Krylov complexity for 1-matrix quantum mechanics

TL;DR

This paper explores Krylov complexity in 1-matrix quantum mechanics, revealing linear growth of Lanczos coefficients and suggesting links to chaos even in integrable systems, thereby advancing understanding of operator growth in quantum models.

Contribution

It introduces the analysis of Krylov complexity in 1-MQM, showing linear growth of Lanczos coefficients and connecting complexity growth to chaotic behavior in an integrable system.

Findings

Lanczos coefficients grow linearly in 1-MQM

Krylov complexity indicates possible chaos in integrable systems

Provides groundwork for studying complexity in holographic theories

Abstract

This paper investigates the notion of Krylov complexity, a measure of operator growth, within the framework of 1-matrix quantum mechanics (1-MQM). Krylov complexity quantifies how an operator evolves over time by expanding it in a series of nested commutators with the Hamiltonian. We analyze the Lanczos coefficients derived from the correlation function, revealing their linear growth even in this integrable system. This growth suggests a link to chaotic behavior, typically unexpected in integrable systems. Our findings in both ground and thermal states of 1-MQM provide new insights into the nature of complexity in quantum mechanical models and lay the groundwork for further studies in more complex holographic theories.