The Gravity Duals of Modular Hamiltonians

TL;DR

This work links modular Hamiltonians for spatial regions in states with gravity duals to bulk geometric data by leveraging the first law of entanglement entropy and RT/HRT prescriptions. It develops a framework to compute the modular response as metric deformations and analyzes how these deformations propagate from extremal surfaces, revealing cases where the modular evolution acts locally while in general it acts as a precursor probing spacelike bulk regions. The results show that, beyond symmetric setups, modular Hamiltonians encode bulk information in a manner that challenges bulk locality under standard holographic entanglement entropy prescriptions, suggesting possible refinements to HRT or the need to account for the bulk state more explicitly. Overall, the paper provides a systematic holographic characterization of H acting on its defining state and nearby excitations, with implications for bulk causality and the interpretation of modular flow in AdS/CFT.

Abstract

In this work, we investigate modular Hamiltonians defined with respect to arbitrary spatial regions in quantum field theory states which have semi-classical gravity duals. We find prescriptions in the gravity dual for calculating the action of the modular Hamiltonian on its defining state, including its dual metric, and also on small excitations around the state. Curiously, use of the covariant holographic entanglement entropy formula leads us to the conclusion that the modular Hamiltonian, which in the quantum field theory acts only in the causal completion of the region, does not commute with bulk operators whose entire gauge-invariant description is space-like to the causal completion of the region.

Figures (6)

• Figure 1: Cartoon of the diffeomorphism-invariant support of the modular response $\partial_{{\alpha}}h\approx (g_{{\alpha}}-g)/\alpha$, as computed using the HRT prescription. The entanglement wedge of $R$, whose intersection with the AdS boundary is $\mathcal{C}(R)$, is delineated in pink. Left: for configurations $(\rho, R)$ with special symmetry, the response is causal from $\partial R$. Right: for generic $(\rho, R)$ the response is causal from the entire extremal surface associated with $R$. To avoid clutter here we have only drawn the upper half of time evolution.
• Figure 2: Riemannian sheets for Euclidean path integrals corresponding to the operator $\rho_{R}^{k}\otimes I$, left, and $\rho_{R}$, right.
• Figure 3: Depiction of a geodesic in Poincaré AdS which computes the expectation value of an equal-time two-point function $\braket{\mathcal{O}(x)\mathcal{O}(y)}$, with $x$ and $y$ in Rindler regions $\mathcal{C}(\bar{R})$ and $\mathcal{C}(R)$, where $R$ is the half-space $x_1\geq 0$. (Note we have $d=2$ in the figure for ease of drawing, but all considerations in the paper are in $d \geq 3$ for which there are local gravitational excitations in the bulk.) The light-cone and boundary light-cone from $\partial R$ are shown in solid beige and transparent purple, respectively. The solid purple line is the extremal surface $E$. All contributions to $\partial_{{\alpha}}\braket{\mathcal{O}(x)\mathcal{O}(y)}$ come from the intersection of the geodesic with the boundary light-cone from $\partial R$.
• Figure 4: Tests of the scaling of $I$ in \ref{['dell']} in $d=3$ and at $\varepsilon=10^{-3}$. The fixed parameters in each plot are as follows - left: $y^{1}=-x^{1}=11$, $\Delta x^2=0$, center: $t=1$, $x^{1}=-3/2$, $\Delta x^2=0$, right: $t=1$, $y^{1}=-x^{1}=3/2$.
• Figure 5: A component of the Weyl response on an interior point of $E$ for an $R$ which is a slab in $d=3$ with finite extent $-1 \leq x^1 \leq 1$ and extending infinitely in remaining boundary spatial coordinate $x^2$. Left: The point on $E$ at which we measure the Weyl response. We show a $t=t_R$, $x^{2}=const.$ cross section of the bulk space-time $g$ which is Poincaré AdS. Right: The component $\partial_{{\alpha}}C_{z2t2}$\ref{['Weylcomp']} on the point specified as a function of the light-cone cutoff $\varepsilon$. It diverges as $\varepsilon \to 0$.