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In the field of ergodicity-breaking phases, it has been recognized thatquantum avalanches can destabilize many-body localization at a wide range ofdisorder strengths. This has in particular been demonstrated by the numericalstudy of a toy model, sometimes simply called the ''avalanche model'' or the''quantum sun model'' [Phys. Rev. Lett. 129, 060602 (2022)], which consists ofan ergodic seed coupled to a perfectly localized material. In this paper, weconnect this toy model to a well-studied model in random matrix theory, theultrametric ensemble. We conjecture that the models share the followingfeatures. 1) The location of the critical point is predicted sharply byanalytics. 2) On the localized site, both models exhibit Fock spacelocalization. 3) There is a manifold of critical points. On the criticalmanifold, the eigenvectors exhibit nontrivial multifractal behaviour that canbe tuned by moving on the manifold. 4) The spectral statistics is intermediatebetween Poisson statistics and random matrix statistics, also tunable on thecritical manifold. We confirm numerically these properties.

Similarity between a many-body quantum avalanche model and the ultrametric random matrix model