Operator Growth in Disordered Spin Chains: Indications for the Absence of Many-Body Localization

We consider the spreading of a local operator $A$ in one-dimensional systems with Hamiltonian $H$ by calculating the $k$-fold commutator $[H,[H,[...,[H,A]]]]$. We derive bounds for the operator norm of this commutator in free and interacting systems with and without disorder thus directly connecting the operator growth hypothesis with questions of localization. We analytically show that an almost factorial growth of the operator norm - as recently proven for the random Ising model - is inconsistent with an exponential localization of $A$. Assuming that a quasi-local unitary $U$ exists which maps $H$ onto an effective Hamiltonian $\tilde H=UHU^\dagger=\sum_n E_n \tau^z_n +\sum_{i,j} J_{ij} \tau^z_i\tau^z_j+\dots$, we show that $\tilde A=UAU^\dagger$ is a quasi-local operator which in the many-body case does not remain exponentially localized in general leading to an almost factorial norm growth. Therefore the unitary $U$ in many-body systems with maximal norm growth either does not exist and such systems are always ergodic or unusual non-ergodic phases described by $\tilde H$ do exist which violate the operator growth hypothesis and in which operators spread, implying that transport will eventually set in. We analytically and symbolically verify our results for the Anderson and Aubry-Andr\'e models. For the XXX case, the symbolic calculations are consistent with a maximal norm growth. Furthermore, we find no indication of a weakened exponential localization of $A$, expected for strong disorder and low commutator orders if the unitary $U$ does exist. Finally, we try to perturbatively construct $U$ by consecutive Schrieffer-Wolff transformations. While it is straightforward to show that this construction converges in the Anderson case, we find no indications for a convergence in the interacting case, suggesting that $U$ does not exist and that many-body localization is absent.