A critical Dyson hierarchical model for the Anderson localization transition

A Dyson hierarchical model for Anderson localization, containing non-random hierarchical hoppings and random on-site energies, has been studied in the mathematical literature since its introduction by Bovier [J. Stat. Phys. 59, 745 (1990)], with the conclusion that this model is always in the localized phase. Here we show that if one introduces alternating signs in the hoppings along the hierarchy (instead of choosing all hoppings of the same sign), it is possible to reach an Anderson localization critical point presenting multifractal eigenfunctions and intermediate spectral statistics. The advantage of this model is that one can write exact renormalization equations for some observables. In particular, we obtain that the renormalized on-site energies have the Cauchy distributions for exact fixed points. Another output of this renormalization analysis is that the typical exponent of critical eigenfunctions is always $\alpha_{typ}=2$, independently of the disorder strength. We present numerical results concerning the whole multifractal spectrum $f(\alpha)$ and the compressibility $\chi$ of the level statistics, both for the box and the Cauchy distributions of the random on-site energies. We discuss the similarities and differences with the ensemble of ultrametric random matrices introduced recently by Fyodorov, Ossipov and Rodriguez [J. Stat. Mech. L12001 (2009)].