Numerically Probing the Universal Operator Growth Hypothesis

Recently, a hypothesis on the complexity growth of unitarily evolving operators was presented. This hypothesis states that in generic, non-integrable many-body systems the so-called Lanczos coefficients associated with an autocorrelation function grow asymptotically linear, with a logarithmic correction in one-dimensional systems. In contrast, the growth is expected to be slower in integrable or free models. In the paper at hand, we numerically test this hypothesis for a variety of exemplary systems, including 1d and 2d Ising models as well as 1d Heisenberg models. While we find the hypothesis to be practically fulfilled for all considered Ising models, the onset of the hypothesized universal behavior could not be observed in the attainable numerical data for the Heisenberg model. The proposed linear bound on operator growth eventually stems from geometric arguments involving the locality of the Hamiltonian as well as the lattice configuration. We investigate such a geometric bound and find that it is not sharply achieved for any considered model.