Divergences in the rate of complexification

TL;DR

This work tests the proposed bounds on holographic complexity growth by perturbing a holographic CFT with a time-dependent relevant operator of dimension $\Delta=3$. By mapping the quench to a time-dependent bulk scalar field with backreaction in AdS$_5$, the authors compute the rate of complexification under both the complexity=action (CA) and complexity=volume (CV) proposals. They find UV divergences in the rates: $\frac{d}{d\mathfrak{t}}\mathcal{C}_A(\mathfrak{t})$ contains a $\frac{\log\delta}{\delta}$ term proportional to $\frac{d}{d\mathfrak{t}}\lambda^2(\mathfrak{t})$, and $\frac{d}{d\mathfrak{t}}\mathcal{C}_V(\mathfrak{t})$ contains a $\frac{1}{\delta}$ term plus a $\log\delta$ term proportional to $\frac{d}{d\mathfrak{t}}\lambda^3(\mathfrak{t})$, while the instantaneous energy remains UV finite. Consequently, neither CA nor CV obeys the Lloyd bound in this setup, demonstrating a strong violation of the proposed complexity-energy bound in a time-dependent holographic quench. The CA result is universal, tied to the central charge, whereas the CV result depends on bulk three-point data $\kappa_3$, underscoring nonuniversality in CV. These findings further illuminate the limits of holographic complexity proposals for dynamical, out-of-equilibrium processes.

Abstract

It is conjectured that the average energy provides an upper bound on the rate at which the complexity of a holographic boundary state grows. In this paper, we perturb a holographic CFT by a relevant operator with a time-dependent coupling, and study the complexity of the time-dependent state using the \textit{complexity equals action} and the \textit{complexity equals volume} conjectures. We find that the rate of complexification according to both of these conjectures has UV divergences, whereas the instantaneous energy is UV finite. This implies that neither the \textit{complexity equals action} nor \textit{complexity equals volume} conjecture is consistent with the conjectured bound on the rate of complexification.