The disordered XXZ model is a prototype model of the many-body localization
transition (MBL). Despite numerous studies of this model, the available
numerical evidence of multifractality of its eigenstates is not very conclusive
due severe finite size effects. Moreover it is not clear if similarly to the
case of single-particle physics, multifractal properties of the many-body
eigenstates are related to anomalous transport, which is observed in this
model. In this work, using a state-of-the-art, massively parallel, numerically
exact method, we study systems of up to 24 spins and show that a large fraction
of the delocalized phase flows towards ergodicity in the thermodynamic limit,
while a region immediately preceding the MBL transition appears to be
multifractal in this limit. We discuss the implication of our finding on the
mechanism of subdiffusive transport.Comment: 13 pages, 8 figure
