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arXiv · 2404.08207

Dissecting Quantum Many-body Chaos in the Krylov Space

Liangyu Chen·Baoyuan Mu·Huajia Wang·Pengfei Zhang
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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.

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Liangyu Chen, Baoyuan Mu, Huajia Wang, Pengfei Zhang. 2024-04-12. Dissecting Quantum Many-body Chaos in the Krylov Space. https://arxiv.org/abs/2404.08207

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