Resonance Proliferation Across Localization Transitions

Models of many-body localization (MBL) exhibit slow numerical drifts towards delocalization with increasing system size, for which no satisfactory theory exists. Numerical results indicate that these drifts are driven by the proliferation of many-body resonances at intermediate disorder strengths. We develop a statistical method to predict the distribution of resonance oscillation frequencies, accounting for how resonances formed at larger frequency scales affect the subsequent formation of resonances at lower frequencies. Working within the statistical Jacobi approximation (SJA), we derive a flow equation for a power-law exponent θ(w) characterizing the density of resonances at frequency scale w. A localized phase is described by a line of fixed points with θ(w)>0, while an instability of the flow signals resonance proliferation and the onset of thermalization. The predicted θ(w) agrees with numerical results for the Anderson model on random regular graphs and the Lévy--Rosenzweig--Porter random matrix ensemble, both of which host resonance-driven delocalization transitions. We further connect the flow to eigenstate properties such as the participation ratio and to dynamical observables such as the return probability. The predicted θ(w) also agrees with numerical measurements in real-space models of MBL at intermediate disorder strengths, providing a step towards explaining the finite-size drifts observed in MBL.