Many-Body Localization from the perspective of the Fock-Space propagator

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, Δ_t 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 D_s extracted from Δ_t jumps discontinuously across the transition, from D_s<1 in the MBL phase to D_s=1 in the thermal phase. Moreover, Δ_t 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 W_c and controls the continuous vanishing of Δ_t as W_c is approached. In contrast, a correlation length (ξ) extracted from Δ_t exhibits a power-law divergence on approaching W_c from the MBL phase. We also comment on the similarities and differences between the form of the Fock space propagator for systems with disorder and those with quasiperiodic potentials.