  We study spectral and wavefunction statistics for many-body localization
transition in systems with long-range interactions decaying as $1/r^\alpha$
with an exponent $\alpha$ satisfying $ d \le \alpha \le 2d$, where $d$ is the
spatial dimensionality. We refine earlier arguments and show that the system
undergoes a localization transition as a function of the rescaled disorder $W^*
= W / L^{2d-\alpha} \ln L$, where $W$ is the disorder strength and $L$ the
system size. This transition has much in common with that on random regular
graphs. We further perform a detailed analysis of the inverse participation
ratio (IPR) of many-body wavefunctions, exploring how ergodic behavior in the
delocalized phase switches to fractal one at the critical point and on the
localized side of the transition. Our analytical results for the scaling of the
critical disorder $W$ with the system size $L$ and for the scaling of IPR in
the delocalized and localized phases are supported and corroborated by exact
diagonalization of spin chains.
