Dissecting Quantum Many-body Chaos in the Krylov Space

TL;DR

This paper introduces the Krylov metric to better identify quantum many-body chaos, clarifying when Krylov complexity indicates chaos versus non-chaotic growth, supported by theoretical and model-based evidence.

Contribution

It proposes the Krylov metric as a new tool to distinguish chaotic from non-chaotic systems, refining criteria for quantum chaos detection in Krylov space.

Findings

Krylov metric $K_{mn}$ probes Krylov basis size.

Universal criteria for fast scramblers include exponential Krylov complexity growth.

The ratio $h=\varkappa / 2\alpha$ links quantum Lyapunov exponent and Krylov exponent.

Abstract

The growth of simple operators is essential for the emergence of chaotic dynamics and quantum thermalization. Recent studies have proposed different measures, including the out-of-time-order correlator and Krylov complexity. It is established that the out-of-time-order correlator serves as the signature of quantum many-body chaos, while the Krylov complexity provides its upper bound. However, there exist non-chaotic systems in which Krylov complexity grows exponentially, indicating that the Krylov complexity itself is not a witness of many-body chaos. In this letter, we introduce the missing ingredient, named as the Krylov metric $K_{mn}$, which probes the size of the Krylov basis. We propose that the universal criteria for fast scramblers include (i) the exponential growth of Krylov complexity, (ii) the diagonal elements $K_{nn}\sim n^h$ with $h\in(0,1]$, and (iii) the negligibility of off-diagonal elements $K_{mn}$ with $m\neq n$. We further show that $h=\varkappa / 2\alpha$ is a ratio between the quantum Lyapunov exponent $\varkappa$ and the Krylov exponent $\alpha$. This proposal is supported by both generic arguments and explicit examples, including solvable SYK models, Luttinger Liquids, and many-body localized systems. Our results provide a refined understanding of how chaotic dynamics emerge from the Krylov space perspective.