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

Gábor Sárosi, Tomonori Ugajin

TL;DR

The paper develops a purely kinematic expansion to compute the entanglement entropy and modular Hamiltonian for slightly excited states in any CFT, reduced to a ball, by expressing corrections as integrals along the vacuum modular flow of the reference state. It shows that at quadratic order the entanglement entropy corresponds to the bulk canonical energy of the HKLL-reconstructed field dual to the excitation, and reinterprets modular Hamiltonian corrections in holographic terms as a sum of boundary OPE blocks and bulk contributions, in line with JLMS-like ideas. The approach yields concrete formulas for higher-order terms in the OPE coefficients, connects to the replica trick, and provides explicit small-subsystem results that reproduce known holographic and CFT data, offering a versatile toolkit beyond large-$N$ approximations. This framework clarifies how bulk physics emerges from CFT data via OPE blocks and modular flow, with potential for systematic resummations and applications to relative entropy and bulk reconstruction.

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.

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

TL;DR

The paper develops a purely kinematic expansion to compute the entanglement entropy and modular Hamiltonian for slightly excited states in any CFT, reduced to a ball, by expressing corrections as integrals along the vacuum modular flow of the reference state. It shows that at quadratic order the entanglement entropy corresponds to the bulk canonical energy of the HKLL-reconstructed field dual to the excitation, and reinterprets modular Hamiltonian corrections in holographic terms as a sum of boundary OPE blocks and bulk contributions, in line with JLMS-like ideas. The approach yields concrete formulas for higher-order terms in the OPE coefficients, connects to the replica trick, and provides explicit small-subsystem results that reproduce known holographic and CFT data, offering a versatile toolkit beyond large- approximations. This framework clarifies how bulk physics emerges from CFT data via OPE blocks and modular flow, with potential for systematic resummations and applications to relative entropy and bulk reconstruction.

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 or other special properties of the CFT and therefore they are purely kinematic.

Paper Structure

This paper contains 27 sections, 133 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Modular Hamiltonian
  3. The Casini-Huerta-Myers map
  4. Rindler representation of the density operator
  5. Expansion of the noncommutative logarithm
  6. The kernel
  7. The contour: part 1
  8. Entanglement entropy
  9. Expansion for the entanglement entropy
  10. The contour: part 2
  11. Question of convergence
  12. The contour: part 3
  13. Quadratic order in the OPE coefficient
  14. Comparison with the replica trick
  15. Vacuum Fisher information
  16. ...and 12 more sections

Figures (2)

  • Figure 1: Left: Ordering of operators in \ref{['eq:entropyexpansion']}. The red crosses are the Euclidean projections of the operator insertions, and the right ordering is achieved from the Euclidean time ordering by picking the $s_k$ integral contours in \ref{['eq:entropyexpansion']} accroding to \ref{['eq:contour']}. Right: Illustration of the integration contours (grey) in \ref{['eq:entropyexpansion']} prescribed in \ref{['eq:contour']}.
  • Figure 2: Branch cut structure of the correlator \ref{['eq:3ptfunc']}, along with the original (red) and deformed (blue) contour in \ref{['eq:triop1a']}.