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arXiv · 1812.08657

A Universal Operator Growth Hypothesis

Daniel E. Parker·Xiangyu Cao·Alexander Avdoshkin·Thomas Scaffidi·Ehud Altman
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Abstract

We present a hypothesis for the universal properties of operators evolving under Hamiltonian dynamics in many-body systems. The hypothesis states that successive Lanczos coefficients in the continued fraction expansion of the Green's functions grow linearly with rate $\alpha$ in generic systems, with an extra logarithmic correction in 1d. The rate $\alpha$ --- an experimental observable --- governs the exponential growth of operator complexity in a sense we make precise. This exponential growth even prevails beyond semiclassical or large-$N$ limits. Moreover, $\alpha$ upper bounds a large class of operator complexity measures, including the out-of-time-order correlator. As a result, we obtain a sharp bound on Lyapunov exponents $\lambda_L \leq 2 \alpha$, which complements and improves the known universal low-temperature bound $\lambda_L \leq 2 \pi T$. We illustrate our results in paradigmatic examples such as non-integrable spin chains, the Sachdev-Ye-Kitaev model, and classical models. Finally we use the hypothesis in conjunction with the recursion method to develop a technique for computing diffusion constants.

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BibTeXRIS

Daniel E. Parker, Xiangyu Cao, Alexander Avdoshkin, Thomas Scaffidi, Ehud Altman. 2018-12-20. A Universal Operator Growth Hypothesis. https://doi.org/10.1103/physrevx.9.041017

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