Holographic Evolution of Entanglement Entropy

TL;DR

The paper uses AdS3-Vaidya to model far-from-equilibrium dynamics in a 2D CFT and computes time-dependent entanglement entropy via spacelike geodesics. It finds that entanglement propagates with maximal velocity v=1, with a locally relaxing but globally unitary evolution, and reveals a nuanced role for the apparent horizon in behind-horizon geodesic contributions. The results reproduce known quench-like behavior in a holographic setting, while highlighting long-range initial entanglement and providing a framework to interpret thermalization times for occupation numbers. A generalized entanglement-density model is proposed to describe the evolving structure of entangled excitations in this strongly coupled system.

Abstract

We study the evolution of entanglement entropy in a 2-dimensional equilibration process that has a holographic description in terms of a Vaidya geometry. It models a unitary evolution in which the field theory starts in a pure state, its vacuum, and undergoes a perturbation that brings it far from equilibrium. The entanglement entropy in this set up provides a measurement of the quantum entanglement in the system. Using holographic techniques we recover the same result obtained before from the study of processes triggered by a sudden change in a parameter of the hamiltonian, known as quantum quenches. Namely, entanglement in 2-dimensional conformal field theories propagates with velocity v^2=1. Both in quantum quenches and in the Vaidya model equilibration is only achieved at the local level. Remarkably, the holographic derivation of this last fact requires information from behind the apparent horizon generated in the process of gravitational collapse described by the Vaidya geometry. In the early stages of the evolution the apparent horizon seems however to play no relevant role with regard to the entanglement entropy. We speculate on the possibility of deriving a thermalization time for occupation numbers from our analysis.

Figures (10)

• Figure 1: Derivation of the entanglement entropy of an interval according to the model proposed in Calabrese:2005in. Only excitations originating from the shaded region at $t\!=\!0$ contribute to the entanglement entropy of region $A$ at time $t$.
• Figure 2: Position of the event (solid) and the apparent (dashed) horizons of the Vaidya metric \ref{['vaidya']} for $d\!=\!2$ and $m(v)\!=\!(\tanh v\!+\!1)/2$.
• Figure 3: ${\tilde{L}}(l,t)$ as a function of $l$ for $t=0,1,..,8$ from bottom to top, for $a\!=\!1/3$ (left) and $a\!=\!2$ (right). The dashed line corresponds to thermal equilibrium and the dot-dashed line to the vacuum result.
• Figure 4: For $a\!=\!1/3$ and $t\!=\!0,2,..,10$: (left) value of $v$ at the geodesic midpoint as a function of the interval size, and on the inset its derivative $\partial_l v_\ast$; (right) $b$ in \ref{['larger']} as a function of $l$ (solid lines) and its asymptotic value $b_\infty$ (dashed lines), whose evolution is plotted on the inset.
• Figure 5: Profile of geodesics for different $l$ at $t\!=\!6$ and $a\!=\!1/3$. Projection into the (a) $r\!-\!x$ plane, (b) $r\!-\!v$ plane and (c) $v\!-\!x$ plane. (d) Full profile of the geodesics. The dashed line in (b) and the shaded surface in (d) signal the location of the apparent horizon. In (a) the dashed lines represent the position of the apparent horizon at the value of $v$ defined by the trajectory of each geodesic: $\sqrt{m(v(x))}$.