Superfluid Black Holes

We present what we believe is the first example of a “$\lambda$-line” phase transition in black hole thermodynamics. This is a line of (continuous) second order phase transitions which in the case of liquid ${}^4\mathrm{He}$ marks the onset of superfluidity. The phase transition occurs for a class of asymptotically anti–de Sitter hairy black holes in Lovelock gravity where a real scalar field is conformally coupled to gravity. We discuss the origin of this phase transition and outline the circumstances under which it (or generalizations of it) could occur.

Figure 1

Thermodynamic behavior near the $\lambda$ transition. Left: Plot of the Gibbs free energy versus temperature for three distinct pressures chosen so that critical temperatures are $t_c=3$, 5, 7, corresponding to the red, blue, and black curves. The dotted lines highlight the points where the second derivative of the Gibbs free energy diverges. Center: Plot of the specific heat $c_p=-t({\partial }^{2}g/\partial t^2)$ for the case $t_c=3$. Right: $p-t$ parameter space. The black line shows the locus of critical points, i.e., a line of second order phase transitions known as the “$\lambda$” line in the context of superfluidity. These plots are for $d=7$.

Figure 2

Thermodynamic properties of ${}^4\mathrm{He}$. Top: The $P-T$ phase diagram for ${}^4\mathrm{He}$. The $\lambda$ line corresponds to a line of critical points where a second order phase transition occurs marking the onset of superfluidity. To the left of the $\lambda$ line, the liquid begins to exhibit remarkable properties. Bottom: The specific heat of liquid ${}^4\mathrm{He}$. As the $\lambda$ line is approached, the specific heat spikes taking the shape of the greek letter “$\lambda$.” These plots have been reprinted with permission from Ref. [32].