Deformations of JT Gravity and Phase Transitions

TL;DR

<3-5 sentence high-level summary> The paper studies a family of two-dimensional dilaton gravity theories that asymptote to JT gravity by requiring W(φ) ~ 2φ at infinity and analyzes their classical black hole solutions and thermodynamics. It derives explicit static solutions, establishes the first-law relations among energy, entropy, and horizon data, and shows that entropy bounded below requires W to dip negative somewhere. The authors demonstrate that black holes with negative specific heat imply the existence of a same-temperature stable branch with positive heat capacity and lower free energy, and that phase structure features Hawking-Page-like first-order transitions and possible zero-temperature ground-state changes when varying W. They also construct compact Euclidean (closed-universe) solutions and discuss their Lorentzian continuations, highlighting a rich interplay between horizon physics, phase transitions, and global spacetime topology, with potential links to matrix-model descriptions.

Abstract

We re-examine the black hole solutions in classical theories of dilaton gravity in two dimensions. We consider an arbitrary dilaton potential such that there are black hole solutions asymptotic at infinity to the nearly $\mathrm{AdS}_2$ solutions of JT gravity, and such that the black hole energy and entropy are bounded below. We show that if there is a black hole solution with negative specific heat at some temperature $T$, then at the same temperature there is a black hole solution with lower free energy and positive specific heat. As the temperature is increased from 0 to infinity, the black hole energy and entropy increase monotonically but not necessarily continuously; there can be first order phase transitions, similar to the Hawking-Page transition. These theories can also have solutions corresponding to closed universes.

Figures (5)

• Figure 1: Here we assume that there is a black hole at $\phi_h=\phi_1$ with $W'(\phi_1)=0$ and we explore its themodynamic stability. (a) In this example, there is some $\phi_2>\phi_1$ with with $W(\phi_1)=W(\phi_2)$ and $W(\phi)<W(\phi_1)$ for $\phi_1<\phi<\phi_2$. A black hole with $\phi_h=\phi_1$ is then always thermodynamically disfavored compared to a black hole at $\phi_h=\phi_2$ (if such a black hole exists). They have the same temperature, but the one with $\phi_h=\phi_2$ has lower free energy. The free energy difference between the two black holes is the area of the shaded region (times $2\pi$). (b) In this example, there is $\phi_0<\phi_1$ with $W(\phi_0)=W(\phi_1)$ and $W(\phi)>W(\phi_1)$ for $\phi_0<\phi<\phi_1$. With our assumptions, such a $\phi_0$ always exists and there is a black hole with $\phi_h=\phi_0$. The black hole at $\phi_h=\phi_0$ is always thermodynamically favored compared to the one at $\phi_h=\phi_1$; the free energy difference between them is determined by the area of the shaded region.
• Figure 2: This figure depicts a function $W(\phi)$ with the properties needed to illustrate an argument in the text. The lower horizontal line intersects points with $W=0$ and the upper one intersects points with $W=W(\phi_1)>0$. We have $W(\phi_1)=W(\phi_2)=W(\phi^*)=W(\phi_3)$ with $\phi_1<\phi_2<\phi^*<\phi_3$; moreover $W'(\phi_1), W'(\phi^*)<0$ while $W'(\phi_2),W'(\phi_3)>0$. For simplicity, we have assumed that the region in which $\phi>\phi_1$ and $W(\phi)<0$ is a connected interval with upper end-point $\phi'$. This ensures that for $\phi_h$ equal to $\phi_1$ or $\phi_2$, the condition (\ref{['mod']}) for existence of a black hole solution with given $\phi_h$ only has to be checked for $\phi=\phi'$. If $W$ is such that $\int_{\phi_1}^{\phi'}\mathrm d \phi \,W(\phi)>0$ but $\int_{\phi_2}^{\phi'}\mathrm d\phi, W(\phi)<0$, then there is a black hole with $\phi_h=\phi_1$ but none with $\phi_h=\phi_2$. As explained in the text, in this case the black hole at $\phi_h=\phi_3$ is thermodynamically favored over the one at $\phi_h=\phi_1$.
• Figure 3: A function $W(\phi)$ that is positive for $\phi>\phi_0$ but such that $W'(\phi)$ has two zeroes in that region. Horizontal lines intersect points with the same value of $\phi_h$, corresponding to black holes with the same temperature. Starting at $\phi_0$ and moving up the curve, the black hole remains thermodynamically stable up to and beyond the point labeled $\alpha$. Thermodynamic stability is lost at the point $\beta$, which is characterized by the fact that the shaded regions above and below the horizontal line $\beta\beta'$ have the same area. As the black hole temperature is increased, the point that labels the black hole solution moves continuously along the curve from $\phi_0$ up to $\beta$, jumps to $\beta'$, and then continues upwards from there. The jump from $\beta$ to $\beta'$ is a first order phase transition, analogous to the Hawking-Page phase transition.
• Figure 4: In this example, the function $W(\phi)$ vanishes precisely at $\phi_0,\phi_1$, and $\phi^*$. It is positive for $\phi>\phi_1$ and negative for $\phi<\phi_0$. There is always an extremal black hole solution of zero temperature with $\phi_h=\phi_1$. There is an extremal black hole solution of zero temperature with $\phi_h=\phi_0$ if and only if the shaded area above the horizontal line exceeds the shaded area below the horizontal line. (This is the condition for eqn. (\ref{['mod']}) to hold at $\phi_h=\phi_0$.) If a black hole solution exists at $\phi_h=\phi_0$, it is the true ground state. If one varies $W(\phi)$ so that the shaded area above the horizontal line no longer exceeds the shaded area below the horizontal line, then there is a first order phase transition and the ground state jumps to $\phi_h=\phi_1$.
• Figure 5: In the model of fig. \ref{['Triple']}, the zero temperature ground state is at $\phi_h=\phi_0$. As one increases the temperature, one reaches a first order phase transition at which the system jumps across the "gap" that consists of the semi-open interval $[\phi^*,\phi_1)$, where there is no classical black hole solution. The phase transition occurs at a point $\beta$ characterized by the fact that the shaded regions above and below the horizontal line $\beta\beta'$ have the same area. Thermodynamically stable black holes start at $\phi_0$ at zero temperature, follow the curve from $\phi_0$ to $\beta$ as the temperature is increased, jump to $\beta'$, and then follow the curve upward as the temperature is increased further. The interval $(\beta,\beta')$ across which the system jumps contains the interval $[\phi^*,\phi)$ where there is no black hole solution, but it is strictly larger: it also contains intervals in which classical black hole solutions exist but are not thermodynamically stable.