Multifractal dimension of high-energy eigenstates in Fock and spin configuration bases across the MBL transition for the random-field Heisenberg chain. Two typical eigenstates on a small L=14 system for (i) the delocalized regime (ETH h=0.5, blue) and (ii) for MBL (h=10, green) are graphically represented, with circle sizes proportional to the wave function amplitudes in the spin basis.

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 Hilbert space, the support of the eigenstates. In a recent work, published in Physical Review Letters, N. Macé, F. Alet, and N. Laflorencie [Phys. Rev. Lett. 123, 180601 (2019)], have used state-of-the art exact diagonalization techniques to address the ergodicity properties in the underlying =≈×6). Fully ergodic eigenstates are observed in the delocalized phase (irrespective of the computational basis), while the MBL regime features a generically (basis-dependent) multifractal behavior, delocalized but nonergodic. The MBL transition is signaled by a nonuniversal jump of the multifractal dimensions. The finite size scaling analysis shows a linear scaling behavior in the MBL regime. In contrast, a volumic scaling is found in the ergodic phase, with an emerging non-ergodicity volume Λ, as previously unveiled by I. García-Mata, O. Giraud, B. Georgeot, J. Martin, R. Dubertrand, and G. Lemarié for the Anderson transition on random graphs in Phys. Rev. Lett. 118, 166801 (2017).