Holographic Entanglement Entropy on Generic Time Slices

TL;DR

This work analyzes holographic entanglement entropy and mutual information on generic time slices in both relativistic and non-relativistic holographic theories. Using AdS/CFT and extremal surface techniques, it finds a universal divergence of I(A,B) in the light-like limit of boosted subsystems in the AdS$_3$/CFT$_2$ and AdS black brane cases, while S_A remains ill-defined for generic boosts in Lifshitz and hyperscaling-violating geometries unless a bound is satisfied. It derives a precise bound on $Δt/(Δx)^ν$, with explicit values for ν=2 and several d, and provides an analytic solution for the special case d=ν=2. Together, these results imply that real-space factorization of the Hilbert space on noncanonical time slices is not generally possible in non-relativistic theories, indicating boost-induced temporal nonlocality with potential experimental relevance.

Abstract

We study the holographic entanglement entropy and mutual information for Lorentz boosted subsystems. In holographic CFTs at zero and finite temperature, we find that the mutual information gets divergent in a universal way when the end points of two subsystems are light-like separated. In Lifshitz and hyperscaling violating geometries dual to non-relativistic theories, we show that the holographic entanglement entropy is not well-defined for Lorentz boosted subsystems in general. This strongly suggests that in non-relativistic theories, we cannot make a real space factorization of the Hilbert space on a generic time slice except the constant time slice, as opposed to relativistic field theories.

Figures (8)

• Figure 1: The left picture explains entanglement entropy and mutual information can be defined on any space-like time slices in relativistic field theories. A constant time slice is described as $\Sigma_1$, while generic one (deformed one) by $\Sigma_2$. We can define the entanglement entropy $S_{A_1}$, $S_{A_2}$, $S_{B_1}$ and $S_{B_2}$ and their mutual informations. The right picture describes a special setup with a Lorentz boosted interval and an unboosted interval. The dotted curve $\Sigma$ represents the time slice in this setup.
• Figure 2: The sketch of the transition in the holographic computation of $S_{A \cup B}$. When the invariant length of subsystem $A$ gets smaller, the extremal surface for the computation of holographic entanglement entropy changes into the disconnected ones (the right picture) and the mutual information becomes vanishing. A similar phase transition occurs when the invariant length between $Q_A$ and $P_B$ changes.
• Figure 3: We plotted $S_A$ (\ref{['sabtz']}) as a function of $x$ and $t$, which are the space and time-like width of the interval $A$ for the BTZ black hole $d=2$. We subtracted the holographic entanglement entropy for a disconnected geodesic from the one for connected one: $\Delta S_{A}=S_{A}-S_{A}^{{\rm (dis)}}$ to remove the UV divergence. Note that the interval $A$ has to be space-like and therefore we need to require $x>t$. We set the parameters $z_{H}=R=G_{N}=1$.
• Figure 4: In the left graph, we describe the setup of the two intervals parametrized by $\epsilon_2$ and $\theta$, defined as $(x,t)=(r\cos\theta,r\sin\theta)$ and $b=x+t+\epsilon_2$. The dashed line denotes a light-like surface. In the right graph, we plotted the mutual information $I(A,B)$ (\ref{['mutbtz']}) as a function of $(\epsilon_2,\theta)$ fixing $r=0.5$. The horizontal coordinate and depth coordinate are $\epsilon_2\cos\theta$ and $\epsilon_2\sin\theta$.
• Figure 5: In the left graph, we plotted $S_A$ as a function of $\Delta x$ and $\Delta t$, which are the space and time-like width of the strip $A$ for the AdS$_4$ black brane $d=3$. We again subtracted the disconnected entropy from the connected one $\Delta S_{A}=S_{A}-S_{A}^{{\rm (dis)}}$ to remove the UV divergence. Note that the strip $A$ has to be space-like and therefore we need to require $\Delta x>\Delta t$. We set the parameters $z_{H}=R=G_{N}=1$. In the right graph, we plotted the mutual information $I(A,B)$ with the choice of subsystems (\ref{['itv']}) for the AdS$_4$ black brane as a function of $(\epsilon_2,\theta)$ defined by $(x,t)=(r\cos\theta,r\sin\theta)$ and $b=x+t+\epsilon_2$ fixing $r=0.5$, as in the Fig.\ref{['fig:miplotBTZ']}.