Dynamics of Holographic Entanglement Entropy Following a Local Quench

TL;DR

We address how holographic entanglement entropy evolves after a local quench in 2+1D CFTs using a full numerical solution of Einstein's equations in AdS$_4$ with a boundary scalar source. We solve for extremal surfaces to compute $S_A$ for strip regions, revealing an emergent light-cone with velocity $v_E$ bounded by $v_E^{*}(3)\le v_E \le 1$, where the lower bound matches the tsunami velocity for broad quenches and the upper bound applies to well-localized pulses. We identify universal high-temperature behavior where $v_E\approx1$ and show that $v_E$ increases with quench amplitude and nonlinearity, while finite-size effects shift it toward the tsunami bound. We also find exponential relaxation of entanglement to equilibrium, contrasting with slower relaxations seen in some lattice models, and discuss implications for scrambling and information propagation.

Abstract

We discuss the behaviour of holographic entanglement entropy following a local quench in 2+1 dimensional strongly coupled CFTs. The entanglement generated by the quench propagates along an emergent light-cone, reminiscent of the Lieb-Robinson light-cone propagation of correlations in non-relativistic systems. We find the speed of propagation is bounded from below by the entanglement tsunami velocity obtained earlier for global quenches in holographic systems, and from above by the speed of light. The former is realized for sufficiently broad quenches, while the latter pertains for well localized quenches. The non-universal behavior in the intermediate regime appears to stem from finite-size effects. We also note that the entanglement entropy of subsystems reverts to the equilibrium value exponentially fast, in contrast to a much slower equilibration seen in certain spin models.

Figures (10)

• Figure 1: Evolution profile of the (a) radial shift $\lambda(x,t)$, and (b) $T_{00}(x,t)$ component of the stress tensor, for $\alpha=0.5$, $M = 0.1$, $\sigma=2$. The field $\lambda$ determines the evolution of the entropy in our solution.
• Figure 2: Evolution of the total energy on the boundary $E = \int T_{00} \,dx$ after a quench described by parameters $\alpha=0.5$, $M = 0.1$, $\sigma=2$.
• Figure 3: I The growth of the apparent horizon (in blue) as a function of boundary time, for $\alpha = 0.3$ and $M = 0.1$. We also overlay the plot for the total energy $\int \,T_{00} dx$ produced by quenching the system in red for direct comparison.
• Figure 4: Evolution of the geodesics' radial depth for a quench; $\alpha=0.5$, $M=0.1$, $L=0.8$, $\sigma=2$. The blue data points represent the radial depth $u^*$ in the fixed, gauged coordinate system, whereas the red data points represent the ungauged radial depth $U^*$.
• Figure 5: Maximum of the entanglement entropy $S_\mathcal{A}(t)$ as a function of $L$, for $\alpha = \{0.1, 0.2, 0.3, 0.4, 0.5\}$ starting from the bottom, and $M=0.1$. The slopes of these curves depend quadratically on the amplitude $\alpha$ of the scalar field.