We study four coupled Sachdev–Ye–Kitaev (SYK) models and nearly AdS₂ gravities. In the SYK model side, we construct a model that couples two copies of two coupled SYK models. In nearly AdS₂ 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 AdS₂ 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 where we do not have 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.

中文翻译：

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四个耦合 SYK 模型和近似 AdS2 重力：可穿越虫洞和 bra-ket 虫洞中的相变

我们研究了四个耦合的 Sachdev–Ye–Kitaev (SYK) 模型和近 AdS ₂引力。在 SYK 模型端，我们构建了一个耦合两个耦合 SYK 模型的两个副本的模型。在近 AdS ₂引力侧，我们将物质场纠缠在两个可穿越虫洞副本中。在这两种情况下，系统通过改变耦合在零温度下显示一阶相变，这被理解为可穿越虫洞配置的交换。在近 AdS ₂重力情况下，通过交换空间和时间的作用，虫洞被解释为 bra-ket 虫洞。在洛伦兹签名中，这些 bra-ket 虫洞导致两个相互纠缠的封闭宇宙以及我们没有动态引力的平坦空间中的物质场。我们研究了平面空间上物质的投射或纠缠操作的影响，它们在小虫洞中引起相变，从而导致封闭宇宙的双湮灭。使用这些 bra-ket 虫洞，我们讨论了将 2D 全息 CFT 中的状态嵌入到许多 2D 自由场的 Hilbert 空间中的方法。