SYK wormhole formation in real time

TL;DR

The paper investigates real-time formation of the ground state of two coupled SYK models, showing that cooling through a negative-specific-heat regime yields a wormhole-like ground state in a time that is independent of the system size. Using large-N Kadanoff–Baym equations with a bath, the authors demonstrate smooth, quasi-adiabatic evolution that tracks microcanonical equilibrium configurations and culminates in a state near the thermofield double. A Schwarzian effective theory explains the low-temperature cold/hot wormhole branches and their connection to JT gravity, while a gravity discussion clarifies how boundary-condition flows can facilitate topology change in the wormhole formation. The results illuminate how wormhole formation can occur efficiently in SYK-like systems and highlight qualitative differences with more generic gravity theories, with implications for preparing entangled, low-energy states and for holographic understanding of wormholes.

Abstract

We study the real time formation of the ground state of two coupled SYK models. This is a highly entangled state which is close to the thermofield double state and can be viewed as a wormhole. We start from a high temperature state, we let it cool by coupling to a cold bath. We numerically solve for the large N dynamics. Our main result is that the system forms a wormhole by going through a region with negative specific heat, taking time that is independent of N. The dynamics is smooth everywhere and it seems to follow the equilibrium thermodynamic configurations of the microcanonical ensemble. Also we comment on the relation between this coupled SYK model and Jackiw-Teitelboim gravity theory with bulk fields.

Figures (14)

• Figure 1: Dots: energy vs beta obtained by numerically solving DS equations for two coupled SYK models with $J=0.5, \mu=0.0053$. Blue dots correspond to the "two black holes phase", whereas green dots correspond to the "cold wormhole phase". Red dashed line: curve for the "hot wormhole" phase expected from a low energy analytic analysis. The question mark "?" reminds us that we were not able to find it as a solution of the euclidean DS equations.
• Figure 2: Euclidean Green function $G_{LR}$. The blue points correspond to the exact solution, and the red ones to the conformal answer (\ref{['glr_conf']}). Left: $\beta=20$. Right: $\beta=53$. For this values of parameters the transition to the wormhole phase happens around $\beta_{\rm 2 BH} \sim 61$. The approximation is better for higher temperatures.
• Figure 3: A reprint of the phase diagram obtained numerically in MQ for $J=0.5$. The right solid black curve indicates $T_{\rm WH}$, purple line $T_c$ and left solid black line $T_{\rm 2BH}$. The dashed horizontal line is at $\mu=0.05$ , the value of $\mu$ we will use in our real time numerical simulation. In this case $\beta_{\rm 2 BH} \sim 61, ~\beta_c \sim 54,~ \beta_{\rm WH} \sim 49$.
• Figure 4: The fit for $T_{\rm WH}$ using the numerical data from MQ in Figure \ref{['mq_phase']}, using only data points with $\mu < 0.03$. The fit is consistent with the analytical prediction $\mu \sim T^{3/2}$.
• Figure 5: The fit for $T_{\rm 2BH}$ using the numerical data from MQ in Figure \ref{['mq_phase']}, using only data points with $\mu < 0.03$. The fit is consistent with the analytical prediction $\mu \sim \sqrt{T}$.