On Shape Dependence and RG Flow of Entanglement Entropy

TL;DR

This work analyzes how entanglement entropy responds to geometric shape and RG flow, using both field-theoretic and holographic tools. It reveals that singular entangling surfaces in 3+1D CFT generate additional divergences, while in 2+1D gapped theories EE admits a mass-expansion where coefficients are governed by extrinsic-curvature data. The CGLP M-theory background provides a concrete holographic RG flow from a UV CFT to an IR gapped phase, with the renormalized EE monotone along the flow and topology-change phenomena. Finally, the study shows universal shape-dependence features for wedges and cones in 3+1D CFT, linking cusp and cone data to anomaly coefficients and holographic results. Collectively, these results deepen the connection between geometry, RG flow, and entanglement in strongly coupled and free theories.

Abstract

We use a mix of field theoretic and holographic techniques to elucidate various properties of quantum entanglement entropy. In (3+1)-dimensional conformal field theory we study the divergent terms in the entropy when the entangling surface has a conical or a wedge singularity. In (2+1)-dimensional field theory with a mass gap we calculate, for an arbitrary smooth entanglement contour, the expansion of the entropy in inverse odd powers of the mass. We show that the shape-dependent coefficients that arise are even powers of the extrinsic curvature and its derivatives. A useful dual construction of a (2+1)-dimensional theory, which allows us to exhibit these properties, is provided by the CGLP background. This smooth warped throat solution of 11-dimensional supergravity describes renormalization group flow from a conformal field theory in the UV to a gapped one in the IR. For this flow we calculate the recently introduced renormalized entanglement entropy and confirm that it is a monotonic function.

Figures (5)

• Figure 1: (a) Numerical solutions to the equation of motion for the holographic entangling surface, given by $r(\tau)$, in the CGLP theory. The dotted red line indicates the critical value $R_{\text{crit}}$, where the solutions change from disk-type to cylinder-type. (b) A zoomed-in plot of the UV region, with disk-type solutions, where we plot the AdS approximation in \ref{['UVfcn']} in dotted black. (c) A zoomed-in plot of the IR region, with cylinder-type solutions, with the analytic approximation given by $\delta(\tau)$ in \ref{['asympInt']} plotted in dotted black.
• Figure 2: The renormalized entanglement entropy ${\cal F}(R)$ along the RG flow in the CGLP theory plotted in orange. The left dotted black curve is the asymptotic UV approximation to ${\cal F}(R)$ given in \ref{['Fuv']}. The right dotted black curve is the IR approximation to ${\cal F}(R)$ given in \ref{['IRFCGLP']}. The dotted red line marks the value $R = R_{\text{crit}}$.
• Figure 3: (a) A plot of an example entangling surface $\Sigma_1$, described by the function $R_{\text{UV}}(\theta)$ in polar coordinates, in the CGLP theory. (b) The extrinsic curvature $\kappa_{\text{UV}} (\theta)$ for the entangling surface $R_{\text{UV}}(\theta)$. The extrinsic curvature is small over the whole curve.
• Figure 4: The function $R_{\text{UV}}(\theta) - R_{\text{IR}}(\theta)$ in the CGLP theory, with $R_{\text{UV}}(\theta)$ plotted in fig. \ref{['curvProf']}. The solid orange curve is computed by numerically solving the equation of motion for the holographic entangling surface $\Sigma_2$. The dotted black curve is an analytic approximation, which is equal to $- \delta(\tau = 0, \theta)$, with $\delta(0,\theta)$ given in \ref{['asympIntRth']}.
• Figure 5: The entanglement entropy for the wedge has a $1/\epsilon$ divergent term, whose coefficient depends on the angle of the wedge. This coefficient $f_\text{wedge}^\text{(hol)}(\Omega)$ is given explicitly in \ref{['omg0']} and \ref{['fom']}, and its normalization depends on the UV cut-off $\epsilon$. The black dotted line is the normalized function $\tilde{f}_\text{wedge}^\text{(hol)}(\Omega)= a f_\text{wedge}^\text{(hol)}(\Omega)$ for the wedge with $a\sim1.11$ and the orange line is the function $f_\text{cusp}^\text{(hol)}(\Omega)$ for the cusp in $(2+1)$-dimensions.