Logarithmic light cone, slow entanglement growth, and quantum memory

Abstract: Effective light cones, characterized by Lieb-Robinson bounds, emerge in non-relativistic local quantum systems. Here, we present several analytical results derived from logarithmic light cones (LLCs), which can arise in the one-dimensional XXZ model with random fields and in a phenomenological model of many-body localization (MBL). In the LLC regime, we prove that the entanglement growth is upper-bounded by logarithmic time with an additional subleading double-logarithmic term stemming from a real asymptotic solution of the \emph{Lambert W} function. In the context of the XXZ model, recent numerical results suggest that the double-logarithmic term correlates with number entropy. We also show that information scrambling is logarithmically slow in the LLC regime. Furthermore, we demonstrate that the LLC supports long-lived quantum memories under unitary time evolution as quantum codes with macroscopic code distance and an exponentially scaling lifetime. Our analytical results provide benchmarks for future numerical explorations of the MBL regime on large time scales.