Expanding the Black Hole Interior: Partially Entangled Thermal States in SYK

TL;DR

This work constructs and analyzes partially entangled thermal states (PETS) in the SYK model, realized by Euclidean evolution with a local scaling operator and interpolating between thermo-field double and thermal pure states. In the Schwarzian/low-energy limit, the holographic dual comprises two AdS$_2$ regions glued along the worldline of a massive particle, with the entanglement entropy governed by the minimal dilaton among competing horizons. The authors compute the interior size and entanglement entropy as functions of the operator dimension and the left/right temperatures, and argue that one-sided bulk reconstruction can access the black hole interior via entanglement-wedge/QEC perspectives. They also develop graphical and tensor-network formalisms to capture the bulk geometry, discuss generalizations to multiple insertions and coarse-grained ensembles, and propose how phase structure and tripartite entanglement arise in these generalized PETS, highlighting potential broader implications for bulk reconstruction and holographic entanglement in AdS$_2$.

Abstract

We introduce a family of partially entangled thermal states in the SYK model that interpolates between the thermo-field double state and a pure (product) state. The states are prepared by a euclidean path integral describing the evolution over two euclidean time segments separated by a local scaling operator $\mathcal{O}$. We argue that the holographic dual of this class of states consists of two black holes with their interior regions connected via a domain wall, described by the worldline of a massive particle. We compute the size of the interior region and the entanglement entropy as a function of the scale dimension of $\mathcal{O}$ and the temperature of each black hole. We argue that the one-sided bulk reconstruction can access the interior region of the black hole.

Figures (16)

• Figure 1: The euclidean (left), lorentzian (right) space-time associated with the thermo-field double state (top) and thermal pure state (bottom). The middle column shows the total geometry describing the state preparation and real time evolution, obtained by gluing the euclidean and lorentzian geometry together along the equator of the disc
• Figure 2: The euclidean and lorentzian space-time dual to the partially entangled states \ref{['ostate']}. The worldline of the massive bulk particle created by the operator insertion ${\cal O}_\ell$ is indicated by the red line. It divides the space-time into two AdS${}_2$ regions. For a sufficiently massive particle, the worldline is hidden behind two horizons. In this figure, the left-horizon is the true 'extremal surface' with minimal value $\Phi_{{\! \space}_{{}^{L}}}\!\space$ of the dilaton.
• Figure 3: The curve that maximizes the action with two operator insertion (red dots) at $\tau_1=\tau$ and $\tau_2=\beta-\tau$. The horizons of each side are located at the black dots.
• Figure 4: Backreaction generated by an operator insertion (red dots) of dimension $\ell$ when $\ell\to0$ and $\ell\to\infty$. We indicate the backreaction by depicting the deformations of the boundary curve in the Euclidean Poincare disk. We indicate the (local) horizons by a black dot.
• Figure 5: Dilaton profile. The values of the dilaton in arbitrary units go from large positive values of $\Phi$ at the boundary (red end of color spectrum) to large negative values ending at the inner horizon (blue end of color spectrum): We show $\ell \sim 0$ (left), $\ell \sim N/\beta J$ (middle) and $\ell \sim N$ (right).