On entanglement spreading from holography

TL;DR

The paper investigates entanglement spreading after global quenches in holographic theories, establishing quantitative bounds and universal features of entanglement growth. It develops a scaling framework for large regions to derive analytic EE evolution for spheres, demonstrates a universal early-time linear growth with velocity $v_E$, and connects saturation dynamics to the butterfly velocity $v_B$ via both one- and two-sided information measures. By analyzing arbitrary shapes, it provides a general bound on entropy growth and an early-time expansion, linking geometric data of HRT surfaces to chaotic spreading. Together with a companion work, these results illuminate how holographic models encode chaos and fast scrambling through entanglement dynamics that are robust to quench details and geometric specifics.

Abstract

A global quench is an interesting setting where we can study thermalization of subsystems in a pure state. We investigate entanglement entropy (EE) growth in global quenches in holographic field theories and relate some of its aspects to quantities characterizing chaos. More specifically we obtain four key results: 1. We prove holographic bounds on the entanglement velocity $v_E$ and the butterfly effect speed $v_B$ that arises in the study of chaos. 2. We obtain the EE as a function of time for large spherical entangling surfaces analytically. We show that the EE is insensitive to the details of the initial state or quench protocol. 3. In a thermofield double state we determine analytically the two-sided mutual information between two large concentric spheres separated in time. 4. We derive a bound on the rate of growth of EE for arbitrary shapes, and develop an expansion for EE at early times. In a companion paper arXiv:1608.05101, we put these results in the broader context of EE growth in chaotic systems: we relate EE growth to the chaotic spreading of operators, derive bounds on EE at a given time, and compare the holographic results to spin chain numerics and toy models. In this paper, we perform holographic calculations that provide the basis of arguments presented in that paper.

Figures (17)

• Figure 1: Entropy growth in a global quench for a sphere of radius $R$ in the limit \ref{['Limit']} in $d=4$. The solid blue curve is \ref{['AnalyticCurve']}, the dashed blue line is the saturation value of the entropy given by a static RT surface, and the dotted red lines indicate \ref{['LinGrowth']} and \ref{['tsRes']} respectively.
• Figure 2: Left: Vaidya quench model. The infalling null shell is drawn by an orange arrow. Below the shell the geometry is pure AdS, above the shell it is a static black brane. We get time evolution of the entropy in the boundary theory because the HRT surface lives in both parts of the geometry and passes through the null shell. Right: The boundary state model of a quench is dual to an end of the world brane cutting the eternal black hole in half. The lower portion of the figure illustrates how the Euclidean path integral prepares a short-range entangled initial state from the boundary state Hartman:2013qma. The time evolution of the entropy comes from the HRT surface entering the black brane horizon and ending on the brane.
• Figure 3: Entropy growth for a strip of width $2R$ from a $d=4$ Schwarzschild black hole in the limit of large region sizes. The above rescaled curve is exactly linear with slope $v_E$, which gives a saturation time $t_S={R\over v_E}$. This saturation time satisfies the bound \ref{['tSBound']}, if we use that $v_E\leq v_B$\ref{['vEvB']}.
• Figure 4: Extremal surfaces in the $d=3$ Schwarzschild geometry. Left: In the Vaidya setup we show the portion of the extremal surfaces that are in the black brane region for $R=15$. Earlier times are drawn by darker colors. The red line shows \ref{['ztExpansion2']}, which is an expansion for $z_c$ for large $\rho_c$. Right: In the end of the world brane setup we chose $R=25$, and the red line is \ref{['ztExpansion']}.
• Figure 5: HRT surfaces with $R=4$ in the Schwarzschild geometry corresponding to $d=3$ boundary dimensions. On the left we plotted the HRT surfaces $(v(\rho),z(\rho))$ in coordinates that correspond to the Penrose diagram shown on the right. On the right only the part of the Penrose diagram is shown that is covered by the coordinates $(z,v)$. Top: HRT surfaces in the Vaidya setup, with the null shell shown with green and the horizon with blue. Darker color correspond to earlier times. At very early times, much of the surface is in the pure AdS region, and is an almost perfect hemisphere in the coordinates \ref{['Metric']}, deformed by the conformal mapping that gives the Penrose diagram on the right. As time evolves the surface goes behind the horizon, most of it lies on $z_\text{HM}$, and near saturation it climbs out from behind the horizon and the entropy saturates, when the surface is only barely touching the shell. Bottom: HRT surfaces in the end of the world brane setup. At early times the HRT surface is a tube connecting the boundary theory entangling surface $\Sigma$ to the brane, and the image on the brane is $\Sigma_\text{im}\approx\Sigma$. The linear regime of entropy growth takes place, when $\Sigma_\text{im}$ migrates up to $z_\text{HM}$. We can clearly see that as we go to later times (lighter color) $\Sigma_\text{im}$ is shrinking, but staying at $z_\text{HM}$. Finally $\Sigma_\text{im}$ migrates towards the bifurcation surface, but this is an $O(\beta/R)$ effect. For similar figures, see Hubeny:2013dea.