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For short-ranged disordered quantum models in one dimension, theMany-Body-Localization is analyzed via the adaptation to the Many-Body context[M. Serbyn, Z. Papic and D.A. Abanin, PRX 5, 041047 (2015)] of the Thoulesspoint of view on the Anderson transition : the question is whether a localinteraction between two long chains is able to reshuffle completely theeigenstates (Delocalized phase with a volume-law entanglement) or whether thehybridization between tensor states remains limited (Many-Body-Localized Phasewith an area-law entanglement). The central object is thus the level ofHybridization induced by the matrix elements of local operators, as comparedwith the difference of diagonal energies. The multifractal analysis of thesematrix elements of local operators is used to analyze the correspondingstatistics of resonances. Our main conclusion is that the critical point ischaracterized by the Strong-Multifractality Spectrum $f(0 \leq \alpha \leq2)=\frac{\alpha}{2}$, well known in the context of Anderson Localization inspaces of effective infinite dimensionality, where the size of the Hilbertspace grows exponentially with the volume. Finally, the possibility of adelocalized non-ergodic phase near criticality is discussed.

Many-Body-Localization Transition : strong multifractality spectrum for matrix elements of local operators