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We treat disordered Heisenberg model in 1D as the "standard model" ofmany-body localization (MBL). Two independent order parameters stemming purelyfrom the half-chain von Neumann entanglement entropy $S_{\textrm{vN}}$ areintroduced to probe its eigenstate transition. From symmetry-endowed entropydecomposition, they are probability distribution deviation $|d(p_n)|$ and vonNeumann entropy $S_{\textrm{vN}}^{n}(D_n\!=\!\mbox{max})$ of themaximum-dimensional symmetry subdivision. Finite-size analyses reveal that$\{p_n\}$ drives the localization transition, preceded by a thermalizationbreakdown transition governed by $\{S_{\textrm{vN}}^{n}\}$. For noninteractingcase, these transitions coincide, but in interacting situation they separate.Such separability creates an intermediate phase region and may helpdiscriminate between the Anderson and MBL transitions. An obstacle whosesolution eludes community to date is the violation of Harris criterion innearly all numeric investigations of MBL so far. Upon elucidating the mutuallyindependent components in $S_{\textrm{vN}}$, it is clear that previous studiesof eigenspectra, $S_{\textrm{vN}}$, and the like lack resolution to pinpoint(thus completely overlook) the crucial internal structures of the transition.We show, for the first time, that after this necessary decoupling, theuniversal critical exponents for both transitions of $|d(p_n)|$ and$S_{\textrm{vN}}^{n}(D_n\!=\!\mbox{max})$ fulfill the Harris criterion:$\nu\approx2.0\ (\nu\approx1.5)$ for quench (quasirandom) disorder. Our workputs forth "symmetry combined with entanglement" as the missing organizationprinciple for the generic eigenstate matter and transition.

Inner Structure of Many-Body Localization Transition and Fulfillment of Harris Criterion