Relaxation of OTOC: effect of symmetries in integrable and non-integrable local Hamiltonians

Out-of-time ordered correlators (OTOCs) help characterize the scrambling of quantum information. We show that emergence of slow, algebraic, relaxation is expected in local Hamiltonians when the operators in the OTOC have non-zero diagonal elements in the energy basis, something that occurs when these operators have overlap with conserved quantities. Furthermore, the long-time slow relaxation of the OTOC can be predicted from studying simpler two-time correlator, and it is more strongly determined by the symmetry rather than integrability. Our numerical results corroborate our analytical expectations both for diagonal and off-diagonal elements of the OTOCs' local operators in the energy eigenbasis.