Counter-examples to the correlated stability conjecture

TL;DR

The paper challenges the Correlated Stability Conjecture (CSC) by constructing explicit counter-examples where horizon instabilities arise from near-horizon phase transitions rather than negative thermodynamic susceptibilities. It develops three classes of models—magnetically charged branes, an AdS5 two-scalar system, and finite-temperature deformations of N=4 SYM to N=1*—to realize Gregory-Laflamme–type instabilities and horizon hair, with a common infrared signature in the operator χ of dimension Δ̄χ = 1/2. A key finding is that a second-order phase transition can occur at finite temperature well above the confinement scale, accompanied by a GL instability on the disordered branch. The work suggests refining CSC to require a unique uniform background for given charges or to include asymptotic scalar data in the thermodynamic stability analysis, thereby linking dynamical horizon stability more closely to horizon-scale phase structure.

Abstract

We demonstrate explicit counter-examples to the Correlated Stability Conjecture (CSC), which claims that the horizon of a black brane is unstable precisely if that horizon has a thermodynamic instability, meaning that its matrix of susceptibilities has a negative eigenvalue. These examples involve phase transitions near the horizon. Ways to restrict or revise the CSC are suggested. One of our examples shows that N=1* gauge theory has a second order chiral symmetry breaking phase transition at a temperature well above the confinement scale.

Figures (5)

• Figure 1: Propagator for $\phi_0$$=$$0.95$$\phi_c$, where $\phi_c = 0.428$. The lack of a singularity indicates the absence of a normalizable mode.
• Figure 2: Propagator for $\phi_0=1.05 \phi_c$. The singularity indicates a normalizable, stationary mode, and the existence of a GL instability for $k < k_c$.
• Figure 3: Plot showing non-zero critical values of $k_c$ for $\phi_0 > \phi_c$ (the ordered phase). $k_*$ is the value of $k$ at which $G(k) = 0$.
• Figure 4: Inverse propagator for $(\phi_0-\phi_c)/\phi_0 \approx 10^{-6}$.
• Figure 5: The thick blue lines show the values $(\phi_0,\chi_0)$ of the scalars at the horizons we were able to produce numerically. Dark blue indicates stable solutions. Light blue indicates solutions with a GL instability. The thin green trajectories are the holographic RG flows (\ref{['SeveralFlows']}). They are found most simply as the gradient flows of a superpotential $W$, whose contours are shown in red. The aspect ratio of this figure is not 1:1, so it is not readily apparent that the green trajectories are orthogonal to the red contours. The $C=0$ trajectory is asymptotic to the black curve, which is part of the locus where the gradient of $W$ is parallel to the gradient of $V$. The $C<0$ trajectories are shown in solid green, and the $C>0$ trajectories are shown in dashed green.