Finite-deformation stability and thermodynamic preference of hairy black branes

Determine whether the finite-deformation scalar-hairy black-brane solution develops a genuine region with negative speed of sound squared, indicating a spinodal thermodynamic instability, and establish whether the hairy phase is thermodynamically preferred over the bald AdS$_4$–Schwarzschild phase.

Background

The paper computes the speed of sound only through quadratic order in the scalar-charge parameter. The resulting truncated expression decreases from the conformal value cs2=1/2c_s^2=1/2 and formally reaches zero at approximately ϵ≈0.46\epsilon\approx0.46, beyond which it becomes negative. A negative cs2c_s^2 would imply negative specific heat and a long-wavelength sound-mode instability of the homogeneous hairy plasma.

The authors emphasize that the perturbative expansion is uncontrolled when the quadratic correction becomes comparable to the leading term. Reorganizing the same second-order thermodynamic data without re-expanding the ratio produces a different expression that remains positive, so the perturbative calculation cannot determine whether a spinodal region actually exists. They state that finite-ϵ\epsilon numerical black-brane solutions are required both to settle the stability question and to determine whether the hairy phase is thermodynamically preferred relative to the bald AdS4_4–Schwarzschild solution.

References

Our expansion cannot decide whether this happens. At $\epsilon\approx0.46$ the $\mathcal{O}(\epsilon{2})$ correction in~eq:cs2_exact equals $100\%$ of the conformal value $\tfrac12$, so the neglected $\mathcal{O}(\epsilon{4})$ term is of the same size and the series is no longer controlled. The ambiguity is explicit: organizing the same $\mathcal{O}(J{2})$ information without re-expanding the ratio, i.e. evaluating $(dp/dT){J}/(d\varepsilon/dT){J}$ directly from~eq:thermo_full-eq:thermo_eps, gives a $c_{s}{2}$ that decreases monotonically from $\tfrac12$ to $1/(\sqrt{33}-1)=(1+\sqrt{33})/32\approx0.211$ and never vanishes, since $(dp/dT){J}$ and $(d\varepsilon/dT){J}$ are then manifestly positive sums. Two equally legitimate treatments of the same data therefore disagree about the existence of the zero, which settles the matter: the sign of $c_{s}{2}$ at $\epsilon\gtrsim0.46$ is not determined at this order. We therefore quote $\epsilon\lesssim0.46$ as the range over which $c_{s}{2}$ is quantitatively reliable--consistent with, and marginally sharper than, the thermodynamic bound $\epsilon\lesssim0.5$ of Sec.~\ref{sec:disc}--and we make no claim of a spinodal instability. Settling that question requires the finite-$\epsilon$ numerical solution discussed in Sec.~\ref{sec:disc}, which would also determine whether the hairy phase is thermodynamically preferred.

eq:cs2_exact:

cs2=12+33 (3−33)24 χ0 J2 h33−3+O(J4).\boxed{ c_{s}^{2} = \frac{1}{2} + \frac{\sqrt{33}\,(3-\sqrt{33})}{24}\,\chi_{0}\, J^{2}\,h^{\sqrt{33}-3} + \mathcal{O}(J^{4}). }

— Perturbative Hairy Black Branes in the $G_{2}$-Invariant Sector of Dyonic ISO(7) Gauged Supergravity  (2608.18959 - Yun, 19 Aug 2026) in Section 5.3, subsection “On the apparent zero of $c_s^2$”; see also Section 6, subsection “Outlook,” paragraph (i)