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The relaxation of out-of-time-ordered correlators (OTOCs) has been studied asa mean to characterize the scrambling properties of a quantum system. We showthat the presence of local conserved quantities typically results in, at thefastest, an algebraic relaxation of the OTOC provided (i) the dynamics is localand (ii) the system follows the eigenstate thermalization hypothesis. Ourresult relies on the algebraic scaling of the infinite-time value of OTOCs withsystem size, which is typical in thermalizing systems with local conservedquantities, and on the existence of finite speed of propagation of correlationsfor finite-range-interaction systems. We show that time-independence of theHamiltonian is not necessary as the above conditions (i) and (ii) can occur intime-dependent systems, both periodic or aperiodic. We also remark that ourresult can be extended to systems with power-law interactions.

From ETH to algebraic relaxation of OTOCs in systems with conserved quantities