# A Universal Operator Growth Hypothesis

## TL;DR

This paper proposes a universal hypothesis linking operator growth rates in many-body systems to exponential complexity growth, providing bounds on chaos measures and new methods for computing diffusion constants.

## Contribution

It introduces a universal operator growth hypothesis with a growth rate  that bounds chaos and complexity measures, applicable across various models and systems.

## Key findings

- Lanczos coefficients grow linearly with rate  in generic systems
- The growth rate  bounds Lyapunov exponents as   ; a universal bound
- The hypothesis is validated in spin chains, SYK model, and classical models.

## 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.