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arXiv:2605.05445

Resonance Proliferation Across Localization Transitions

Carlo Vanoni, David M. Long, Anushya Chandran

Published May 6, 2026• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2605.05445

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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)θ(w)θ(w) characterizing the density of resonances at frequency scale www. A localized phase is described by a line of fixed points with θ(w)>0θ(w)>0θ(w)>0, while an instability of the flow signals resonance proliferation and the onset of thermalization. The predicted θ(w)θ(w)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)θ(w)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.
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Topology and many-body localization (MBL) have opened new avenues for preserving quantum information at finite energy density. Resonant delocalization plays a crucial role in destabilizing these phenomena. In this paper, we study the statistical properties of many-body resonances in a disordered interacting Ising model---which can host symmetry-protected topological order---using a Clifford circuit encoding of the real-space renormalization group which allows the resonant properties of the wave functions to be efficiently characterized. Our findings show that both the trivial and topologically ordered MBL phases remain stable in the presence of the resonances but, in the vicinity of the transition between them, localization is destabilized by resonance proliferation. Diverging susceptibility towards the development of an avalanche instability suggests an intervening ergodic phase. We are also able to access the local integrals of motion in the MBL phases and identify the topological edge-mode operators in the ordered phase. Our results have important implications for the stability of MBL and phase transitions between distinct MBL phases with and without symmetries.
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