Holographic entanglement entropy and thermodynamic instability of planar $R$-charged black holes

The holographic entanglement entropy of an infinite strip subsystem on the asymptotic anti–de Sitter (AdS) boundary is used as a probe to study the thermodynamic instabilities of planar $R$-charged black holes (or their dual field theories). We focus on the single-charge AdS black holes in $D=5$, which correspond to spinning $D3$-branes with one nonvanishing angular momentum. Our results show that the holographic entanglement entropy indeed exhibits the thermodynamic instability associated with the divergence of the specific heat. When the width of the strip is large enough, the finite part of the holographic entanglement entropy as a function of the temperature resembles the thermal entropy, as is expected. As the width becomes smaller, however, the two entropies behave differently. In particular, there exists a critical value for the width of the strip, below which the finite part of the holographic entanglement entropy as a function of the temperature develops a self-intersection. We also find similar behavior in the single-charge black holes in $D=4$ and 7.

Figure 1

Thermal entropy $S_{\text{fin}}(T)$. Left: The canonical ensemble with fixed $Q=2$. The vertical line corresponds to $T_{\min }\approx 0.32$. Right: The grand canonical ensemble with fixed $\varphi =4$. The first vertical line corresponds to $T_{\min }\approx 1.8$, and the second corresponds to $T_1\approx 1.84$, where the specific heat of the lower branch changes sign.

Figure 2

The Helmholtz free energy $F(T)$ with $Q=2$.

Figure 3

The Gibbs free energy $W(T)$ and its second order derivative $-W^{\prime \prime }(T)$ with $\varphi =4$. The vertical line denotes the point $T_1$ where the second derivative $W^{\prime \prime }(T)$ of the upper branch changes sign.

Figure 4

$S_{\text{fin}}(T)$ in the grand canonical ensemble with fixed $\varphi =4$ and $l=0.2–0.6$. The $r_{*}$ vs $r_{+}$ plot for $l=0.6$ shows the linear relation between the two, as $l$ is large enough. The arrows indicate the direction of increasing $r_{+}$. Curves with $l=0.2$, 0.25, 0.3, from bottom to top, are plotted together to exhibit the transition.

Figure 5

$S_{\text{fin}}(T)$ in the grand canonical ensemble with fixed $\varphi =4$. The minimum moves from $T\approx 1.82$ to $T\approx 1.83$, as $l$ is increased from 1 to 2. The vertical line to the right of $T=1.84$ is the position $T_1$, where the thermal entropy develops a minimum.

Figure 6

$S_{\text{fin}}(l)$ in the grand canonical ensemble with fixed $\varphi =4$, with $T=73/(\sqrt{65}\pi )\approx 2.88$ (red lines) and $T=14/(\sqrt{5}\pi )\approx 1.99$ (blue lines) from top to bottom. The solid curves correspond to the large black hole branches, and the dashed curves correspond to the small black hole branches. The thick solid black curve corresponds to $T_{\min }\approx 1.80$, where the two branches merge into one.

Figure 7

$S_{\text{fin}}(T)$ in the canonical ensemble with fixed $Q=2$, $l=0.3$ and 1. The arrows indicate the direction of increasing $r_{+}$. The vertical line denotes the temperature at which the specific heat diverges.

Figure 8

$S_{\text{fin}}(l)$ in the canonical ensemble with fixed $Q=2$ and, from top to bottom, $l=0.8,0.7,0.65,0.6$.

Figure 9

$S_{\text{fin}}(l)$ in the canonical ensemble with fixed $Q=2$, $T=0.51$ (blue lines), and $T=0.45$ (green lines) from top to bottom (as seen from the upper-right part). The solid curves correspond to the large black hole branches, and the dashed curves correspond to the small black hole branches. The thick solid black curve corresponds to $T_{\min }\approx 0.32$, where the two branches merge into one.

Figure 10

$S_{\text{fin}}(T)$ in ${\mathrm{AdS}}_7$ in the grand canonical ensemble with fixed $\varphi =4$, $l=0.1$ and 0.6.

Figure 11

$S_{\text{fin}}(T)$ in ${\mathrm{AdS}}_7$ in the canonical ensemble with fixed $Q=2$. Left: Transition from $l=0.3$ (bottom blue line), $l=0.4$ (middle black line), and $l=1$ (top red line). Right: $l=1$.

Figure 12

$S_{\text{fin}}(T)$ in ${\mathrm{AdS}}_4$ in the grand canonical ensemble with fixed $\varphi =4$, $l=0.2$ and 0.6.

Figure 13

$S_{\text{fin}}(T)$ in ${\mathrm{AdS}}_4$ in the canonical ensemble with fixed $Q=2$. Left: $S_{\text{fin}}(T)$ plots with $l=2$, 0.6, and 0.2, from top to bottom. Right: $S_{\text{th}}(T)$, starting from the minimum $T_{\min }=\frac{\sqrt{Q}}{2\pi }\approx 0.225079$, $S_{\text{th}}\approx 12.56$.

Figure 14

$S_{\text{fin}}(l)$ in ${\mathrm{AdS}}_4$ in the canonical ensemble with fixed $Q=2$ and, from top to bottom (as seen from the upper-right part), $T=0.72$ (red line), 0.49 (blue line), 0.25 (green line), 0.22509 (brown line), and $T_{\min }$ (thick black line).

Figure 15

Illustration for the critical value $l_c$. From top to bottom, $T=2.88$ (red lines), 1.99 (blue lines), and 1.83 (green lines), and $T_{\min }=1.80$ (thick black line, which almost overlaps with the $T=1.83$ curves). The solid curves correspond to the large black hole branch and dashed ones to the small black hole branch. The dots denote the intersection points. For a fixed $l\ge l_c$, $S_{\text{fin}}$ for the small black holes is always lower than that of the large black holes.