Modular Hamiltonians of excited states, OPE blocks and emergent bulk fields

Abstract

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.

40 pages, 2 figures. v3: some typos corrected, references added, extended discussion on convergence and holographic interpretation

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• Quantum hypothesis testing in many-body systems
• Holographic relations for OPE blocks in excited states
• Properties of Modular Hamiltonians on Entanglement Plateaux
• Explicit reconstruction of the entanglement wedge via the Petz map
• Bulk reconstruction in AdS and Gel'fand-Graev-Radon Transform
• Gravity from Entanglement and RG Flow in a Top-down Approach
• Holographic computation of quantum corrections to the bulk cosmological constant
• Endpoint contributions to excited-state modular Hamiltonians
• Light-ray moments as endpoint contributions to modular Hamiltonians
• Virasoro algebras, kinematic space and the spectrum of modular Hamiltonians in CFT${}_2$
• Perturbative expansions of Rényi relative divergences and holography
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