Evolution of Complexity Following a Global Quench

TL;DR

The paper tests the complexity equals action (CA) conjecture and the associated Lloyd bound for a global quantum quench in holographic CFTs by modeling the quench with AdS-Vaidya, computing the Wheeler-deWitt patch action, and deriving the time dependence of complexity. It provides exact results in d=2 and a general rate formula for higher dimensions, showing the plane where the WdW patch intersects the infalling shell never crosses the horizon, thereby obeying the bound, with the growth rate saturating soon after local equilibrium is reached. The analysis confirms a nontrivial consistency between CA and dynamical quench dynamics, and identifies the timescale t ~ z_h as when the rate approaches its maximum. These results strengthen the case for CA as a holographic measure of quantum complexity and clarify its late-time behavior after quenches.

Abstract

The rate of complexification of a quantum state is conjectured to be bounded from above by the average energy of the state. A different conjecture relates the complexity of a holographic CFT state to the on-shell gravitational action of a certain bulk region. We use 'complexity equals action' conjecture to study the time evolution of the complexity of the CFT state after a global quench. We find that the rate of growth of complexity is not only consistent with the conjectured bound, but it also saturates the bound soon after the system has achieved local equilibrium.

Figures (3)

• Figure 1: The Penrose diagram of the time dependent geometry as a result of a collapse of a null shell (shown as a double line). The dashed line is the event horizon, and the shaded region denotes the WdW patch corresponding to boundary time $\mathfrak{t}$. The intersection of the past null boundary of the WdW patch and the collapsing null shell is denoted by a black dot and is labeled by $P$.
• Figure 2: The Penrose diagrams for AdS-Schwarzschild black hole (left) and vacuum AdS (right). The shaded region on the left/right corresponds to the part of the WdW inside/outside the collapsing null shell in Fig. (\ref{['fig-1']}). We have also included a cut-off surface at $z=\delta$ (shown as a blue line) in the Penrose diagram of AdS-Schwarzschild. The intersections of the WdW patch with the cut-off surface are labeled $A$ and $B$, whereas the intersection of the past null boundary of the WdW patch and the collapsing null shell is denoted by a black dot and is labeled $P$.
• Figure 3: The plots showing the position of the plane $P$ as a function of time for $d = \{3,4,5,6\}$. It is evident from these plots that $z_{P}(\mathfrak{t}) \to z_{h}$ when $\mathfrak{t} \sim z_{h}$.