New dynamical instability in asymptotically anti–de Sitter spacetime

We present fully dynamical solutions to Einstein-scalar theory in asymptotically anti–de Sitter spacetime with a scalar potential containing particularly rich physics. Depending on one parameter in the potential, we find an especially interesting regime, which exhibits a dynamically unstable black brane, already at zero momentum, while nevertheless having positive specific heat. We show this using the nonlinear dynamics and give a clear interpretation in terms of the spectrum of linearized perturbations. Our results translate directly to their dual strongly coupled nonconformal field theories, whereby we in particular provide two mechanisms to obtain equilibration times much larger than the inverse temperature.

Figure 1

(a) We show the dynamical solution for $\tilde{\mathrm{\Phi }}$ [as in Eq. (8)] with initial conditions given by $j=0.2$, $a_4(0)=-1$ and $\tilde{\mathrm{\Phi }}(z,0)=1.06822$ for the potential (2) with $\gamma =4\sqrt{2}/3$. Interestingly, this evolution finds both an unstable and a stable solution. The dynamics hence goes through three regimes, as more clearly shown in (b). There, we show the difference $\delta ⟨\mathcal{O}⟩(t)$ between $⟨\mathcal{O}⟩(t)$ and the equilibrium values $⟨\mathcal{O}⟩(1.55/T_f)$ and $⟨\mathcal{O}⟩(7/T_f)$ of the unstable and stable phases in solid and dashed black respectively. Both plots show time in units of the final temperature $T_f$. The system first equilibrates to the unstable phase, dominated by two QNMs of frequencies $({\omega }_X,{\omega }_{\mathrm{\Delta }})/\pi T=(3.23-1.93i,-2.18i)$. After this, the unstable QNMs becomes dominant, and this mode grows exponentially with ${\omega }_{\square }/\pi T=0.92i$. Lastly, this mode leaves the linear regime, and the solution finally rings down to the stable solution with ${\omega }_O/\pi T=1.82-1.79i$.

Figure 2

We show the first two QNM of both the stable (blue) and unstable (red) solutions as a function of the source over the temperature. The unstable solution always contains a mode with a positive imaginary part (solid red), which goes to zero for the maximum source allowed. At this point, the stable solution also has a vanishing imaginary part, which corresponds to a diverging relaxation time. Interestingly, the lowest QNM of the stable phase has vanishing real part for $j/T>0.95$, after which the imaginary parts separate. This is needed for the modes to meet at the maximum source, where both phases coalesce.

Figure 3

(a) We show the expectation of the scalar operator as a function of the source over the temperature for both the stable (blue) and unstable (red) phase. There are also two other branches shown dashed, which we comment on in Sec. 3. (b) For the maximum source, the equilibration time diverges, and the system equilibrates in a time $\gg 1/T$. (c) When evolving initial data slightly different from Fig. 1, the coefficient of the unstable QNM changes sign, and the solution grows instead of decreasing at late times.

Figure 4

(a) We show the unstable QNM as a function of $\gamma$. It is especially interesting that the relaxation time diverges as $\gamma$ approaches the critical value from above. (b, c) We show the entropy and free energy as a function of the temperature over the minimum temperature found for fixed source $j$. Both curves are normalized with their value at the minimum temperature. Interestingly, both the dynamically unstable (red) and stable (black) black brane phases have positive specific heat. Third and fourth branches are shown dashed, which smoothly connect to the thermal gas phase for high temperature, whereby as usual the small branch has negative specific heat.

Figure 5

Here, the first eight QNMs of Fig. 2 are shown on the complex plane as a function of $j$. The lowest stable QNM (blue) hits the imaginary axis at $j/T=0.95$, after which the modes separate on the axis to meet the unstable QNMs at $j=j_{\max }$.