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Anderson localization on tree-like graphs such as the Bethe lattice, Cayleytree, or random regular graphs has attracted attention due to its apparentmathematical tractability, hypothesized connections to many-body localization,and the possibility of non-ergodic extended regimes. This behavior has beenconjectured to also appear in many-body localization as a "bad metal" phase,and constitutes an intermediate possibility between the extremes of ergodicquantum chaos and integrable localization. Despite decades of research, acomplete consensus understanding of this model remains elusive. Here, we usecages, maximally tree-like structures from extremal graph theory; and numericalcontinuous unitary Wegner flows of the Anderson Hamiltonian to develop anintuitive picture which, after extrapolating to the infinite Bethe lattice,appears to capture ergodic, non-ergodic extended, and fully localized behavior.

Anderson Localization on the Bethe Lattice using Cages and the Wegner Flow