Quantum information spreading in random spin chains with topological order

Quantum information spreading and scrambling in many-body systems attract interest these days. Tripartite mutual information (TMI) based on operator-based entanglement entropy (EE) is an efficient tool for measuring them. In this paper, we study random spin chains that exhibit phase transitions accompanying nontrivial changes in topological properties. In their phase diagrams, there are two types of many-body localized (MBL) states and one thermalized regime intervening these two MBL states. Quench dynamics of the EE and TMI display interesting behaviors, providing an essential perspective concerning the encoding of quantum information. In particular, one of the models is self-dual, but information spreading measured by the TMI does not respect this self-duality. We investigate this phenomenon from the viewpoint of spatial structure of the stabilizers. In general, we find that knowledge of the phase diagram corresponding to the qubit system is useful for understanding the nature of quantum information spreading in that system. The connection between the present paper and random circuit of projective measurements and also topological Majorana quantum memory is remarked upon.