A critical Dyson hierarchical model for the Anderson localization transition

Abstract

A Dyson hierarchical model for Anderson localization, containing non-random hierarchicalhoppings and random on-site energies, has been studied in the mathematicalliterature since its introduction by Bovier (1990 J. Stat. Phys. 59 745), with theconclusion that this model is always in the localized phase. Here we show that if oneintroduces alternating signs in the hoppings along the hierarchy (instead of choosingall hoppings of the same sign), it is possible to reach an Anderson localizationcritical point presenting multifractal eigenfunctions and intermediate spectralstatistics. The advantage of this model is that one can write exact renormalizationequations for some observables. In particular, we obtain that the renormalized on-siteenergies have the Cauchy distributions for exact fixed points. Another output of thisrenormalization analysis is that the typical exponent of critical eigenfunctions is alwaysαtyp = 2, independentlyof the disorder strength. We present numerical results concerning the whole multifractal spectrumf(α) and thecompressibility χ of the level statistics, for both the box and the Cauchy distributions of the random on-siteenergies. We discuss the similarities and differences with the ensemble of ultrametricrandom matrices introduced recently by Fyodorov, Ossipov and Rodriguez (2009 J. Stat.Mech. L12001).