Multifractality and self-averaging at the many-body localization transition

We study the one-dimensional (1D) Heisenberg spin-1/2 model with onsite disorder at the transition to the many-body localized phase and show that the states satisfy an analytical expression obtained for the multifractal dimensions of the eigenstates of the 1D Fibonacci lattice. These results suggest that the spin model may be multifractal even before the transition point to localization. We also discuss the difficulties in reaching definite answers about the presence or absence of multifractality in this system due to the lack of self-averaging. One risks arriving at wrong conclusions if the number of disorder realizations used in numerical studies is reduced as the system size grows.