Bra-Ket Wormhole Phase Transition

• Bra-ket wormhole phase transition is a sharp change in the dominant semiclassical saddle in gravitational path integrals, marked by a switch from disconnected to connected geometries.
• It relies on the competition between attractive negative Casimir energy and repulsive Casimir entropy from gauge holonomy integration, modeled via matrix models.
• This transition underpins factorization restoration, entanglement island formation, and holographic code subspace embedding in low-dimensional gravitational models.

A bra-ket wormhole phase transition refers to a sharp, non-analytic change in the dominant semiclassical saddle contributing to the gravitational path integral describing a quantum state prepared via Euclidean gravity, notably in low-dimensional AdS or dS models with matter. The transition demarcates a regime where the dominant configuration is a disconnected geometry (two disks or boundaries) from one where a nontrivial connected geometry—a "bra–ket wormhole" supported by negative Casimir energy—dominates. The underlying mechanism involves the competition between attractive Casimir energy and repulsive “Casimir entropy” associated with integrating over gauge-holonomy moduli, yielding a transition with precise matrix-model and random matrix product state duals. This phenomenon governs the restoration of factorization and strong subadditivity, underpins the embedding of gravitational code subspaces, and manifests in diverse gravitational, tensor network, and quantum chaotic model frameworks.

1. Gravitational Path Integral Setup and Saddle Structure

The archetypal setting for bra-ket wormhole transitions is two-dimensional Jackiw–Teitelboim (JT) gravity (or its de Sitter analog), where the gravitational path integral is performed over metrics with specified boundary conditions for gravity (fixed dilaton at the “gluing” surface) while matter fields have transparent (non-fixed) boundary conditions. For example, the total Euclidean action in JT gravity with matter is: $$I_{\text{total}} = I_{\text{JT}} + I_{\text{matter}}$$ where

$$I_{\text{JT}}[g, \phi] = -\frac{S_0}{4\pi} \left[\int R + 2\int K \right] - \frac{1}{4\pi}\left[\int \phi (R+2) + 2\phi_b \int K \right]$$

Matter is typically a 2d CFT of large central charge ($c\sim N$), propagating on a disk glued to a flat strip. The state norm or correlator receives contributions from two dominant topologies:

• Disconnected (double disk): No wormhole, factorized boundary.
• Bra–ket wormhole (annulus/connected): An $S^1$ throat joins the two boundaries, supported by negative Casimir energy of the matter fields (Milekhin et al., 2022, Jung et al., 12 Dec 2025).

The evaluation of the full path integral involves integrating gravitational moduli (e.g., the throat size $b$) and gauge field holonomy moduli ($\theta_i$ for $U(N)$ or related groups), with the Haar measure for flat connections: $$\mu_{\rm Haar}(\{\theta_i\}) = \prod_{i<j} \sin^2\left(\tfrac{\theta_i-\theta_j}{2}\right)$$ This measure is crucial in determining the entropy cost (Casimir entropy) of holonomy integration and acts as a repulsive effect in the total action (Milekhin et al., 2022).

2. Competition of Casimir Energy and Casimir Entropy

The core dynamical mechanism for the phase transition is the interplay between:

• Negative Casimir energy ($E_C$): For transparent boundary conditions and clumped holonomies ($\theta_i=0$), the Casimir energy is strongly negative and stabilizes the throat. Explicitly, for $N$ Dirac fermions: $$E_C[\{\theta_i\}] = -\frac{\pi}{6d} \sum_{i=1}^N \left[1 - 12\left(\frac{\theta_i}{2\pi}\right)^2\right]$$
• Casimir entropy ($S_C$): Arises from integrating over holonomy moduli,

$$S_C = -\int_{[-\pi,\pi]^N} d^N\theta \ \mu_{\rm Haar}(\{\theta_i\}) \ln \mu_{\rm Haar}(\{\theta_i\})$$

$S_C$ scales as $N^2$, heavily penalizing configurations with aligned holonomies, and thus driving $E_C$ towards zero for large $N$ (Milekhin et al., 2022).

The total effective action, after integrating geometry to a modulus $b$ and holonomies, exhibits a matrix model structure: $$Z = \int db \prod_{i=1}^N d\theta_i \exp \left[-I_{\rm grav}(b) - b E_C[\{\theta_i\}] + S_C \right]$$ The balance of $O(N)$ attractive Casimir energy and $O(N^2)$ repulsive entropy is what triggers the sharp bra-ket wormhole transition (Milekhin et al., 2022, Jung et al., 12 Dec 2025, Numasawa, 2020).

3. Matrix Model, Critical Line, and Universal Features

The phase structure reduces, at large $N$, to the solution of a unitary matrix model. Introducing

$f(b) = \frac{b}{6(1+\frac{2\tau b}{\pi L})}$

one finds that a nontrivial clumped (wormhole) solution exists only for

$\frac{f(b)}{N} \gtrsim 0.27$

The corresponding critical line is determined by inequalities in system parameters, e.g. $N\tau/L < 0.1$, $L/\phi_r \gtrsim 33$. The phase transition is characterized by:

• Third-order (Gross–Witten–Wadia type): Order parameter (throat size $b$) increases as $b\sim (f/N-0.27)^{1/2}$ above threshold.
• Continuous opening of eigenvalue support ($\theta_c < \pi$) as one crosses the critical line.
• A sharp transition in the phase diagram in $(N\tau/L, L/\phi_r)$ space (Milekhin et al., 2022).

Tensor-network duals in the form of random matrix product states (RMPS) precisely reproduce this phenomenology: the bra-ket wormhole phase corresponds to a gapped transfer matrix ($\Delta=\lambda_1-\lambda_2>0$), while in the gapless (disconnected) phase, correlators do not decay exponentially. The critical point is at boundary length $$L= L_{\text{crit}} \sim 256 \phi_0 \phi_r/(\pi c^2)$$ (Jung et al., 12 Dec 2025).

4. Kinetics, Universality, and Analogies

In the SYK framework and quantum gravity duals, the transition is sharply first-order when viewed across couplings/conjugate order parameters, e.g., in the free energy landscape of coupled SYK or four-coupled SYK models:

• The landscape exhibits multiple minima (wormhole and two-black-hole), with nucleation between them governed by Kramers escape theory.
• Scaling of barrier heights and kinetic times matches mean-field (Landau) predictions: escape time $\tau \sim \exp[\Delta F(T)/k_B T]$, and $\Delta F(T)\propto|T-T_c|^{3/2}$ (Li et al., 2021).
• The analogy to the Van der Waals gas-liquid transition is precise: coexistence, spinodals, and critical points are all sharply defined, with coexistence given by a Maxwell-like equal-area rule.

No exotic universality class is observed; the transition follows standard mean-field critical exponents (Li et al., 2021).

5. Physical Interpretation, Observables, and Entropy Bounds

The physical implications of the phase transition are manifest in correlation functions and entropies:

• In the disconnected (double-disk) phase, matter correlators are thermal and exhibit violations of strong subadditivity if the island rule is naively applied.
• In the wormhole phase, the negative Casimir energy opens a finite throat, restoring factorization and compliance with strong subadditivity.
• The transition restores eternal traversability and allows embedding of holographic CFT code subspaces into free-field Hilbert spaces via bra-ket wormholes (Milekhin et al., 2022, Numasawa, 2020).
• A "species bound" on extremal entropy $S_0 \gtrsim \frac{N}{6}\ln N$ is required to avoid paradoxical entropy growth.

Observable manifestations include:

• Ramp-and-plateau behavior in correlators at late times, characterized by a crossover time $t_c\sim (1/H)\ln S_{dS}$ in dS analogs, controlled by the competition between mode counting and topological (wormhole) suppression (Jung et al., 15 Dec 2025).
• Long-distance correlation offsets and area-law entropy saturation in the wormhole phase, contrasted with unbounded (volume-law) growth circumvented by the phase transition (Jung et al., 12 Dec 2025, Miyaji, 2021).

6. Generalizations: dS, Measurement-Induced, and Tensor-Network Approaches

Generalizations and further applications include:

• de Sitter Bra-Ket Wormholes: In dS$_2$ JT gravity, a sharp transition exists between the Hartle–Hawking (no-wormhole) and connected bra–ket wormhole saddle, characterized by the Wigner phase space distribution. For large average boundary length $L\gg L_{\text{crit}} \sim e^{2\phi_0/3} \phi_b$, the wormhole dominates, ensuring a semiclassical cosmological ensemble (Fumagalli et al., 15 Aug 2024, Jung et al., 15 Dec 2025).
• Measurement-Induced Transitions: In both continuous projection and unitary coupling protocols (Kitaev–Yoshida / Gao–Jafferis–Wall), there is a critical rate/coupling ($\gamma_c$, $g_c$) at which an eternal traversable (bra–ket) wormhole forms, signaled by the saturation of teleportation order parameters and static wormhole throat (Milekhin et al., 2022).
• Tensor-Network/RMPS Approaches: RMPS and cMPS provide an exact topological expansion for gravitationally prepared quantum states, where the spectral gap of the transfer matrix directly encodes the existence of a wormhole. Finite-$N$ corrections reproduce off-shell wormhole contributions as cumulants of the leading eigenvalue (Jung et al., 12 Dec 2025).

7. Entanglement Islands, Factorization, and Quantum Information

Analysis of entanglement entropy across the transition reveals:

• For large subregions, entropy is bounded either by the initial-state boundary (finite entropy S$_B$ from an EOW brane) or by the emergence of an island (entanglement wedge), but cannot grow indefinitely. The transition between these regimes aligns with the bra–ket wormhole phase transition, which itself is first-order (Miyaji, 2021).
• The transition solves the factorization puzzle at leading order, restoring normal expectations for entanglement structure in gravitationally prepared states, while phase transitions between competing saddles provide a precise quantum information-theoretic control (Miyaji, 2021, Numasawa, 2020).

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References:

• (Milekhin et al., 2022): "Bra-ket wormholes and Casimir entropy"
• (Jung et al., 15 Dec 2025): "Two-point correlators in de Sitter-prepared states with bra-ket wormholes"
• (Jung et al., 12 Dec 2025): "Random matrix product state models of gravitationally prepared states"
• (Fumagalli et al., 15 Aug 2024): "De Sitter Bra-Ket Wormholes"
• (Li et al., 2021): "Free energy landscape and kinetics of phase transition in two coupled SYK models"
• (Numasawa, 2020): "Four coupled SYK models and Nearly AdS$_2$ gravities: Phase Transitions in Traversable wormholes and in Bra-ket wormholes"
• (Milekhin et al., 2022): "Measurement-induced phase transition in teleportation and wormholes"
• (Miyaji, 2021): "Island for Gravitationally Prepared State and Pseudo Entanglement Wedge"