Many-body localization edge in the random-field Heisenberg chain

We present a large-scale exact diagonalization study of the one-dimensional spin-1/2 Heisenberg model in a random magnetic field. In order to access properties at varying energy densities across the entire spectrum for system sizes up to L = 22 spins, we use a spectral transformation which can be applied in a massively parallel fashion. Our results allow for an energy-resolved interpretation of the many-body localization transition including the existence of an extensive many-body mobility edge. The ergodic phase is well characterized by Gaussian orthogonal ensemble statistics, volume-law entanglement, and a full delocalization in the Hilbert space. Conversely, the localized regime displays Poisson statistics, area-law entanglement, and nonergodicity in the Hilbert space where a true localization never occurs. We perform finite-size scaling to extract the critical edge and exponent of the localization length divergence.