Bulk Emergence and the RG Flow of Entanglement Entropy

TL;DR

The paper develops a perturbative framework to compute entanglement entropy (EE) for CFTs deformed by relevant operators, revealing universal second-order terms in the deformation that agree with holographic predictions. By combining replica-trick and direct approaches, EE corrections are expressed as nonlocal CFT data that reorganize into a local bulk description in emergent $AdS_{d+1}$, with a dual scalar field $\phi$ and, for spatially varying couplings, a linearized bulk metric satisfying Einstein equations. This bulk emergence implies EE can be computed as the area of a bulk RT surface, linking nonlocal modular Hamiltonians to a gravitational bulk map and enabling efficient evaluation of EE integrals. The framework extends to nonuniform couplings, yielding a consistent holographic picture and a path toward real-time generalizations and deeper insights into the gravity-EE connection in QFTs.

Abstract

We further develop perturbative methods used to calculate entanglement entropy (EE) away from an interacting CFT fixed point. At second order we find certain universal terms in the renormalized EE which were predicted previously from holography and which we find hold universally for relevant deformations of any CFT in any dimension. We use both replica methods and direct methods to calculate the EE and in both cases find a non-local integral expression involving the CFT two point function. We show that this integral expression can be written as a local integral over a higher dimensional \emph{bulk} modular hamiltonian in an emergent $AdS$ space-time. This bulk modular hamiltonian is associated to an emergent scalar field dual to the deforming operator. We generalize to arbitrary spatially dependent couplings where a linearized metric emerges naturally as a way of efficiently encoding the field theory entanglement: by demanding that Einstein's equations coupled to the bulk scalar field are satisfied, we show that EE can be calculated as the area of this metric. Not only does this show a direct emergence of a higher dimensional gravitational theory from any CFT, it allows for effective evaluation of the the integrals required to calculate EE perturbativly. Our results can also be interpreted as relating the non-locality of the modular hamiltonian for a spherical region in non-CFTs and the non-locality of the holographic bulk to boundary map.

Figures (7)

• Figure 1: Contour integration used to do the sum in \ref{['tosum2']} for the case $n=3$. The general branch cut structure of $G_n( - is)$ is shown for any thermal Euclidean greens functions in the complex time plane (the structure repeats periodically in the imaginary direction with period $2\pi n$.) The original contour $\mathcal{C}$ encircles the $n$ poles of $\left(e^{ s- i \tau_{ba}} -1 \right)^{-1}$ between the cuts. We then deform so the contour $\mathcal{C}'$ lies just above/below the cuts of $G_n$ (dashed lines.) This method of contour integration for doing replica sums is very similar to the methods applied to free field theories in Casini:2009sr.
• Figure 2: The calculation of EE forces us into real times, where we integrate over the position of one of the operators after a real time modular flow transformation by an amount $s$. This modular flow pushes the operator into the causal development of $A$ pictured here (thick lines are lines of constant $s$.) This is contrasted with the modular flow by an amount $\tau$ in Euclidean time (lighter lines are lines of constant $\tau$.)
• Figure 3: Pictorial argument showing that $I(\tau_a,\tau_b)$ is analytic near $\tau_{ba} = \tau_b - \tau_a = 0$. The position of the poles in these pictures lies along the imaginary axis displaced from the vertical location of the branch cuts by an amount $\tau_{ba}$. As we move $\tau_{ba}$ towards zero from below, we come close to the prescirbed integration contour in \ref{['inI']} for $\tau_{ba} < 0$ at $s = i \epsilon$. We can then deform this contour upwards towards $s = 2\pi i - i \epsilon$ being careful not to hit the next pole above the integration contour. Once we have done this we can use the imaginary time periodicity of the integrand to arrive at the prescribed integration contour in \ref{['inI']} for $\tau_{ba}>0$.
• Figure 4: The $\ell_B$ integral in \ref{['finI']} in the complex $\ell_B$ plane at fixed $Y_B$. We denote the integration contour by $\mathcal{C}$. The two branch points are located at $e^{i \tau_{a,b}}/(-2 Y_{a,b}\cdot Y_B)$ . As long as these branch points do not cross the real axis there is no discontinuity and so the answer is analytic in the domain $0 < \tau_{a,b} < 2\pi$.
• Figure 5: Future part of the bulk rindler horizon. This is also the horizon associated to the hyperbolic black hole which was discussed in Casini:2011kv.