Operator and entanglement growth in nonthermalizing systems: Many-body localization and the random singlet phase

We characterize the growth and spreading of operators and entanglement in two paradigmatic nonthermalizing phases - the many-body localized (MBL) phase and the random singlet phase (RSP) - using the entanglement contour and multipartite operator entanglement measures. The entanglement contour characterizes the spacetime spreading of entanglement and reveals logarithmically growing entanglement light cones in the MBL and RSP phases, sharply contrasting the linear light cones of clean, thermalizing systems. The operator entanglement characterizes scrambling, i.e., the delocalization of information. We find slow scrambling behavior in the MBL phase; the late-time value of the tripartite mutual information scales linearly with system size, but is submaximal. The tripartite logarithmic negativity is also negative and nonzero, but smaller in magnitude, revealing an intriguing distinction between classical and quantum information scrambling in the MBL phase. This is in contrast with the RSP, which, as a noninteracting model, is nonscrambling.