Hilbert space properties of the many-body localization problem: from full ergodicity to multifractality

In contrast with Anderson localization where a genuine localization is observed in real space, the many-body localization (MBL) problem is much less understood in the Hilbert space, support of the eigenstates. In this work, using exact diagonalization techniques up to L=24 spin-1/2 particles (i.e. Hilbert space of size N=2.7 millions) we address the ergodicity properties in the underlying N-dimensional complex networks spanned by various computational bases. We report fully ergodic eigenstates in the delocalized phase (irrespective of the computational basis), while the MBL regime features a generically (basis-dependent) multifractal behavior, delocalized but non-ergodic. The MBL transition is signaled by a non-universal jump of the multifractal dimensions.