Spread complexity in saddle-dominated scrambling

TL;DR

This paper investigates spread complexity in integrable systems with saddle-dominated scrambling, revealing chaotic-like features and analyzing its relation to spectral form factors and transition probabilities.

Contribution

It demonstrates that spread complexity can mimic chaotic behavior in integrable models and provides analytical and numerical insights into its early-time quadratic growth.

Findings

Spread complexity shows ramp-peak-slope-plateau pattern in studied models

Analytical confirmation of early-time quadratic behavior

Complexity features resemble chaos despite integrability

Abstract

Recently, the concept of spread complexity, Krylov complexity for states, has been introduced as a measure of the complexity and chaoticity of quantum systems. In this paper, we study the spread complexity of the thermofield double state within \emph{integrable} systems that exhibit saddle-dominated scrambling. Specifically, we focus on the Lipkin-Meshkov-Glick model and the inverted harmonic oscillator as representative examples of quantum mechanical systems featuring saddle-dominated scrambling. Applying the Lanczos algorithm, our numerical investigation reveals that the spread complexity in these systems exhibits features reminiscent of \emph{chaotic} systems, displaying a distinctive ramp-peak-slope-plateau pattern. Our results indicate that, although spread complexity serves as a valuable probe, accurately diagnosing true quantum chaos generally necessitates additional physical input. We also explore the relationship between spread complexity, the spectral form factor, and the transition probability within the Krylov space. We provide analytical confirmation of our numerical results, validating the Ehrenfest theorem of complexity and identifying a distinct quadratic behavior in the early-time regime of spread complexity.