Four coupled SYK models and Nearly AdS$_2$ gravities: Phase Transitions in Traversable wormholes and in Bra-ket wormholes

TL;DR

This work analyzes four coupled SYK models and four coupled JT gravities to understand phase transitions between traversable wormholes and bra-ket wormholes as couplings are varied, with zero-temperature first-order transitions revealing a $Z_2$-type order parameter. The authors develop large-$N$ Schwinger-Dyson equations and Schwarzian-based effective actions to characterize the wormhole saddles, including LR and 1-2 couplings, and show how boundary interactions can lengthen or shorten wormholes depending on the phase. By Wick-rotating to braket-wormhole configurations, they explore state preparation, projections, and entangling operations that can annihilate braket wormholes, tying these dynamics to unitarity in flat-space regions and island-based entanglement entropy. They furthermore embed holographic CFTs and free CFTs within these setups, discuss partial couplings and multiple matter sectors, and connect to replica-wormhole concepts, thereby providing a broad framework for entanglement-geometry interplay in both SYK-like quantum systems and nearly AdS$_2$ gravities.

Abstract

We study four coupled SYK models and nearly AdS$_2$ gravities. In the SYK model side, we construct a model that couples two copies of two coupled SYK models. In nearly AdS$_2$ gravity side, we entangle matter fields in two copies of traversable wormholes. In both cases, the systems show first order phase transitions at zero temperature by changing couplings, which is understood as the exchange of traversable wormhole configurations. In nearly AdS$_2$ gravity cases, by exchanging the role of space and time the wormholes are interpreted as bra-ket wormholes. In Lorentzian signature, these bra-ket wormholes lead to two closed universes that are entangled with each other as well as matter fields in the flat space without dynamical gravity. We study the effect of projection or entangling operation for matters on flat spaces and they cause phase transitions in bra-ket wormholes, which leads to the pair annihilation of closed universes. Using these bra-ket wormholes, we discuss the way to embed states in 2d holographic CFTs into Hilbert space of many 2d free fields.

Figures (36)

• Figure 1: The schematic form of two traversable wormholes in four dimensions. Each traversable wormhole is the one in Maldacena:2018gjkMaldacena:2020sxe. We model this 4d setup both in SYK models and Nearly AdS$_2$ gravities.
• Figure 2: We plot the $E_g$ and the spin operator expectation value $\frac{1}{2}\braket{S_{LR}} = |\braket{\psi_L\psi_R}|$ for several $\mu$ and compare with the conformal limit results (\ref{['eq:MQEgapConformal']}). Here we take the temperature to be $T=0.001$, which is very low and essentially the system at zero temperature. The conformal limit is a good approximation for small $\mu$. The spin operator, which is given by $\frac{1}{2}\braket{S_{LR}} = -i\braket{\psi_L(0)\psi_R(0)} =-iG_{LR}(0)$, behaves as $-iG_{LR}(0) \approx c_{\Delta}(\frac{t'}{2})^{2\Delta}$ in the conformal limit.
• Figure 3: The schematic form of the interaction. Left: On each dot we have a copy of the SYK model. The blue line is the interaction between $L$ and $R$. The red line is the interaction between $1$ and $2$. Middle: We can think of the 4 coupled model as the two coupled Maldacena-Qi model. Right: Other way to think of the 4 coupled model as the two coupled Maldacena-Qi model.
• Figure 4: The phase diagram of the four coupled model for $\mu_{LR}= 0.03$, $T=0.001$, and varying $\mu_{12}$.
• Figure 5: The phase diagram at zero temperature in the $\mu_{LR} - \mu_{12}$ plane. The diagram is symmetric under the reflection along $\mu_{12} = \mu_{LR}$ line. The blue line and the orange line meet at about $\mu_{12} = \mu_{LR} \approx 0.154$. Beyond this critical point, the different wormhole phases are continuously connected.