On The Evolution Of Operator Complexity Beyond Scrambling

TL;DR

The paper investigates operator complexity beyond the scrambling time using K-complexity, defined via Krylov-space dynamics and Lanczos coefficients. Under ETH, the post-scrambling evolution exhibits a linear growth of K-complexity with a rate of order $\lambda S$ and saturates at $\mathcal{O}(e^{S})$ on exponentially long times, linking to bulk evolution concepts. It introduces K-entropy to quantify operator randomization, finding linear growth during scrambling and logarithmic growth in the post-scrambling regime, with saturation at $\log(n_{ m max}) = O(S)$. The results support interpreting K-complexity as a meaningful holographic quantity and provide a framework to study long-time operator dynamics in finite systems. Numerical and analytic analyses (Airy-function and Bessel-function tails) demonstrate how randomization emerges in the post-scrambling era, suggesting deep connections to bulk dynamics and information scrambling in AdS/CFT.

Abstract

We study operator complexity on various time scales with emphasis on those much larger than the scrambling period. We use, for systems with a large but finite number of degrees of freedom, the notion of K-complexity employed in arXiv:1812.08657 for infinite systems. We present evidence that K-complexity of ETH operators has indeed the character associated with the bulk time evolution of extremal volumes and actions. Namely, after a period of exponential growth during the scrambling period the K-complexity increases only linearly with time for exponentially long times in terms of the entropy, and it eventually saturates at a constant value also exponential in terms of the entropy. This constant value depends on the Hamiltonian and the operator but not on any extrinsic tolerance parameter. Thus K-complexity deserves to be an entry in the AdS/CFT dictionary. Invoking a concept of K-entropy and some numerical examples we also discuss the extent to which the long period of linear complexity growth entails an efficient randomization of operators.

Figures (11)

• Figure 1: Qualitative form of the Lanczos sequence for a fast scrambler with $S$ degrees of freedom and Lyapunov exponent $\lambda$. It shows the linear growth characteristic of a fast scrambler, the long constant regime and a sharp turnoff at saturation.
• Figure 2: Evolution of K-complexity for a fast scrambler of size $S$, featuring an exponential law in the pre-scrambling era $t<t_* \sim \lambda^{-1} \,\log\,S$, followed by a a linear law in the post-scrambling era, up to $t_K \sim e^{O(S)} / \lambda S$, when the complexity finally saturates.
• Figure 3: Plot of the Airy function (\ref{['appt']}) for $t=40$ (red), $t=200$ (blue) and $t=500$ (magenta). Notice the very efficient randomization.
• Figure 4: The amplitude (\ref{['exactg']}) for a very narrow initial pulse, $\delta = 10^{-2}$ in units of the Lyapunov exponent, is very similar to the amplitude for an initial delta-function pulse.
• Figure 5: The amplitude (\ref{['exactg']}) for a wide initial pulse, with $\delta =1$ in units of the Lyapunov exponent, exhibits an exponentially damped tail.