Exotic Hairy Black Holes

TL;DR

This work studies finite-temperature critical phenomena in a holographic setting: a deformed $2+1$-dimensional CFT realized by a bulk action in asymptotically $AdS_4$ spacetime with scalar hair. The authors engineer finite-temperature RG flows by including a relevant operator $O_r$ (dim 1 or 2) and an irrelevant operator $O_i$ (dim 4) that mixes with coupling $g$, and they analyze Schwarzschild black holes to extract the dual plasma thermodynamics and verify the first law. They reveal a rich phase structure that includes high-temperature symmetry-broken phases metastable relative to the symmetric phase, undergoing a mean-field transition with $\langle O_i \rangle ∼ (T-T_c)^{1/2}$. They further show the existence of a critical mixing $g_c<0$, whose variation controls the presence and extent of the broken phases, and discuss avenues for analytic proofs, string theory embeddings, and connections to condensed-matter systems.

Abstract

We study black hole solutions in asymptotically AdS_4 spacetime with scalar hair. Following AdS/CFT dictionary these black holes can be interpreted as thermal states of 2+1 dimensional conformal gauge theory plasma, deformed by a relevant operator. We discover a rich phase structure of the solutions. Surprisingly, we find thermodynamically stable phases with spontaneously broken global symmetries that exist only at high temperatures. These phases are metastable, and join the stable symmetric phase via a mean-field second-order phase transition.

Figures (4)

• Figure 1: The free energy density of hairy black holes in asymptotic $AdS_4$ geometry for different deformations of the UV fixed point. The red curves correspond to symmetric phases with $\langle {\cal O}_i\rangle =0$. The other curves correspond to phases with spontaneously broken ${\mathbb Z}_2$ symmetry, $\langle {\cal O}_i\rangle \ne 0$. The purple curves represent symmetry broken phases with the holographic condensate wavefunction $\chi(x)$ not having any nodes for $x\in [0,1]$. The green curves represent symmetry broken phases with the holographic condensate wavefunction $\chi(x)$ having exactly one node for $x\in [0,1]$.
• Figure 2: The energy density of hairy black holes in asymptotic $AdS_4$ geometry for different deformations of the UV fixed point. The color coding is as in figure \ref{['figure1']}.
• Figure 3: The speed of sound of hairy black holes in asymptotic $AdS_4$ geometry for different deformations of the UV fixed point. The color coding is as in figure \ref{['figure1']}.
• Figure 4: The difference between the free energy density of the "purple" symmetry broken phase and the symmetric phase (see figure \ref{['figure1']}) for $\frac{\Lambda}{T}\to 0$ as a function of $\ln(-g)$. The straight line fit to the data produces the following functional dependence: $\Delta\approx 2.32861 - 1.01493 \ln(-g)$.