Probing phase transitions of holographic entanglement entropy with fixed area states

TL;DR

The paper probes corrections to holographic entanglement entropy near RT/HRT phase transitions by decomposing the bulk state into fixed-area sectors and proposing a diagonal approximation. It shows that, near transitions, the entanglement includes a leading $O(G^{-1/2})$ term arising from fluctuations in the fixed-area data, while corrections away from the transition are exponentially suppressed, converting sharp RT phase transitions into crossovers. The authors validate the framework through explicit AdS$_3$/BTZ examples and demonstrate consistency with Murthy-Srednicki ETH predictions and with Penington's quantum RT transition results. They also analyze cutoff-dependence of RT-area fluctuations and discuss the broader implications for bulk reconstruction and the smoothing of entanglement wedges. The findings point to a robust bulk mechanism for smoothing entanglement transitions beyond bulk field entanglement, with potential generalization to HRT and more complex states.

Abstract

Recent results suggest that new corrections to holographic entanglement entropy should arise near phase transitions of the associated Ryu-Takayanagi (RT) surface. We study such corrections by decomposing the bulk state into fixed-area states and conjecturing that a certain `diagonal approximation' will hold. In terms of the bulk Newton constant $G$, this yields a correction of order $O(G^{-1/2})$ near such transitions, which is in particular larger than generic corrections from the entanglement of bulk quantum fields. However, the correction becomes exponentially suppressed away from the transition. The net effect is to make the entanglement a smooth function of all parameters, turning the RT `phase transition' into a crossover already at this level of analysis. We illustrate this effect with explicit calculations (again assuming our diagonal approximation) for boundary regions given by a pair of disconnected intervals on the boundary of the AdS$_3$ vacuum and for a single interval on the boundary of the BTZ black hole. In a natural large-volume limit where our diagonal approximation clearly holds, this second example verifies that our results agree with general predictions made by Murthy and Srednicki in the context of chaotic many-body systems. As a further check on our conjectured diagonal approximation, we show that it also reproduces the $O(G^{-1/2})$ correction found Penington et al for an analogous quantum RT transition. Our explicit computations also illustrate the cutoff-dependence of fluctuations in RT-areas.

Figures (10)

• Figure 1: The manifold $M_{CFT}$ (bottom) that we use in the Euclidean path integral to prepare our holographic state $|\psi\rangle$ and the CPT-conjugate manifold $M^\dagger_{CFT}$ (top). Sewing the two together along their boundaries defines the manifold $M_{double} : = M_{CFT}^\dagger M_{CFT}$. If the state is time-symmetric, then $M^\dagger_{CFT}$ is equivalent to $M_{CFT}$, the two manifolds are exchanged by the relevant notion of time-reversal, and this symmetry leaves invariant the boundary $\partial M_{CFT} = \partial M^\dagger_{CFT}$ of $M_{CFT}, \partial M^\dagger_{CFT}$. The surface $\partial M_{CFT}$ is partitioned into regions $R$ (red) and $\bar{R}$ (blue).
• Figure 2: An illustration of two competing RT surfaces $\gamma_1$ and $\gamma_2$ near a phase transition. In our convention, we always let $\Sigma_{1R}\subset \Sigma_{2R}$, as a result $\Sigma_{1R}$ and $\Sigma_{2\bar{R}}$ are non-overlapping.
• Figure 3: Left: A two-dimensional projection of an $n=1$ solution with two extremal surfaces $\gamma_1,\gamma_2$ (having areas $A_1$ and $A_2$) and a surface $\Sigma_{int}$ stretching between them. Right: An $n$-fold cover of the figure at left for the case $n=3$ after cutting open a slit along $\Sigma_{int}$. The $2n$ copies of $\Sigma_{int}$ are labeled $\Sigma^{(i)}_{int\pm}$, where $i=1,\dots,n$. Saddles for the Rényi entropy are formed by identifying $\Sigma^{(i)}_{int+}$ with $\Sigma^{(\pi(i))}_{int-}$ for some permutation $\pi$. After making such identifications, the number $N_2 = n-n_2$ of copies of $\gamma_2$ that remain is the number $C(\pi)$ of cycles generated by $\pi$, while the corresponding $N_1 = n-n_1$ is $C(\tau \circ \pi)$ where $\tau$ is the counterclockwise cyclic permutation. The asymmetry is due to the numbering of replicas, which breaks the natural symmetry between the dashed lines (separating replicas) and dotted lines (separating the two halves of each replica).
• Figure 4: Using the same projection as for the $n=1$ figure at left in figure \ref{['fig:RBsaddles']}, we show two pieces of a corresponding saddle that for $A_1\neq A'_1$ describes an off-diagonal contribution to the second Rényi entropy $S_2$. The full saddle is constructed by sewing the two pieces together along edges of the same color; i.e., we may first identify the two green edges and then identify the two orange edges. Note that this particular saddle contains only one copy of the surface $\gamma_2$ and so cannot be 'off-diagonal' in $A_2$.
• Figure 5: Two competing extremal surfaces when $R= R_1 \cup R_2$ is a pair of intervals on the boundary of the $AdS_3$ vacuum. The homotopic RT surface $\gamma_d = \gamma_{11} \cup \gamma_{22}$ and the non-homotopic RT surface $\gamma_o = \gamma_{12} \cup \gamma_{21}$ are shown respectively in red and blue. The case shown sits precisely at the RT phase transition, where $\gamma_d$ and $\gamma_o$ are related by a $\pi/2$ rotation.