Spectral multifractality and emergent energy scales across the many-body localization transition

We present a scaling theory of the many-body localization transition in terms of emergent, characteristic energy scales. The analysis is based on the decomposition of the eigenstates in the basis of trivially localized states, resolved in the energies of the latter, which we refer to as the spectral decomposition of the eigenstates. The characteristic energy scales emerge when the multifractal properties, or lack thereof, of the spectral decomposition are studied at different scales. These characteristic scales correspond to the ones, above which the spectral decompositions exhibit their global behavior, namely full ergodicity in the ergodic phase and multifractality in the many-body localized phase. On the other hand, at scales below the characteristic ones, the decomposition in the ergodic phase shows finer (multi)fractal structures whereas in the localized phase, the decomposition picks out well-separated, localized resonant peaks. The scaling of these characteristic energy scales across the many-body localization transition admits a scaling theory consistent with a Kosterlitz-Thouless-type scenario and bears striking resemblances to that of inverse participation ratios of eigenstates.