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

## TL;DR

This paper develops a formal expansion for entanglement entropy and modular Hamiltonians of excited states in conformal field theories, connecting CFT data with holographic bulk fields without large N assumptions.

## Contribution

It provides a universal, kinematic framework relating CFT OPE blocks and bulk canonical energy for excited states, independent of large N or specific CFT details.

## Key findings

- Quadratic order contributions match holographic expectations via bulk canonical energy.
- Modular Hamiltonian contributions are expressed through OPE blocks integrated along modular flow.
- Results are purely kinematic, not relying on large N assumptions.

## 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.