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arXiv:2006.13317 · Journal of High Energy Physics

Endpoint contributions to excited-state modular Hamiltonians

Daniel Kabat, Gilad Lifschytz, Phuc Nguyen, Debajyoti Sarkar

Published Dec 1, 2020• 5 citations
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Abstract

Abstract We compute modular Hamiltonians for excited states obtained by perturbing the vacuum with a unitary operator. We use operator methods and work to first order in the strength of the perturbation. For the most part we divide space in half and focus on perturbations generated by integrating a local operator J over a null plane. Local operators with weight n ≥ 2 under vacuum modular flow produce an additional endpoint contribution to the modular Hamiltonian. Intuitively this is because operators with weight n ≥ 2 can move degrees of freedom from a region to its complement. The endpoint contribution is an integral of J over a null plane. We show this in detail for stress tensor perturbations in two dimensions, where the result can be verified by a conformal transformation, and for scalar perturbations in a CFT. This lets us conjecture a general form for the endpoint contribution that applies to any field theory divided into half-spaces.

DOI
10.1007/jhep12(2020)128

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Abstract We compute modular Hamiltonians for excited states obtained by perturbing the vacuum with a unitary operator. We use operator methods and work to first order in the strength of the perturbation. For the most part we divide space in half and focus on perturbations generated by integrating a local operator J over a null plane. Local operators with weight n ≥ 2 under vacuum modular flow produce an additional endpoint contribution to the modular Hamiltonian. Intuitively this is because operators with weight n ≥ 2 can move degrees of freedom from a region to its complement. The endpoint contribution is an integral of J over a null plane. We show this in detail for stress tensor perturbations in two dimensions, where the result can be verified by a conformal transformation, and for scalar perturbations in a CFT. This lets us conjecture a general form for the endpoint contribution that applies to any field theory divided into half-spaces.
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.
Abstract 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.
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.
Abstract We compute the modular Hamiltonians of regions having the future horizon lying on a null plane. For a CFT this is equivalent to regions with a boundary of arbitrary shape lying on the null cone. These Hamiltonians have a local expression on the horizon formed by integrals of the stress tensor. We prove this result in two different ways, and show that the modular Hamiltonians of these regions form an infinite dimensional Lie algebra. The corresponding group of unitary transformations moves the fields on the null surface locally along the null generators with arbitrary null line dependent velocities, but act non-locally outside the null plane. We regain this result in greater generality using more abstract tools on the algebraic quantum field theory. Finally, we show that modular Hamiltonians on the null surface satisfy a Markov property that leads to the saturation of the strong sub-additive inequality for the entropies and to the strong super-additivity of the relative entropy.
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.
The vacuum modular Hamiltonian KKK of the Rindler wedge in any relativistic quantum field theory is given by the boost generator. Here we investigate the modular Hamiltonian for more general half-spaces which are bounded by an arbitrary smooth cut of a null plane. We derive a formula for the second derivative of the modular Hamiltonian with respect to the coordinates of the cut which schematically reads ${K}^{\ensuremath{'}\ensuremath{'}}={T}_{vv}$. This formula can be integrated twice to obtain a simple expression for the modular Hamiltonian. The result naturally generalizes the standard expression for the Rindler modular Hamiltonian to this larger class of regions. Our primary assumptions are the quantum null energy condition---an inequality between the second derivative of the von Neumann entropy of a region and the stress tensor---and its saturation in the vacuum for these regions. We discuss the validity of these assumptions in free theories and holographic theories to all orders in 1/N1/N1/N.