Modular Hamiltonians for Deformed Half-Spaces and the Averaged Null Energy Condition

TL;DR

Faulkner et al. develop a perturbative method to compute the modular Hamiltonian for deformed half-spaces in relativistic QFT, showing first-order shape deformations contribute horizon-bound integrals of $T_{++}$ and $T_{--}$. Using monotonicity of relative entropy, they derive the Averaged Null Energy Condition for excited states and recover Hofman-Maldacena bounds on CFT three-point functions. They also connect to AdS/CFT via the JLMS prescription, demonstrating boundary and bulk modular energies agree under small deformations through a bulk shock-wave analysis. The work strengthens the link between entanglement structure and energy conditions, with implications for holography and shape-dependent entanglement dynamics.

Abstract

We study modular Hamiltonians corresponding to the vacuum state for deformed half-spaces in relativistic quantum field theories on $\mathbb{R}^{1,d-1}$. We show that in addition to the usual boost generator, there is a contribution to the modular Hamiltonian at first order in the shape deformation, proportional to the integral of the null components of the stress tensor along the Rindler horizon. We use this fact along with monotonicity of relative entropy to prove the averaged null energy condition in Minkowski space-time. This subsequently gives a new proof of the Hofman-Maldacena bounds on the parameters appearing in CFT three-point functions. Our main technical advance involves adapting newly developed perturbative methods for calculating entanglement entropy to the problem at hand. These methods were recently used to prove certain results on the shape dependence of entanglement in CFTs and here we generalize these results to excited states and real time dynamics. We also discuss the AdS/CFT counterpart of this result, making connection with the recently proposed gravitational dual for modular Hamiltonians in holographic theories.

Figures (5)

• Figure 1: We deform the half space ${A_0}$ (solid blue line) inwards into the region $A$ (solid red line), such that $\mathcal{D}(A)$ (darker shaded region) is contained inside $\mathcal{D}({A_0})$ (lighter shaded region). Also shown are the Rindler horizons $\mathcal{H}_{\pm}$ corresponding to the regions ${A_0}$ and ${A_0^c}$. (The transverse directions $\vec{x}$ are implicit.)
• Figure 2: The path integral construction for matrix elements of the reduced density matrix for the state $|\psi_{\alpha}\rangle$, over the original half space ${A_0}$ (solid blue line). The operator insertions are marked at $x_E^0 = \pm \tau$. The black dot is the entangling surface (with transverse directions $\vec{x}$ implicit).
• Figure 3: We split the region of integration into two parts: the region inside the dotted line is $R_b$, and the region outside is $\widetilde{R}$. Also shown is the brach-cut $\partial \widetilde{R}_{\pm}$.
• Figure 4: The contribution from inside the region $R_b$ can be written in real time as an integral over the shaded region.
• Figure 5: The path integral construction of the regulated reduced density matrix for an excited state. We cut out a cylindrical region of radius $r=a$ around the entangling surface, with brick-wall-like boundary conditions. Also shown is the fictitious cutoff surface of radius $r=b$.