Holographic Entanglement Entropy and Renormalization Group Flow

TL;DR

The paper investigates how holographic entanglement entropy, computed via the Ryu-Takayanagi prescription, encodes the Renormalization Group flow between UV and IR fixed points in strongly coupled field theories. It develops and analyzes a sharp domain-wall model to isolate geometric contributions and verifies the predictions in explicit holographic RG flows in four and three spacetime dimensions, using both analytic and numerical methods. Key findings include a wall-controlled area law, a shift of the universal logarithmic term to the IR central charge, and an IR/UV scale structure (tilde{\ell}, tilde{\epsilon}, and Omega) that organizes how degrees of freedom change across the flow. The results provide a robust diagnostic of RG flow in holographic theories and yield intuition for how entanglement encodes scale-separated physics in strongly coupled systems.

Abstract

Using holography, we study the entanglement entropy of strongly coupled field theories perturbed by operators that trigger an RG flow from a conformal field theory in the ultraviolet (UV) to a new theory in the infrared (IR). The holographic duals of such flows involve a geometry that has the UV and IR regions separated by a transitional structure in the form of a domain wall. We address the question of how the geometric approach to computing the entanglement entropy organizes the field theory data, exposing key features as the change in degrees of freedom across the flow, how the domain wall acts as a UV region for the IR theory, and a new area law controlled by the domain wall. Using a simple but robust model we uncover this organization, and expect much of it to persist in a wide range of holographic RG flow examples. We test our formulae in two known examples of RG flow in 3+1 and 2+1 dimensions that connect non-trivial fixed points.

Figures (12)

• Figure 1: Diagrams of the two shapes we will consider for region $\mathcal{A}$. This is the case of AdS$_4$, and here, $z$ denotes the radial direction in AdS$_4$. In one dimension higher we will generalize these shapes to a box and a round ball, and in one dimension fewer, we will consider an interval.
• Figure 2: Samples of the domain wall behaviour. The lowest (blue solid) curve corresponds to $b_1 \to - 3$), next lowest (green dot--dashed) curve is with $b_1 = -2$, and the top (red dashed curve) is with $b_1 = - 1$. The UV is to the right, and the mass increases with increasing (toward the positive) $b_1$.
• Figure 3: Blue solid curve is pure AdS$_5$ result (corresponds to $b_1 \to - \infty$), green dot--dashed curve is with $b_1 = -2$, and the red dashed curve is with $b_1 = - 1$
• Figure 4: Value of the asymptotic ($\ell/R \to \infty$) entanglement entropy for changing multiplet mass. Recall that the multiplet mass is proportional to $e^{a(0)} a_0$.
• Figure 5: The adjusted entanglement entropy. Blue solid curve is pure AdS$_5$ result (corresponds to $b_1 \to - \infty$), green dot--dashed curve is with $b_1 = -2$, and the red dashed curve is with $b_1 = - 1$. The curves do not perfectly overlap when inspected at a higher magnification.