Inner structure of the many-body localization transition and the fulfillment of the Harris criterion

We treat the disordered Heisenberg model in one dimension as the standard model of many-body localization (MBL). Two new and independent order parameters stemming solely from the half-chain von Neumann entanglement entropy ${S}_{\text{vN}}$ are introduced to probe the eigenstate phase transition in this model. From the symmetry-endowed entropy decomposition, they are the probability distribution deviation $|d({p}_{n})|$ and the von Neumann entropy ${S}_{\text{vN}}^{n}({D}_{n}=\text{max})$ of the maximally dimensional symmetry subdivision. The finite-size scaling reveals that the probability set ${{p}_{n}}$ drives the localization transition, preceded by a thermalization breakdown transition governed by ${{S}_{\text{vN}}^{n}}$. For the noninteracting case, these transitions coincide, but in the interacting circumstance, they separate. Such separability creates an intermediate phase regime and discriminates between the Anderson and MBL transitions. One obstacle whose solution eludes the community to date concerns the violation of the Harris criterion in most numerical investigations of MBL. Upon elucidating the mutually independent measures comprising ${S}_{\text{vN}}$, it becomes clear that the previous studies may lack the resolution to pinpoint, and thus potentially overlook, the crucial internal structure of the transition. We show that after this necessary decomposition, the universal critical exponents for both transitions of $|d({p}_{n})|$ and ${S}_{\text{vN}}^{n}({D}_{n}=\text{max})$ fulfill the Harris criterion: $\ensuremath{\nu}\ensuremath{\approx}2\phantom{\rule{4pt}{0ex}}(\ensuremath{\nu}\ensuremath{\approx}1.5)$ for quench (quasirandom) disorder. Our work puts forth symmetry combined with entanglement as an organization principle for the generic eigenstate matter and phase transition.