Scaling of the Fock-space propagator and multifractality across the many-body localization transition

We implement a recursive Green's function method to extract the Fock space (FS) propagator and associated self-energy across the many-body localization (MBL) transition, for one-dimensional interacting fermions in a random on-site potential. We show that the typical value of the imaginary part of the local FS self-energy, ₓ, related to the decay rate of an initially localized state, acts as a probabilistic order parameter for the thermal to MBL phase transition and can be used to characterize critical properties of the transition as well as the multifractal nature of MBL states as a function of disorder strength W. In particular, we show that a fractal dimension Dₒ extracted from ₓ jumps discontinuously across the transition, from Dₒ<1 in the MBL phase to Dₒ=1 in the thermal phase. Moreover, ₓ follows an asymmetrical finite-size scaling form across the thermal-MBL transition, where a nonergodic volume in the thermal phase diverges with a Kosterlitz-Thouless--like essential singularity at the critical point W₂ and controls the continuous vanishing of ₓ as W₂ is approached. In contrast, a correlation length () extracted from ₓ exhibits a power-law divergence on approaching W₂ from the MBL phase.