Probing renormalization group flows using entanglement entropy

TL;DR

The paper develops a comprehensive holographic framework to extract the large-$R$ expansion of entanglement entropy for regions with spherical or strip geometry along RG flows. It demonstrates that the approach to an IR fixed point contains contributions that depend on the entire RG trajectory, including non-analytic terms controlled by IR data and analytic terms that encode UV-to-IR RG physics, with distinct behavior for gapped versus scaling IR geometries. By applying UV/IR matching and analyzing domain-wall, scaling, and black-hole geometries, the authors derive explicit asymptotic structures for the renormalized entanglement entropy and show how IR data fixes certain coefficients while full RG information shapes others, such as the 1/R terms in spheres. These results extend the utility of REE as a probe of RG flows and IR physics, including in nontrivial holographic settings like D$p$-brane near-horizon geometries and black holes. Overall, REE provides a sharp, scale-dependent observable to explore IR fixed points, domain-wall transitions, and thermal effects in strongly coupled theories.

Abstract

In this paper we continue the study of renormalized entanglement entropy introduced in [1]. In particular, we investigate its behavior near an IR fixed point using holographic duality. We develop techniques which, for any static holographic geometry, enable us to extract the large radius expansion of the entanglement entropy for a spherical region. We show that for both a sphere and a strip, the approach of the renormalized entanglement entropy to the IR fixed point value contains a contribution that depends on the whole RG trajectory. Such a contribution is dominant, when the leading irrelevant operator is sufficiently irrelevant. For a spherical region such terms can be anticipated from a geometric expansion, while for a strip whether these terms have geometric origins remains to be seen.

Figures (3)

• Figure 1: Sketch of the $R\to\infty$ minimal surface in a domain wall geometry \ref{['1']}--\ref{['pep1']}. The violet and red regions represent the UV and IR regions of the expansion. The UV and IR solutions overlap in the matching region, which is used to determine the parameters of the two expansions.
• Figure 2: Samples of minimal surfaces for IR geometries that fall into four different categories.
• Figure 3: $e_n$ plotted as a function of $n$ for $d=2,3$ and $4$. The vertical dashed lines indicate $n_c$. \ref{['defen']} consists of numerical factors and $\bar{h}$, which is a constant determined by the IR solution, $\bar{\rho}_d$. $\bar{h}$ was obtained by numerically determining $\bar{\rho}_d$ and fitting the small $u$ expansion \ref{['jepexp']}. For $d=2$ we know the exact answer from \ref{['stripan']}; the data points lie exactly on the analytically determined curve. For $d=4$ the dotted part of the line is an extrapolation of the solid line; we do not have reliable numerical results in that region for $\bar{h}$.