  The many-body localization transition (MBLT) between ergodic and many-body
localized phase in disordered interacting systems is a subject of much recent
interest. Statistics of eigenenergies is known to be a powerful probe of
crossovers between ergodic and integrable systems in simpler examples of
quantum chaos. We consider the evolution of the spectral statistics across the
MBLT, starting with mapping to a Brownian motion process that analytically
relates the spectral properties to the statistics of matrix elements. We
demonstrate that the flow from Wigner-Dyson to Poisson statistics is a
two-stage process. First, fractal enhancement of matrix elements upon
approaching the MBLT from the metallic side produces an effective power-law
interaction between energy levels, and leads to a plasma model for level
statistics. At the second stage, the gas of eigenvalues has local interaction
and level statistics belongs to a semi-Poisson universality class. We verify
our findings numerically on the XXZ spin chain. We provide a microscopic
understanding of the level statistics across the MBLT and discuss implications
for the transition that are strong constraints on possible theories.
