Delocalizing Entanglement of Anisotropic Black Branes

TL;DR

This work analyzes entanglement and chaos in holographic, anisotropic black branes by examining mutual information across two boundaries and its disruption by anisotropic shock waves. It establishes a general gravity setup for 5D anisotropic backgrounds, derives shock-wave solutions, and extracts chaos indicators including the maximal Lyapunov exponent $\lambda_L$, scrambling time $t_*$, butterfly velocity $v_B$, and entanglement velocity $v_E$. Mutual information for strip-like regions is computed in both unperturbed and shock-wave spacetimes, revealing how anisotropy modifies the critical width and the disruption dynamics. The results show $\lambda_L$ saturates the chaos bound, while $v_B$ and $v_E$ can violate isotropic bounds but remain tied to infrared theory values, consistent with a UV-to-IR RG flow from AdS to Lifshitz in the Mateos–Trancanelli background.

Abstract

We study the mutual information between pairs of regions on the two asymptotic boundaries of maximally-extended anisotropic black-brane solutions. This quantity characterizes the local pattern of entanglement of thermofield double states which are dual to these geometries. We analyse the disruption of the mutual information in anisotropic shock wave geometries and show that the entanglement velocity plays an important role in this phenomenon. Besides that we compute several chaos-related properties of this system, like the entanglement velocity, the butterfly velocity and the scrambling time. We find that the butterfly velocity and the entanglement velocity violate the upper bounds proposed in 1311.1200 and 1612.00082, but remain bounded by their corresponding values in the infrared effective theory.

Figures (9)

• Figure 1: Penrose diagram for the two-sided black branes we consider.
• Figure 2: Penrose diagram for the shock wave geometry.
• Figure 3: Butterfly velocity as a function of $a/T$ for the MT model. The continuous blue curve represent the result for $\frac{3}{2}v_\textrm{\tiny B}^{\perp\,2}$, while the dashed blue curve represents the result for $\frac{3}{2}v_\textrm{\tiny B}^{||\,2}$. The black horizontal line is the isotropic result $\frac{3}{2}v_\textrm{\tiny B}^{\textrm{\tiny iso}\,2}=1$, whereas the gray horizontal line is the result for a five-dimensional Lifshitz-like geometry, $\frac{3}{2}v_\textrm{\tiny B}^{\textrm{\tiny Lif}\,2}=\frac{33}{32}$.
• Figure 4: Mutual Information (in units of $V_2/G_\textrm{\tiny N}$) as a function of $\ell$ for the MT model. The curves correspond from the right to the left to $a/T = 0$ (black curve), $a/T = 8.56$ (blue curves) and $a/T = 21.57$ (red curves). The continuous/dashed curves represent the result for a strip orthogonal/parallel to the anisotropic direction. Here we have fixed $T=1/\pi$.
• Figure 5: Extremal surface (horizontal, red) in the shock wave geometry. Following Leichenauer-2014, we divide the left half of the surface into three parts, $I$, $II$ and $III$. The segments $II$ and $III$ have the same area and they are separated by the point $u_0$ at which the surface defined by $u=u_0$ (blue, dashed curve) intersects the extremal surface.