Holographic Entanglement Entropy at Finite Temperature

TL;DR

This work probes whether holographic entanglement entropy signals confinement/deconfinement transitions at finite temperature by applying the Ryu–Takayanagi prescription to a range of black hole backgrounds, including BTZ, nonextremal Dp-branes, a dyonic AdS$_4$ black hole, and global AdS geometries. By comparing two minimal-surface configurations (a smooth surface vs. a piecewise surface that probes the horizon), the authors find that the smooth surface typically dominates, yielding no entanglement-entropy-driven phase transition in most cases (with a noted exception for $p=6$). The temperature dependence often factors out through the horizon radius, and in several cases the entanglement entropy aligns with field-theory expectations while behaving as a thermodynamic potential rather than the free energy. The results suggest that while entanglement entropy is a powerful diagnostic, horizon geometry plays a crucial role, and alternative notions of geometric entropy might be required to reveal phase transitions. $S_A = \frac{\mathrm{Area}(\gamma_A)}{4G_N^{d+2}}$ remains the central holographic quantity, and the study highlights the nuanced relationship between entanglement and confinement in finite-temperature holography.

Abstract

Using a holographic proposal for the entanglement entropy we study its behavior in various supergravity backgrounds. We are particularly interested in the possibility of using the entanglement entropy as way to detect transitions induced by the presence horizons. We consider several geometries with horizons: the black hole in $AdS_3$, nonextremal Dp-branes, dyonic black holes asymptotically to $AdS_4$ and also Schwarzschild black holes in global $AdS_p$ coordinates. Generically, we find that the entanglement entropy does not exhibit a transition, that is, one of the two possible configurations always dominates.

Figures (10)

• Figure 1: Two competing configurations for the entanglement entropy in the presence of a black hole horizon. The green surface represents a continuous configuration while the red surface goes straight down from the boundary to the horizon. The subsystem $A$ is given by the blue section. Its characteristic length is $l$.
• Figure 2: Figure \ref{['fig:1-a']} shows the behavior of $l$ as a function of $y_*$ for constant $u_0$. Figure \ref{['fig:1-b']} Shows the difference in area as a function of $l$. These plots show no phase transition the entropy in the nonextremal D3 brane background.
• Figure 3: Figure \ref{['fig:2-a']} shows the behavior of $lu_0^{(p-5)/2}$ as a function of $y_*$ at $p=3,4,5$ (Blue, Green, Orange respectively). Figure \ref{['fig:2-b']} Shows the difference in area as a function of $l$. These plots show no phase transition for the entanglment entropy in the backgrounds of nonextremal D3, D4 and D5 branes.
• Figure 4: Figure \ref{['fig:finiteT6']}(a) shows the behavior of $l/\sqrt{u_0}$ as a function of $y_*$ for constant $u_0$ at $p=6$. Figure \ref{['fig:finiteT6']}(b) Shows the difference in area as a function of $l$. These plots show a transition for the entanglement entropy in the nonextremal D6 brackground.
• Figure 5: This plots shows the behavior of the temperature as a function of $\rho^2$.