Anderson Localization on Husimi Trees and its implications for Many-Body localization

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. These results show that local loops are a crucial structural ingredient for realistic single-particle analogies to many-body Hilbert spaces.

💡 Research Summary

This paper investigates Anderson localization (AL) on the Husimi tree, a hierarchical graph that extends the Bethe lattice by embedding a finite density of local loops of arbitrary but bounded size. The motivation stems from the long‑standing analogy between many‑body localization (MBL) and single‑particle AL on high‑dimensional configuration‑space graphs. While the Bethe lattice captures the tree‑like nature of the Hilbert‑space graph, it lacks two essential features of realistic many‑body problems: (i) a linear growth of local connectivity with the number of degrees of freedom, and (ii) the presence of loops on all scales, which generate correlated return paths and affect disorder statistics. These omissions have led to quantitative discrepancies, such as an over‑estimated critical disorder (W_c) for AL on the tree ((W_c\sim z\ln z)) compared with the effective disorder in MBL ((W_c\sim\sqrt{z})), and a mismatch in the scaling of the inverse participation ratio (IPR).

To bridge this gap, the authors study the Anderson model on a Husimi tree defined by two parameters: the number of cliques attached to each site, (k+1), and the size of each clique, (p+1). Each site thus belongs to ((k+1)) cliques, each of which is a fully connected subgraph of (p+1) vertices. The resulting degree is (z=(k+1)p). When (p=1) the structure reduces to the ordinary Bethe lattice; larger (p) introduces loops of length 3 up to (p+1). The average loop length scales as (\langle\ell\rangle\sim p), providing a controlled knob to vary both the number and size of loops while keeping the overall degree fixed.

The Hamiltonian is the standard tight‑binding model with nearest‑neighbor hopping (t) and on‑site disorder (\epsilon_i) drawn uniformly from (