Gravitational collapse in SYK models and Choptuik-like phenomenon

TL;DR

This work presents a solvable holographic setup for gravitational collapse using a deformed SYK model subjected to a second quantum quench. The boundary dynamics are governed by a modified Schwarzian action with a mass gap, and the bulk dual is JT gravity with matter. A critical deformation Δε_c separates horizonless AdS2 from black-hole spacetimes, yielding a Choptuik-like scaling T_{bh} ∝ (Δε−Δε_c)^{1/2} when a BH forms. Correlation functions exhibit non-analytic behavior at the quench and thermalize in the BH phase, while a two-coupled-SYK extension generalizes the mechanism to richer bulk dynamics. Overall, the paper provides a concrete, tractable framework linking boundary quenches to bulk horizon formation and critical phenomena in a holographic setting.

Abstract

SYK model is a quantum mechanical model of fermions which is solvable at strong coupling and plays an important role as perhaps the simplest holographic model of quantum gravity and black holes. The present work considers a deformed SYK model and a sudden quantum quench in the deformation parameter. The system, as in the undeformed case, permits a low energy description in terms of pseudo Nambu Goldstone modes. The bulk dual of such a system represents a gravitational collapse, which is characterized by a bulk matter stress tensor whose value near the boundary shows a sudden jump at the time of the quench. The resulting gravitational collapse forms a black hole only if the deformation parameter $Δε$ exceeds a certain critical value $Δε_c$ and forms a horizonless geometry otherwise. In case a black hole does form, the resulting Hawking temperature is given by a fractional power $T_{bh} \propto (Δε- Δε_c)^{1/2}$, which is reminiscent of the `Choptuik phenomenon' of critical gravitational collapse.

Figures (13)

• Figure 1: The red dots are the states $\left\lvert{B_s}\right\rangle$ (above) and $\left\langle{B_s}\right\rvert \IfNoValueTF{}{}{}$ (below). If we have insertions of operators which are flip symmetric then the boundary condition (red dots) can be converted to trace boundary condition. In such a case we identify the two blue lines and the Euclidean path of contour becomes a circle.
• Figure 2: Quench protocol used in this paper. A non-zero value of $\hat{\epsilon}=\epsilon_1$ is turned on, as in Kourkoulou:2017zaj, at time $t=0$ (which we call the first quench). After a time $t=T$, a second quench is performed, to $\hat{\epsilon}=\epsilon_2$.
• Figure 3: The plot of exact (blue) and approximate (orange) $P(r)$ for a) $N=20$ and b) $N=100$ (the curves are overlapping). At large $N$ distribution is sharply peaked around the average and the approximation is more accurate.
• Figure 4: Possible shapes of the potential, depending on the sign of $\hat{\epsilon}$. We have bounded or scattering solutions depending on whether $E<0$ or $E>0$. It is assumed for simplicity that $\Delta <\frac{1}{2}$.
• Figure 5: We have taken here $\epsilon_1= \frac{8\pi}{\beta J}$ for simplicity. With this value, the $\phi$-particle sits at the bottom for all times $0<t<T$, with the value of $\phi$ given by $\phi_1$ in (\ref{['phi1']}). At $t=T$, we perform the quench $\epsilon_1 \to \epsilon_2$, which modifies the shape of the potential. Depending on the value of $\epsilon_2$, the line $\phi=\phi_1$ may hit the potential curve at the right turning point below the $\phi$-axis (in which case the geometry remains horizonless) or above (in which case the geometry develops a horizon). At the critical value $\epsilon_2 =\epsilon_{cr}$, (\ref{['ep-cr-special']}), the shape of the potential is such that the line $\phi=\phi_1$ hits the potential curve exactly on the $\phi$-axis. For $\epsilon_2 < \epsilon_{cr}$, this intersection point rises above the $\phi$-axis, corresponding to the formation of a black hole.