Strongly resonating clusters around the ergodic-MBL transition

An interacting quantum system can transition from an ergodic to a many body localized (MBL) phase under the presence of sufficiently large disorder. Both phases are radically different in their dynamical properties, which are characterized by highly excited energy eigenstates. One of the differences between both phases is in the statistics of their energy levels: while in an ergodic phase levels experience repulsion and follow GOE statistics, in MBL they largely lack repulsion and follow Poisson statistics. Here, we argue that the transition between both behaviors is accompanied by the formation of resonating clusters in the eigenstates of the Hamiltonian. Using a basis of local integrals of motion (l-bits), we observe that these clusters take the form of cat states over a subset of l-bits on a 1D spin chain. While such resonances are rare in MBL, they proliferate around the transition, and are absent well into the ergodic phase. Finally, the spatial structure of the resonating clusters suggests correlations that are scale invariant in the chain, which we check numerically for finite size systems.