Operator Growth and the Absence of Many-Body Localization

Many-body localized systems are characterized by an effective form of the Hamiltonian written in terms of local conserved charges with longer-range terms being exponentially suppressed. With this in mind, we discuss the growth of local operators in one-dimensional many-body systems by considering their k-fold commutator with the system's Hamiltonian. By studying bounds on the operator norm of this commutator as well as the spatial structure of its contributions, we are able to directly connect operator growth to the question of localization. We also investigate attempts to perturbatively transform the microscopic Hamiltonian into its effective form through consecutive Schrieffer-Wolff transformations. We present results from both the Heisenberg and mixed-field Ising model that indicate an absence of localization in the many-body interacting case.