, K, while leaving the Weyl mode dynamical. Extending Lewkowycz-Maldacena-Dong's replica construction, we derive the corresponding holographic entanglement entropy formula. We show that the fluctuating Weyl mode does not contribute additional entropy. The entropy of the full boundary is therefore the Bekenstein-Hawking entropy, and the entropy of a boundary subregion continues to obey the Ryu-Takayanagi prescription, namely, the area of a minimal surface divided by 4GN. We carry out explicit calculations for global AdS, rotating and non-rotating BTZ geometries. For an interval in AdS3 we find that the K-dependent holographic entanglement entropy is governed by cm=2GN3ℓ. For a thermal state at high conformal temperatures we find that, the entropy is governed by ceff=2GN3ℓ2Kℓ−K2ℓ2−4, the same effective central charge that governs the Cardy-like density of states, in agreement with previous results in the literature. Finally, we also compute the entanglement entropy directly from the conjectured dual boundary theory - a holographic CFT coupled with time-like Liouville theory and deformed by a marginal TTˉ like operator - and find SEE=3ceffln(ε2Rsinϕ0). This result for the state with no operator insertions (vacuum state ), provides an independent boundary realization of ceff while clarifying that this state is not the state dual to global AdS.