Entanglement and Correlations near Extremality: CFTs dual to Reissner-Nordström AdS${}_5$

TL;DR

We address entanglement structure in charged thermofield double states of $d=4$ CFTs via holography using planar RN-AdS$_5$ black holes. The study computes thermo-mutual information and charged large-dimension two-point functions in the small-$T$ limit, highlighting how an infinite $AdS_2$ throat coexists with finite entropy density. Key findings include a logarithmic scaling of the strip-strips entanglement length $L_{\mathrm{strips}} \sim \frac{\gamma}{\sqrt{6}\mu} \ln(\mu/T)$ near extremality, a finite zero-temperature threshold $L_{s+\mathrm{CFT}} \approx 1.05\,\gamma/\mu$ for a strip vs entire CFT_2, and the existence of extremally charged operators with $\Delta_{\mathrm{IR}}=0$ whose correlators remain finite as $T\to 0$, all tied to IR fixed-point physics and quasinormal mode structure. These results illuminate a mixed picture where some entanglement remains localized while other correlations extend to large scales, challenging localized-quasiparticle pictures of TFD entanglement in holographic plasmas.

Abstract

We use the AdS/CFT correspondence to study models of entanglement and correlations between two $d=4$ CFTs in thermofield double states at finite chemical potential. Our bulk spacetimes are planar Reissner-Nordström AdS black holes. We compute both thermo-mutual information and the two-point correlators of large-dimension scalar operators, focussing on the small-temperature behavior -- an infrared limit with behavior similar to that seen at large times. The interesting feature of this model is of course that the entropy density remains finite as $T \rightarrow 0$ while the bulk geometry develops an infinite throat. This leads to a logarithmic divergence in the scale required for non-zero mutual information between equal-sized strips in the two CFTs, though the mutual information between one entire CFT and a finite-sized strip in the other can remain non-zero even at $T=0$. Furthermore, despite the infinite throat, there can be extremally charged operators for which the two-point correlations remain finite as $T \rightarrow 0$. This suggests an interestingly mixed picture in which some aspects of the entanglement remain localized on scales set by the chemical potential, while others shift to larger and larger scales. We also comment on implications for the localized-quasiparticle picture of entanglement.

Figures (10)

• Figure 1: A conformal diagram of the maximally extended planar AdS-Schwarzschild black hole. This geometry is the bulk dual to the TFD state of two disconnected CFTs living on the two boundaries of the spacetime.
• Figure 2: The relevant portion of the conformal diagram of RN-AdS. The exterior regions are I and III, with their boundaries at $z = 0$. The singularity is at $z = \infty$, and the spacetime has inner and outer horizons at $z = z_0$ and $z = z_+$, respectively. We take the imaginary part of $t$ in regions I-IV to be $0$, $\beta/4$, $\beta/2$, and $-\beta/4$, respectively.
• Figure 3: Assorted entangling surfaces at $t = 0$. The boundary CFTs live on the solid lines, which show the transverse $x^1$ direction on which we define the strips ${\cal A}_1$ and ${\cal A}_2$, which have width $L$. The dashed line is the bifurcation surface. Here we show five surfaces: $\gamma_{ {\cal A}_1 }$ and $\gamma_{ {\cal A}_2 }$ correspond to the entangling surface of each strip; $\gamma_{ {\cal A}_1 {\cal A}_2 }$ runs through the bulk from one strip to the other, and can contribute to the entanglement entropy of the two strips; $\mathcal{H}$ is a surface that runs along the horizon and corresponds to the entangling surface of the entire left CFT; and $\gamma_\infty$ connects $\partial {\cal A}_1$ to infinity, and can contribute to the mutual information between ${\cal A}_1$ and the left CFT.
• Figure 4: Surfaces relevant to the IR regularization used to compute $\mathrm{TMI}( {\cal A}_1 :\mathrm{CFT}_2)$.
• Figure 5: The critical lengths $L_\mathrm{strips}$ (upper curve, red) and $L_{s+\mathrm{CFT}}$ (lower curve, blue) as functions of temperature. $L_\mathrm{strips}$ diverges logarithmically at small $T$, whereas $L_{s+\mathrm{CFT}}$ approaches a constant value $\approx 1.05 \gamma/\mu$.