Complexity measure diagnostics of ergodic to many-body localization transition

We introduce diagnostics of the transition between the ergodic and many-body localization phases, which are based on complexity measures defined via the probability distribution function of the Lanczos coefficients of the tridiagonalized Hamiltonian. We use these complexity measures to analyze the power-law random banded matrix model as a function of the correlation strength and show that the moments and the entropy of the distribution diagnose the ergodic to many-body transition, as well as the distinctive feature of the phases concerning the memory of the initial conditions.