Enhanced corrections near holographic entanglement transitions: a chaotic case study

Abstract

Recent work found an enhanced correction to the entanglement entropy of a subsystem in a chaotic energy eigenstate. The enhanced correction appears near a phase transition in the entanglement entropy that happens when the subsystem size is half of the entire system size. Here we study the appearance of such enhanced corrections holographically. We show explicitly how to find these corrections in the example of chaotic eigenstates by summing over contributions of all bulk saddle point solutions, including those that break the replica symmetry. With the help of an emergent rotational symmetry, the sum over all saddle points is written in terms of an effective action for cosmic branes. The resulting Renyi and entanglement entropies are then naturally organized in a basis of fixed-area states and can be evaluated directly, showing an enhanced correction near holographic entanglement transitions. We comment on several intriguing features of our tractable example and discuss the implications for finding a convincing derivation of the enhanced corrections in other, more general holographic examples.

Figures (9)

• Figure 1: $F_1\left(\mathcal{E}\right)$ and $F_{\text{dom}}\left(\mathcal{E}\right)$ for different regimes of Renyi index $n$. Left: $n-1\gg 1/\sqrt{C_V}$, the saddle points are well separated, there is no enhanced correction; Right: $n-1\sim 1/\sqrt{C_V}$, the saddle points are close within the curvature scale of $F_1$, an $\mathcal{O}\left(\sqrt{V}\right)$ enhanced correction appears, and an approximate flat interval for $F_1\left(\mathcal{E}\right)$ emerges between the saddles.
• Figure 2: Left: Euclidean path-integral for the reduced density matrix $\rho_A(\hat{c})$ from a particular chaotic microstate; right: emergent KMS condition for averaged $\hat{\rho_A}$
• Figure 3: The four closed manifolds $\mathcal{M}_i, i=1,..,5$ that emerge after disorder averaging $\hat{c}$ in $\rho_A^n(\hat{c})$ for $n=3$.
• Figure 4: "Canonical" representations of $\mathcal{M}_2\sim \mathcal{M}_3\sim \mathcal{M}_4$ and $\mathcal{M}_5$ for $n=3$.
• Figure 5: "Kinematics" of the bulk saddles $\mathcal{B}_i$'s corresponding to $\mathcal{M}_i$'s. Green and red arrows indicate how the three bulk time-slices are glued together for each configuration. The dotted circles represent the horizons.