Spectral Multifractality and Emergent Energyscales Across the Many-Body Localisation Transition

We present a scaling theory of the many-body localisation transition in terms of emergent, characteristic energyscales. The analysis is based on the decomposition of the eigenstates in the basis of trivially localised states, resolved in the energies of the latter, which we refer to as the spectral decomposition of the eigenstates. The characteristic energyscales 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 behaviour, namely full ergodicity in the ergodic phase and multifractality in the many-body localised 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 localised phase, the decomposition picks out well-separated, localised resonant peaks. The scaling of these characteristic energyscales across the many-body localisation transition admits a scaling theory consistent with a Kosterlitz-Thouless type scenario and bears striking resemblances to that of inverse participation ratios of eigenstates.