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arXiv:1906.00726 · Journal of High Energy Physics

Spectrum of modular Hamiltonian in the vacuum and excited states

Suchetan Das, Bobby Ezhuthachan

Published Oct 1, 2019• 6 citations
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Abstract

Abstract We study the non-zero eigenmodes for the modular Hamiltonian in the context of AdS3/CFT2. We show how to perturbatively construct zero eigenmodes for the modular Hamiltonian of the class of excited states constructed in [38], using information about the vacuum non-zero eigen modular modes.
DOI
10.1007/jhep10(2019)009

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We consider excited states in a CFT, obtained by applying a weak unitary perturbation to the vacuum. The perturbation is generated by the integral of a local operator J(n) of modular weight n over a spacelike surface passing through x = 0. For |n| ≥ 2 the modular Hamiltonian associated with a division of space at x = 0 picks up an endpoint contribution, sensitive to the details of the perturbation (including the shape of the spacelike surface) at x = 0. The endpoint contribution is a sum of light-ray moments of the perturbing operator J(n) and its descendants. For perturbations on null planes only moments of J(n) itself contribute.
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A bstract We develop a perturbative understanding of the modular Hamiltonian for a 2D CFT, divided into left and right half-spaces, with a weak local perturbation inserted in the future wedge. A formal perturbation series for the modular Hamiltonian is available, but must be properly interpreted in quantum field theory. We work inside correlation functions with spectator operators, and introduce a prescription for defining complex modular flow via analytic continuation to properly resolve singularities. From the correlators, we extract an operator expression for the modular Hamiltonian. It takes the form of a local operator in the future wedge plus contact terms with an unconventional singularity structure. Thanks to this structure the KMS conditions are satisfied, which independently establishes the validity of the results. Similar techniques apply to perturbations inserted in the past wedge. We mention various future directions, including an all-orders speculation for the excited state modular Hamiltonian.