New stability results for Einstein scalar gravity

TL;DR

The paper addresses the stability of asymptotically AdS spacetimes in Einstein gravity coupled to a scalar field with mass near the BF bound, where boundary conditions are encoded by an arbitrary function $W$. It extends the minimum energy theorem to cases where $W$ is unbounded below by introducing a one‑parameter family of superpotentials $P_s(oldsymbol{
})$ and showing energy remains bounded provided $W(oldsymbol{
})+{s_cigl|oldsymbol{
}igr|^3}/{3}$ is bounded from below, connecting this generalized bound to the conjectures on designer gravity. The work then relates the bulk result to the dual field theory, demonstrating that one can add negative multi‑trace deformations (e.g., negative double traces) and still possess a stable vacuum, potentially triggering spontaneous symmetry breaking or novel RG flows on spheres via soliton solutions and domain walls. The analysis extends to other masses and dimensions (with a corresponding bound involving $|oldsymbol{
}|^{d/oldsymbol{
}_-}$) and uncovers a subtle dependence on operator dimensions near unitarity, including a surprising regime where multi‑trace operators qualitatively alter stability. Overall, the results broaden the landscape of stable AdS ground states under general boundary conditions and unveil new holographic mechanisms for symmetry breaking and superconductivity without chemical potential.

Abstract

We consider asymptotically anti de Sitter gravity coupled to a scalar field with mass slightly above the Breitenlohner-Freedman bound. This theory admits a large class of consistent boundary conditions characterized by an arbitrary function $W$. An important open question is to determine which $W$ admit stable ground states. It has previously been shown that the total energy is bounded from below if $W$ is bounded from below and the bulk scalar potential $V(φ)$ admits a suitable superpotential. We extend this result and show that the energy remains bounded even in some cases where $W$ can become arbitrarily negative. As one application, this leads to the possibility that in gauge/gravity duality, one can add a double trace operator with negative coefficient to the dual field theory and still have a stable vacuum.

Figures (4)

• Figure 1: Solutions to (\ref{['Peq']}). Blue curves are generic solutions. The black dashed line is $P_c$, the solution with the largest value of $s_c$ which exists globally. The green and purple dotted lines are $P_\pm$ (see footnote 3). For $V=-3-\phi^2$, $s_c=0.52$. For $V=5/2-6\cosh(\phi/\sqrt{2})+\cosh(\sqrt{2}\phi)/2$, which comes from a consistent supergravity truncation Gauntlett:2009dn, $s_c=.56$. The orange dashed curve in the right plot corresponds to $P^{IR}_+$ (recall $P^{IR}_-=P_c$.) The gray region in the right plot is where $2P'^2-V<0$ and therefore $P$ is not real.
• Figure 2: Plots of $\beta_0(\alpha)$ for various bulk potentials. All soliton curves have the same linear behavior at small $\alpha$ and have asymptotic scale invariance $\beta_0=-s_w\alpha^2$. The left figure is for $V=-3-\phi^2,~s_w= 0.52$ (solid) and the potential used in Gauntlett:2009dn$V=5/2-6\cosh(\phi/\sqrt{2})+\cosh(\sqrt{2}\phi)/2,~s_w=0.56$ (dashed.) The right figure is for (\ref{['genSUGRApot']}) with for $b=c=1,~0.5,~0.25,~0.1$ (solid), $b=c=0$ (dashed.) We find that as $b,c\rightarrow 0$, $s_w=s_c\rightarrow 0$.
• Figure 3: Plot of $P'$ analogous to Fig. 1 for a $\phi^4$ potential with $\Delta=3/5$ and two values of $\lambda$. (left) The $P_-$ curve crosses the axis before the second extremum, showing that there is no minimum energy theorem for Neumann boundary conditions. (right) Here the $P_-$ exists globally and there is a minimum energy theorem for Neumann boundary conditions. Because in this case $\lambda$ is negative there is no second extremum in this potential however the critical curve $P_c$ still exists.
• Figure 4: Plots of $s_c$ as a function of $\Delta_-$ for two different choices of $\lambda$. The solid red curve is for $\lambda = 1$ and the dashed black curve is for $\lambda = -1/2$. Recall that the condition for a minimum energy theorem with $\beta = 0$ boundary condition is $s_c \ge 0$. Close to $\Delta_- = 3/4$ the behavior of $s_c$ is determined by the pole in (\ref{['eq:pole']}). The sign of the coefficient of the pole is different for the two cases presented here.