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Endpoint contributions to excited-state modular Hamiltonians

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 $n \geq 2$ under vacuum modular flow produce an additional endpoint contribution to the modular Hamiltonian. Intuitively this is because operators with weight $n \geq 2$ 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.