Logarithmic entanglement light cone from eigenstate correlations in the many-body localized phase

We investigate the operator entanglement of the time-evolution operator through the framework of eigenstate correlations. Focusing on strongly disordered quantum many-body systems in the many-body localized (MBL) regime, we analyze the operator entanglement across various spatiotemporal cuts, revealing the logarithmic lightcone of entanglement spreading. We demonstrate that this logarithmic lightcone arises directly from a hierarchy of energyscales and lengthscales encoded in eigenstate correlations. By characterizing the statistics of these hierarchical scales, we develop a microscopic theory for the spatiotemporal structure of entanglement spreading in MBL systems---without invoking phenomenological constructs such as $\ensuremath{\ell}$-bits. This approach reveals the fundamental connection between eigenstate correlations and the emergent entanglement structure in MBL systems.