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arXiv:2002.04637 · Physical review. D/Physical review. D.

Modular Hamiltonian for holographic excited states

Raúl Arias, M. Botta Cantcheff, Pedro J. Martínez, Juan Felipe Zárate Chahín

Published Jul 27, 2020• 21 citations
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Abstract

In this work we study the Tomita-Takesaki construction for a family of excited states that, in a strongly coupled CFT---at large $N$---correspond to coherent states in an asymptotically AdS spacetime geometry. We compute the modular flow and modular Hamiltonian associated to these excited states in the Rindler wedge and for a ball shaped entangling surface. Using holography, one can compute the bulk modular flow and construct the Tomita-Takesaki theory for these cases. We also discuss generalizations of the entanglement regions in the bulk and how to evaluate the modular Hamiltonian in a large N approximation. Finally, we extend the holographic Banks, Douglas, Horowitz and Matinec (BDHM) formula to compute the modular evolution of operators in the corresponding CFT algebra, and propose this as a more general prescription.

DOI
10.1103/physrevd.102.026021

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In this work we study the Tomita-Takesaki construction for a family of excited states that, in a strongly coupled CFT---at large NNN---correspond to coherent states in an asymptotically AdS spacetime geometry. We compute the modular flow and modular Hamiltonian associated to these excited states in the Rindler wedge and for a ball shaped entangling surface. Using holography, one can compute the bulk modular flow and construct the Tomita-Takesaki theory for these cases. We also discuss generalizations of the entanglement regions in the bulk and how to evaluate the modular Hamiltonian in a large N approximation. Finally, we extend the holographic Banks, Douglas, Horowitz and Matinec (BDHM) formula to compute the modular evolution of operators in the corresponding CFT algebra, and propose this as a more general prescription.
We study the entanglement entropy and the modular Hamiltonian of slightly excited states reduced to a ball shaped region in generic conformal field theories. We set up a formal expansion in the one point functions of the state in which all orders are explicitly given in terms of integrals of multi-point functions along the vacuum modular flow, without a need for replica index analytic continuation. We show that the quadratic order contributions in this expansion can be calculated in a way expected from holography, namely via the bulk canonical energy for the entanglement entropy, and its variation for the modular Hamiltonian. The bulk fields contributing to the canonical energy are defined via the HKLL procedure. In terms of CFT variables, the contribution of each such bulk field to the modular Hamiltonian is given by the OPE block corresponding to the dual operator integrated along the vacuum modular flow. These results do not rely on assuming large N or other special properties of the CFT and therefore they are purely kinematic.
In the context of the AdS/CFT correspondence, the Jafferis-Lewkowycz-Maldacena-Suh (JLMS) formula relates the boundary modular Hamiltonian to the HRT area operator and the corresponding bulk modular Hamiltonian. In standard discussions of a holographic code built from perturbative excitations on a fixed classical background, the A/4GA/4GA/4G term dominates over the bulk modular Hamiltonian in the limit G→0G \to 0G→0. However, this setup is insufficient to describe the action of boundary modular flows that change the classical background. In contrast, our recent discussion of `large' codes provides a useful framework to obtain a JLMS formula that relates bulk and boundary modular flows acting on appropriate states in the code. Here, one sees that the bulk modular Hamiltonian is not in fact subdominant when the modular flow defined by one state acts on another state peaked around a sufficiently different classical background. Instead, the bulk modular Hamiltonian can be of order 1/G1/G1/G and can dominate over the A/4GA/4GA/4G term. Nevertheless, we show that if the two states have the same classical background except for data conjugate to the HRT area then, under mild assumptions and over a large range of flow parameters, the modular-flowed state is described by a classical background given by the HRT-area flow alone.
We extend the holographic test of entanglement cohomology from d=2d=2d=2 to higher-dimensional conformal field theories. For a ddd-dimensional CFT with a spherical entangling region AAA, the vacuum modular Hamiltonian is known explicitly, and the boundary relative entropy of a primary-operator perturbation can be computed via conformal Ward identities. On the bulk side, the perturbation is dual to a scalar field in AdSd+1_{d+1}d+1​, and the bulk canonical energy is computed using the JLMS relation. We carry out the computation explicitly for d=3d=3d=3 and d=4d=4d=4 and find that the boundary and bulk results agree at first order in the perturbation, with identical coefficients κdΔ2/CTκ_d Δ^2 / C_Tκd​Δ2/CT​, where κ3=π2/12κ_3 = π^2/12κ3​=π2/12 and κ4=3π3/16κ_4 = 3π^3/16κ4​=3π3/16. The result extends the quantitative verification of the holographic dual of entanglement cohomology beyond the d=2d=2d=2 case treated in the companion paper and confirms that the modular Hamiltonian perturbation is the correct holographic image of the first entanglement cohomology class. We discuss the role of the conformal anomaly, the shape of the entanglement wedge, and an outlook on TsT-deformed backgrounds. \keywords{Entanglement entropy \and Cyclic cohomology \and von Neumann algebras \and Holography \and Modular Hamiltonian}
We further develop perturbative methods used to calculate entanglement entropy (EE) away from an interacting CFT fixed point. At second order we find certain universal terms in the renormalized EE which were predicted previously from holography and which we find hold universally for relevant deformations of any CFT in any dimension. We use both replica methods and direct methods to calculate the EE and in both cases find a non-local integral expression involving the CFT two point function. We show that this integral expression can be written as a local integral over a higher dimensional bulk modular hamiltonian in an emergent AdS space-time. This bulk modular hamiltonian is associated to an emergent scalar field dual to the deforming operator. We generalize to arbitrary spatially dependent couplings where a linearized metric emerges naturally as a way of efficiently encoding the field theory entanglement: by demanding that Einstein’s equations coupled to the bulk scalar field are satisfied, we show that EE can be calculated as the area of this metric. Not only does this show a direct emergence of a higher dimensional gravitational theory from any CFT, it allows for effective evaluation of the the integrals required to calculate EE perturbativly. Our results can also be interpreted as relating the non-locality of the modular hamiltonian for a spherical region in non-CFTs and the non-locality of the holographic bulk to boundary map.
Abstract We determine the Tomita-Takesaki modular data for CFTs in double cone and light cone regions in conformally flat spacetimes. This includes in particular the modular Hamiltonian for diamonds in the de Sitter spacetime. In the limit where the diamonds become large, we show that the modular automorphisms become time translations in the static patch. As preparation, we also provide a pedagogical rederivation of the known results for Minkowski spacetime. With our results and using the Araki formula, it becomes possible to compute relative entanglement entropies for CFTs in these regions.
We study half-space/Rindler modular Hamiltonians for excited states created by turning on sources for local operators in the Euclidean path integral in relativistic quantum field theories. We derive a simple, manifestly Lorentzian formula for the modular Hamiltonian to all orders in perturbation theory in the sources. We apply this formula to the case of shape-deformed half spaces in the vacuum state, and obtain the corresponding modular Hamiltonian to all orders in the shape deformation in terms of products of half-sided null energy operators, i.e., stress tensor components integrated along the future and past Rindler horizons. In the special case where the shape deformation is purely null, our perturbation series can be resummed, and agrees precisely with the known formula for vacuum modular Hamiltonians for null cuts of the Rindler horizon. Finally, we study some universal properties of modular flow (corresponding to Euclidean path integral states) of local operators inside correlation functions in conformal field theories. In particular, we show how the flow becomes the local boost in the limit where the operator being flowed approaches the entanglement cut.