A note on the extensivity of the holographic entanglement entropy

TL;DR

This work analyzes the renormalized holographic entanglement entropy using the Ryu–Takayanagi prescription, focusing on when the entropy becomes extensive due to infrared walls in the bulk geometry. The authors show that any geometry capped in the IR, with minimal surfaces saturating at the wall and a nondegenerate cap, yields volume-extensive entropy at large region size, across soft, hard, resolved, and thermal walls, with the cap contribution S_cap dominating the finite part. They provide a general framework for the UV and IR behavior, illustrate with a parametric expression for the entropy S(u_m) and discuss the role of disconnected surfaces and phase transitions. A key example is an extremal dyonic AdS$_4$ black hole dual to a 2+1D CFT with magnetic field and/or electric charge condensates, where extensivity persists down to zero temperature and the large-ℓ entropy scales as S_A ∼ N_eff M_eff^d |A|, with M_eff set by μ and B. The results link IR geometry, confinement-like phenomena, and zero-temperature quantum critical behavior in holographic theories, and suggest avenues for QFT verification of zero-temperature extensivity.

Abstract

We consider situations where the renormalized geometric entropy, as defined by the AdS/CFT ansatz of Ryu and Takayanagi, shows extensive behavior in the volume of the entangled region. In general, any holographic geometry that is `capped' in the infrared region is a candidate for extensivity provided the growth of minimal surfaces saturates at the capping region, and the induced metric at the `cap' is non-degenerate. Extensivity is well-known to occur for highly thermalized states. In this note, we show that the holographic ansatz predicts the persistence of the extensivity down to vanishing temperature, for the particular case of conformal field theories in 2+1 dimensions with a magnetic field and/or electric charge condensates.

Figures (5)

• Figure 1: Schematic plot of the typical configuration of a smooth minimal surface that gives us the entanglement entropy in the presence of infrared walls. Close to the position of the infrared wall, the surface prefers to lean on it, giving an extensive contribution to the entropy. We also show the 'capped cylinder' in dashed lines, which becomes a better and better approximation to the minimal surface as $\ell \rightarrow \infty$.
• Figure 2: Qualitative behavior of the warp factor near a soft wall, with $\gamma'_0 =0$ and $\gamma_0 >0$. On the right, the corresponding UV/IR relation with saturation, $\ell\rightarrow \infty$, at the wall.
• Figure 3: Qualitative behavior of the warp factor near a hard wall, with $\gamma'_0 >0$ and $\gamma_0 >0$. On the right, the corresponding UV/IR relation with a maximal value of $\ell$.
• Figure 4: Qualitative behavior of the warp factor near a resolved wall, with $\gamma'_0 >0$ and $\gamma_0 >0$. On the right, the corresponding UV/IR relation with a maximal value of $\ell$ and two smooth extremal surfaces for each $\ell < \ell_{\rm max}$.
• Figure 5: Numerical plot of the function $f(\ell)$, as defined in Eq. (\ref{['deff']}), in units where $u_0 = M_{\rm eff} =1$. The red continuous line corresponds to the case of a CFT in $2+1$ dimensions in the presence of an external magnetic field and/or charge condensate, with a bulk dual given by the dyonic black hole background. The blue dashed line corresponds to the same CFT without any magnetic field or charge condensate, as given by the AdS$_4$ dual background. We see how extensive behavior, linear in $\ell$, sets it at $M_{\rm eff} \ell \gg 1$ in the presence of magnetic field and/or charge condensate.