Krylov operator complexity in holographic CFTs: Smeared boundary reconstruction and the dual proper radial momentum

TL;DR

The paper develops a holographic (AdS/CFT) framework to quantify Krylov operator complexity in CFT$_d$ via bulk-to-bulk propagators in $\text{Rindler-AdS}_{d+1}$, revealing how non-locality in boundary observables alters Lanczos data and complexity growth. It demonstrates that the rate of Krylov complexity growth $\dot K(t)$ tracks the bulk's proper radial momentum $P_\rho$, with exact matching at the asymptotic boundary and near the horizon, and approximate matching at intermediate radii due to conformal symmetry breaking. The work connects boundary Krylov growth to bulk thermal ensembles and discusses interpretations in terms of holographic RG flow and $T\overline T$ deformations, while outlining the need for a rigorous first-principles derivation of the holographic dictionary entry. It also highlights future avenues, including the switchback effect, precursor insertions in shockwave geometries, and a more precise HKLL-based inner product formulation at finite bulk location. Overall, the results advance Krylov complexity as a robust diagnostic in holography and suggest deep links between operator growth, bulk motion, and the holographic dictionary.

Abstract

Motivated by bulk reconstruction of smeared boundary operators, we study the Krylov complexity of local and non-local primary CFT$_d$ operators from the local bulk-to-bulk propagator of a minimally-coupled massive scalar field in Rindler-AdS$_{d+1}$ space. We derive analytic and numerical evidence on how the degree of non-locality in the dual CFT$_d$ observable affects the evolution of Krylov complexity and the Lanczos coefficients. Curiously, the near-horizon limit matches with the same observable for conformally-coupled probe scalar fields inserted at the asymptotic boundary of AdS$_{d+1}$ space. Our results also show that the evolution of the growth rate of Krylov operator complexity in the CFT$_d$ takes the same form as to the proper radial momentum of a probe particle inside the bulk to a good approximation. The exact equality only occurs when the probe particle is inserted in the asymptotic boundary or in the horizon limit. Our results capture a prosperous interplay between Krylov complexity in the CFT, thermal ensembles at finite bulk locations and their role in the holographic dictionary.

Figures (23)

• Figure 1: Rindler wedges (shown in gray) of global AdS (solid cylinder)
• Figure 2: Penrose diagram of the Rindler-AdS geometry representing the location of the two points $P=(t,r,\mathbf{x})$ and $P'=(i \pi,r,\mathbf{x}')$ such that $\textrm{dist}({\bf x,x'})=0$.
• Figure 3: Lanczos coefficient $b_n$ for $C(t,r)$ (\ref{['eq:afbbp_eps_pi']}) with $\Delta=15$ and $d=4$ for different values of $r=50,2,1.1,1.01,1.0001$. The blue and orange solid curves are the boundary $r\rightarrow \infty$ and horizon $r\rightarrow 1$ Lanczos coefficients respectively in (\ref{['eq:bnLimits']}).
• Figure 4: Krylov complexity $K(t)$ for $C(t,r)$ (\ref{['eq:afbbp_eps_pi']}) with $\Delta=15$ and $d=4$ for different values of $r=50,2,1.1,1.01,1.0001$. The blue and orange solid curves are the boundary $r\rightarrow \infty$ and horizon $r\rightarrow 1$ Krylov complexities respectively in \ref{['eq:KCLimits']}.
• Figure 5: Lanczos coefficient $b_n$ for $C(t,r)$ (\ref{['eq:afbbp_eps_pi']}) with $\Delta=1.2$ and $d=4$ for different values of $r$. The blue and orange solid curves on the left-hand side plot are the boundary $r\rightarrow \infty$ and horizon $r\rightarrow 1$ Lanczos coefficients respectively in (\ref{['eq:bnLimits']}). The left-hand side plot shows the behaviour of $b_{n}$ for $n\in [0,100]$ with $r=12,8,1.5,1.2,1.0001$, while the right-hand side plot shows the behaviour for $n\in [0,15]$ with $r=12,4,1.5,1.0001$.