Kerr-AdS analogue of triple point and solid/liquid/gas phase transition

TL;DR

This work studies the thermodynamics of six-dimensional Kerr–AdS black holes in the canonical ensemble with fixed angular momenta $J_1$ and $J_2$, using the control parameter $q = J_2/J_1$ to classify phase structure. The authors analyze the Gibbs free energy $G = M - TS$ as a function of pressure $P$, temperature $T$, and angular momenta to reveal a sequence of regimes: a reentrant large/small/large transition at $q=0$, a solid/liquid analogue for small $q$, a solid/liquid/gas analogue with a triple (tricritical) point and two critical points for $0.00905<q<0.0985$, and a standard Van der Waals liquid/gas behaviour for $q>0.0985$. The phase diagrams exhibit coexisting small/intermediate/large black hole phases, multiple critical points, and lines terminating at critical points, illustrating a rich parallel with familiar thermodynamic systems. These results have implications for AdS/CFT dual gauge theories and suggest that such complex phase structures may persist in higher dimensions and for equal-spin configurations in odd dimensions.

Abstract

We study the thermodynamic behavior of multi-spinning d=6 Kerr-anti de Sitter black holes in the canonical ensemble of fixed angular momenta J1 and J2. We find, dependent on the ratio q=J2/J1, qualitatively different interesting phenomena known from the `every day thermodynamics' of simple substances. For q=0 the system exhibits recently observed reentrant large/small/large black hole phase transitions, but for 0<q<<1 we find an analogue of a `solid/liquid' phase transition. Furthermore, for 0.00905<q<0.0985 the system displays the presence of a large/intermediate/small black hole phase transition with one tricritical and two critical points. This behavior is reminiscent of the solid/liquid/gas phase transition except that the coexistence line of small and intermediate black holes does not continue for an arbitrary value of pressure (similar to the solid/liquid coexistence line) but rather terminates at one of the critical points. Finally, for q>0.0985 we observe the `standard liquid/gas behavior' of the Van der Waals fluid.

Figures (6)

• Figure 1: Gibbs free energy for $q=0$. Pressures increase from left to right and solid-red/dashed-blue lines correspond to $C_P$ positive/negative respectively; at their joins $C_P$ diverges. Moving from right to left, 'near vertical' solid red curves have an increasingly large negative slope. As $T\to \infty$ they asymptote to $-\infty$; for sufficiently small $T$, each curve joins its blue-dashed counterpart at some large positive value of $G$, the limit of zero temperature is attained in the asymptotically flat case $P\to 0$. As with Schwarzschild-AdS black holes, for $P\geq P_c$, the (lower) large BH branch is thermodynamically stable whereas the upper branch is unstable. For $P=P_c$ we observe critical behavior. For $P\approx 0.0564\in(P_t,P_z)$ we observe a "zeroth-order phase transition": a discontinuity in the global minimum of $G$ at $T=T_0\approx 0.2339\in(T_t,T_z)$ (denoted by the vertical line in the inset) signifying the onset of an reentrant phase transition. For $P<P_t$ only one branch of stable large BHs exists.
• Figure 2: $P-T$ diagram for $q=0$. The coexistence line of the first order phase transition between small and large black holes is depicted by a thick black solid line. It initiates from the critical point $(P_c, T_c)$ and terminates at $(P_t, T_t)$. The red solid line (inset) indicates the 'coexistence line' of small and intermediate black holes, separated by a finite gap in $G$, indicating the reentrant phase transition. It commences from $(T_z, P_z)$ and terminates at $(P_t, T_t$). The "No BH region" is to the left of the dashed oblique curve, containing the $(T_z, P_z)$ point. Below $T_t$, the lower dashed curve terminates at the origin $P=T=0$.
• Figure 3: Gibbs free energy for $q=0.005$ is displayed for decreasing pressures (from top to bottom). The horizon radius $r_+$ increases from left to right. The uppermost isobar corresponds to $P=P_{c}=4.051$; for higher pressures only one branch of (stable) black holes with positive $C_P$ exists. The second uppermost isobar displays the swallowtail behavior and implies the existence of a first order phase transition. For $P=P_v\approx 0.0958$ another critical point emerges but occurs for a branch that does not minimize $G$ globally. Consequently, out of the two swallowtails for $P<P_v$ only one occurs in the branch globally minimizing $G$ and describes a 'physical' first order phase transition.
• Figure 4: Gibbs free energy for $q=0.05$, displayed for various pressures (from top to bottom) $P=0.259, 0.170, 0.087, 0.0641, 0.015$. The horizon radius $r_+$ increases from left to right. The uppermost isobar corresponds to $P=P_{c_1}=0.259$; for higher pressures only one branch of stable black hole with positive $C_P$ exists. The second uppermost isobar displays the swallowtail behavior, implying a first order phase transition. The third isobar corresponds to $P_{c_2}=0.0956<P<P_{c_1}$ for which we have 'two swallowtails'. For such pressures there are two first order phase transitions. The fourth isobar displays the tricritical pressure $P_{tr}=0.087$ where the two swallowtails 'merge' and the triple point occurs. Finally the lower-most isobar corresponds to $P<P_{tr}$.
• Figure 5: $P-T$ diagram for $q=0.05$. The diagram is analogous to the solid/liquid/gas phase diagram. Note however that there are two critical points. That is, the solid-liquid coexistence line does not extend to infinity but rather terminates, similar to the "liquid/gas" coexistence line, in a critical point.