Local Operator Entanglement in Spin Chains

We study the time evolution of bi- and tripartite operator mutual information of the time-evolution operator and Pauli's spin operators in the one-dimensional Ising model with magnetic field and the disordered Heisenberg model. In the Ising model, the early-time evolution qualitatively follows an effective light cone picture, and the late-time value is well described by Page's value for a random pure state. In the Heisenberg model with strong disorder, we find many-body localization prevents the information from propagating and being delocalized. We also find an effective Ising Hamiltonian describes the time evolution of bi- and tripartite operator mutual information for the Heisenberg model in the large disorder regime.