Towards a Refinement of Krylov Complexity: Scrambling, Classical Operator Growth and Replicas

TL;DR

This paper introduces logarithmic Krylov complexity, a new operator growth measure that effectively distinguishes genuine scrambling from saddle effects in quantum systems, and extends the concept to classical dynamics.

Contribution

It proposes and tests a replica-based logarithmic Krylov complexity that refines operator growth analysis, applicable to both quantum and classical systems, capturing integrability and scrambling.

Findings

logK complexity discriminates genuine from saddle-dominated scrambling in quantum models

Refined logK complexity captures integrable properties in certain models

Classical versions of operator growth measures show no false positives from unstable saddles

Abstract

We propose and test logarithmic Krylov (logK) complexity, an operator growth measure akin to Krylov complexity defined through a replica approach, as a viable probe of early-time operator scrambling without false positives. In finite-dimensional quantum systems, such as the Lipkin--Meshkov--Glick (LMG) model and the mixed-field Ising model at the chaotic point, we provide numerical evidence that logK-complexity discriminates between genuine and saddle-dominated scrambling at early times, correctly avoiding the exponential contribution coming from the unstable saddle in the former case, and closely tracking the conventional Krylov complexity in the latter. In integrable quantum systems admitting infinite-dimensional Krylov subspaces, such as the SYK$_{2}$ model and the quantum inverted harmonic oscillator, we show that by modifying the Krylov spreading operator, obtained through generalizing the analytic continuation procedure in the replica trick, the logK complexity can be refined to capture the integrable properties of the theories. We supplement these analyses by extending the Krylov formalism in classical dynamical systems and defining classical versions of these operator growth measures, showing that the false positives arising from unstable saddles in classical phase space are non-existent.