A Universal Operator Growth Hypothesis

• Main Authors: Daniel E. Parker, Xiangyu Cao, Alexander Avdoshkin, Thomas Scaffidi, Ehud Altman
• Format: Article
• Language: English
• Published: American Physical Society 2019-10-01
• Series: Physical Review X
• Online Access: http://doi.org/10.1103/PhysRevX.9.041017

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. Moreover, α 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 λ_{L}≤2α, which complements and improves the known universal low-temperature bound λ_{L}≤2πT. We illustrate our results in paradigmatic examples such as nonintegrable 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.