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.
Highlights
The emergence of ergodic behavior in quantum systems is an old puzzle
We show that under the hypothesis, the 1D quantum mechanics, governed by the Lanczos coefficients bn ∼ αn, captures the irreversible process of simple operators evolving into complex ones
We show in the preceding section that K-complexity provides an upper bound for any q-complexity whatsoever, which includes certain types of order correlation function (OTOC)
Summary
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 α in generic systems, with an extra logarithmic correction in 1D. The rate α—an experimental observable—governs the exponential growth of operator complexity in a sense we make precise. This exponential growth prevails beyond semiclassical or large-N limits. Α upper bounds a large class of operator complexity measures, including the out-of-timeorder correlator. We obtain a sharp bound on Lyapunov exponents λL ≤ 2α, which complements and improves the known universal low-temperature bound λL ≤ 2πT. We use the hypothesis in conjunction with the recursion method to develop a technique for computing diffusion constants
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