Anderson localization on the Bethe lattice: non-ergodicity of extended states

Statistical analysis of the eigenfunctions of the Anderson tight-binding model with on-site disorder on regular random graphs strongly suggests that the extended states are multifractal at any finite disorder. The spectrum of fractal dimensions $f(\alpha)$ defined in Eq.(3), remains positive for $\alpha$ noticeably far from 1 even when the disorder is several times weaker than the one which leads to the Anderson localization, i.e. the ergodicity can be reached only in the absence of disorder. The one-particle multifractality on the Bethe lattice signals on a possible inapplicability of the equipartition law to a generic many-body quantum system as long as it remains isolated.