Spread of entanglement for small subsystems in holographic CFTs

We develop an analytic perturbative expansion to study the propagation of entanglement entropy for small subsystems after a global quench, in the context of the $\mathrm{AdS}/\mathrm{CFT}$ correspondence. Opposite to the large interval limit, in this case the evolution of the system takes place at time scales that are shorter in comparison to the local equilibration scale and, thus, different physical mechanisms govern the dynamics and subsequent thermalization. In particular, we show that the heuristic picture in terms of an “entanglement tsunami” does not apply in this regime. We find two crucial differences: First, that the instantaneous rate of growth of the entanglement is not constrained by causality, but rather its time average and, second, that the approach to saturation is always continuous, regardless of the shape of the entangling surface. Our analytic expansion also enables us to verify some previous numerical results, namely, that the saturation time is nonmonotonic with respect to the chemical potential. All of our results are pertinent to CFTs with a classical gravity dual formulation.

Figure 1

Pictorial representation of the “entanglement tsunami” for a subsystem $A$. The entanglement is carried by a wave that starts from the its boundary $\mathrm{\Sigma }$ (depicted in red) and propagates inwards at a constant speed $v_E$. The shaded region has been covered by the tsunami wavefront (depicted in orange) and is now entangled with the region outside of $A$. The white region is currently not entangled but it will become at a later time.

Figure 2

(a) Evolution of entanglement entropy for $TR=10^2$. For this choice of parameters $x_{\mathrm{loc}}=t_{\mathrm{loc}}/t_{\mathrm{sat}}={10}^{-2}\ll 1$ and the growth of entanglement is approximately linear. (b) Instantaneous rate of growth for $TR=10^2$. We observe that $\mathfrak{R}(t)>1$ for $x_{\mathrm{loc}}<x\in [0.015,0.858]$ which contradicts the conjectured bound on $\max [\mathfrak{R}(t)]$. However, in the strict limit $\mathfrak{l}\to \infty$, $\mathfrak{R}(0\le t\le t_{\mathrm{sat}})\to 1$ (and becomes discontinuous at both $t=0$ and $t=t_{\mathrm{sat}}$).

Figure 3

(a) Evolution of entanglement entropy for $TR={10}^{-2}$. For this choice of parameters $x_{\mathrm{loc}}=t_{\mathrm{loc}}/t_{\mathrm{sat}}=10^2>1$ and the growth of entanglement deviates from a linear behavior. (b) Instantaneous rate of growth for $TR={10}^{-2}$. Our numerical results suggest a maximum rate of $\max [\mathfrak{R}(t)]=3/2$.

Figure 4

Extremal area surfaces in a thin shell Vaidya geometry for two different geometries: (a) the strip and (b) the ball. The shell (depicted in red) moves at the speed of light and eventually collapses into a black hole. The entanglement entropy of region $A$ grows as time evolves until the corresponding extremal surface ${\mathrm{\Gamma }}_A$ grazes the shell at $v=0$. From this point on the whole surface lies entirely in the AdS-RN portion of the geometry so the entanglement entropy saturates to its final value.

Figure 5

(a) Evolution of entanglement entropy for a strip in $d=3$ and $\mu /T=\{0,2,5,10\}$ from bottom to top, respectively. For the plots we have fixed $\ell T_{\mathrm{eff}}={10}^{-1}$ so that the approximation is valid and we have set the overall factor $A_{\mathrm{\Sigma }}/4G_N^{(d+1)}=1$. According to (4.24), the saturation time scales as $t_{\mathrm{sat}}\sim \ell$ which, for our particular choice of parameters, translates into $t_{\mathrm{sat}}\sim 1/T_{\mathrm{eff}}$. Both, the differences in final entropies and saturation times become more pronounced as we increase the number of dimensions, but the behavior is qualitatively similar. In (b) we plot the instantaneous rate of growth for $\mathfrak{R}(x)$ for $d=\{2,3,4,5\}$ from top to bottom, respectively. We observe that the maximum rate growth only exceed the speed of light for $d=2$, and decreases as we increase the number of dimensions.

Figure 6

(a) Evolution of entanglement entropy for a ball in $d=3$ and $\mu /T=\{0,2,5,10\}$ from bottom to top, respectively. For the plots we have set $R{T}_{\mathrm{eff}}={10}^{-1}$ and $A_{\mathrm{\Sigma }}/4G_N^{(d+1)}=1$. In (b) we plot the instantaneous rate of growth for $\mathfrak{R}(x)$ for $d=\{2,3,4,5\}$ from top to bottom, respectively. Again, the maximum rate growth only exceed the speed of light for $d=2$.

Figure 7

Computation of the entanglement entropy growth at early times. All extremal surfaces intersect the boundary of AdS at right angle (regardless of the size or shape of $\mathrm{\Sigma }$), so the leading contribution is simply $A_{\mathrm{\Sigma }}\times z_c(t)$ (where $z_c(t)$ is the position of the shell at time $t$) times a conformal factor that may only depend on $z_c(t)$. The leading correction to the geometry near the boundary comes with a factor of $z_H^{-d}\sim {T_{\mathrm{eff}}}^d\sim \mathcal{E}$. Via dimensional analysis, this fixes the initial time dependence to be $t^2$.