Motivated by the analogy between many-body localization (MBL) and single-particle Anderson localization on hierarchical graphs, we study localization on the Husimi tree, a generalization of the Bethe lattice with a finite density of local loops of arbitrary but finite length. The exact solution of the model provides a transparent and quantitative framework to systematically inspect the effect of loops on localization. Our analysis indicates that local loops enhance resonant processes, thereby reducing the critical disorder with increasing their number and size. At the same time, loops promote local hybridization, leading to an increase in the spatial extent of localized eigenstates. These effects reconcile key discrepancies between MBL phenomenology and its single-particle Anderson analog on hierarchical graphs. These results show that local loops are a crucial structural ingredient for realistic single-particle representations of many-body Hilbert spaces.

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• Anderson localization
• Localization
• Many-body localization
• Phase transitions
• Disordered systems
• Random graphs
• Cavity methods