Light-ray moments as endpoint contributions to modular Hamiltonians

TL;DR

This paper investigates how weak perturbations in a conformal field theory modify the modular Hamiltonian, revealing endpoint contributions characterized by light-ray moments of the perturbing operators, especially in excited states.

Contribution

It introduces a novel connection between endpoint contributions to modular Hamiltonians and light-ray moments of local operators in excited CFT states.

Findings

Endpoint contributions depend on the shape of the spacelike surface at the boundary.

Light-ray moments of the perturbing operator determine the endpoint contributions.

On null planes, only moments of the operator 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 $\vert n \vert \geq 2$ 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.