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We introduce a multi-scale diagonalization scheme to study the transitionbetween the many-body localized and the ergodic phase in disordered quantumchains. The scheme assumes a sharp dichotomy between subsystems that behave aslocalized and resonant spots that obey the Eigenstate Thermalization Hypothesis(ETH). We establish a set of microscopic principles defining thediagonalization scheme, and use them to numerically study the transition invery large systems. To a large extent the results are in agreement with ananalytically tractable mean-field analysis of the scheme: We find that at thecritical point the system is almost surely localized in the thermodynamiclimit, hosting a set of thermal inclusions whose sizes are power-lawdistributed. On the localized side the {\em typical} localization length isbounded from above. The bound saturates upon approach to criticality, entailingthat a finite ergodic inclusion thermalizes a region of diverging diameter. Thedominant thermal inclusions have a fractal structure, implying that averagedcorrelators decay as stretched exponentials throughout the localized phase.Slightly on the ergodic side thermalization occurs through an avalancheinstability of the nearly localized bulk, whereby rare, supercritically largeergodic spots eventually thermalize the entire sample. Their size diverges atthe transition, while their density vanishes. The non-local, avalanche-likenature of this instability entails a breakdown of single parameter scaling andputs the delocalization transition outside the realm of standard criticalphenomena.

A microscopically motivated renormalization scheme for the MBL/ETH transition