Spectral statistics across the many-body localization transition

The many-body localization transition (MBLT) between ergodic and many-body\nlocalized phase in disordered interacting systems is a subject of much recent\ninterest. Statistics of eigenenergies is known to be a powerful probe of\ncrossovers between ergodic and integrable systems in simpler examples of\nquantum chaos. We consider the evolution of the spectral statistics across the\nMBLT, starting with mapping to a Brownian motion process that analytically\nrelates the spectral properties to the statistics of matrix elements. We\ndemonstrate that the flow from Wigner-Dyson to Poisson statistics is a\ntwo-stage process. First, fractal enhancement of matrix elements upon\napproaching the MBLT from the metallic side produces an effective power-law\ninteraction between energy levels, and leads to a plasma model for level\nstatistics. At the second stage, the gas of eigenvalues has local interaction\nand level statistics belongs to a semi-Poisson universality class. We verify\nour findings numerically on the XXZ spin chain. We provide a microscopic\nunderstanding of the level statistics across the MBLT and discuss implications\nfor the transition that are strong constraints on possible theories.\n