Logarithmic light cone, slow entanglement growth, and quantum memory

Effective light cones, characterized by Lieb-Robinson bounds, emerge in nonrelativistic local quantum systems. Here, we present several analytical results derived from logarithmic light cones (LLCs). Possible origins of LLCs include the one-dimensional (1D) disordered XXZ model and a phenomenological model of many-body localization (MBL). In the LLC regime, we prove that, for arbitrary spatial dimensions and any initial pure state, entanglement growth is upper-bounded by logarithmic time with an additional subleading \emph{double-logarithmic} correction -- arising from a real asymptotic solution of the \emph{Lambert W} function -- valid up to the asymptotic time limit. In the context of the 1D disordered XXZ model, this result resolves the ambiguity in distinguishing between logarithmic and power-law fits of entanglement growth in numerical studies; we also propose a falsifiable phenomenological functional form for the entanglement growth that agrees with existing numerical results. We show that information scrambling is logarithmically slow in the LLC regime. Furthermore, we demonstrate that the LLC supports long-lived quantum memories -- quantum codes with macroscopic code distance and lifetimes that scale exponentially with system size -- under unitary time evolution. Our analytical results provide benchmarks for future numerical studies of the MBL regime at large time scales.