Influence of inhomogeneities on holographic mutual information and butterfly effect

TL;DR

This work analyzes how bulk inhomogeneity introduced by the graviton mass in massive gravity affects holographic mutual information (MI) and butterfly-like chaotic dynamics in a 2+1D boundary theory. It combines static and dynamical MI calculations: static MI for two strips in MG backgrounds and dynamical MI in shock-wave geometries modeling energy perturbations, leveraging $I(x_0)=S_A+S_B-S_{A\cup B}$ and holographic minimal-surface prescriptions. The key findings show that near-homogeneous regimes (small $m$) can induce MI growth with increasing $m$ for short strips, while strongly inhomogeneous regimes (larger $m$) suppress MI, especially for longer strips; in dynamical settings, larger perturbations and greater inhomogeneity accelerate MI disruption, with a reduced scrambling-like threshold $h_{crit}$. Collectively, the results reveal that spatial inhomogeneity controlled by the bulk graviton mass governs entanglement structure and scrambling in holographic systems, providing insights into momentum-relaxation effects and lattice-like physics in strongly coupled field theories.

Abstract

We study the effect of inhomogeneity, which is induced by the graviton mass in massive gravity, on the mutual information and the chaotic behavior of a 2+1-dimensional field theory from the gauge/gravity duality. When the system is near-homogeneous, the mutual information increases as the graviton mass grows. However, when the system is far from homogeneity, the mutual information decreases as the graviton mass increases. By adding the perturbations of energy into the system, we investigate the dynamical mutual information in the shock wave geometry. We find that the greater perturbations disrupt the mutual information more rapidly, which resembles the butterfly effect in chaos theory. Besides, the greater inhomogeneity reduces the dynamical mutual information more quickly just as in the static case.

Figures (8)

• Figure 1: Penrose diagrams for an eternal black hole without (left panel) and with (right panel) a perturbation. $h$ is the shift on the horizon between the left and right Kruskal coordinate $\nu$.
• Figure 2: Left: The relation between the width of the strip $x_{0}$ and $r_{\rm min}$. Right: The relation between the mutual information $I(x_{0})$ and the position of the turning point $r_{\rm min}$. For both cases, we set $m= 0.6$, $Q=2$, and $r_h=1$.
• Figure 3: The relation between the mutual information $I(x_{0})$ and width of the strip $x_0$ for $m= 0.6$, $Q=2$, and $r_h=1$.
• Figure 4: Left: The relation between $I(x_0)$ and black hole charge $Q$ while fixing $m=0.6$, $r_h$=1. The curves from top to down correspond to $r_{\rm min}$ increasing from 1.21 to 1.27 with steps 0.02; Right: The relation between the mutual information $I(x_0)$ and the graviton mass $m$ by fixing $Q=2$, $r_h$=1. Curves from top to down correspond to $r_{\rm min}$ increasing from 1.21 to 1.27 with steps 0.02.
• Figure 5: Left: Relation between the critical width $x_{0c}$ and graviton mass $m$ by fixing charge $Q=2$. Curves from top to down correspond to $r_h$ increasing from 1 to 5 with steps 2, respectively. Right: Relation between critical width $x_{0c}$ and the charge $Q$ while fixing the graviton mass $m=0.6$. Curves from top to down correspond to $r_h$ increasing from 1 to 2 with steps 0.4, respectively.