The emergence of ergodic behavior in quantum systems is an old puzzle. Quantum mechanical time-evolution is local and unitary, but many quantum systems are successfully described by irreversible hydrodynamics. I will present a hypothesis for how operators grow in strongly-interacting many-body systems and thence give rise to hydrodynamics. The hypothesis states that the Lanczos coefficients in the continued fraction expansion of the Green's function growth linearly with a "universal growth rate" $\alpha$ in chaotic quantum systems.
I will describe the extensive analytical and numerical evidence for this hypothesis, as well as three of its consequences. (1) Operator growth can diagnose free, integrable, and chaotic dynamics. (2) The growth rate --- an experimental observable --- gives rise to a quantity called the "K-complexity". The K-complexity quantifies the "amount of chaos" in any quantum system, and reduces to the Lyapunov exponent in semiclassical limits. (3) Assuming the hypothesis, one can accurately compute diffusion coefficients and other hydrodynamical data with minimal computational effort.
This talk is based on arXiv: 1812.08657.