Statistics of renormalized on-site energies and renormalized hoppings for Anderson localization in two and three dimensions

摘要

For Anderson localization models, there exists an exact real-space renormalization procedure at fixed energy which preserves the Green's functions of the remaining sites [H. Aoki, J. Phys. C 13, 3369 (1980)]. Using this procedure for the Anderson tight-binding model in dimensions d=2,3, we study numerically the statistical properties of the renormalized on-site energies e and of the renormalized hoppings V as a function of the linear size L. We find that the renormalized on-site energies e remain finite in the localized phase in d=2,3 and at criticality (d=3), with a finite density at ∈=0 and a power-law decay 1/∈~2 at large |∈|. For the renormalized hoppings in the localized phase, we find: In V_L approx= —ξ_(loc)/L+L~ωu, where ξ/oc is the localization length and u a random variable of order one. The exponent w is the droplet exponent characterizing the strong disorder phase of the directed polymer in a random medium of dimension 1 +(d-1), with ω(d=2) = 1/3 and ω(d=3) approx= 0.24. At criticality (d=3), the statistics of renormalized hoppings V is multifractal, in direct correspondence with the multifractality of individual eigenstates and of two-point transmissions. In particular, we measure ρ_(typ) - 1 for the exponent governing the typical decay In V_L approx= -ρ_(typ) In L, in agreement with previous numerical measures of α_(typ) = d+ρ_(typ)approx=4 for the singularity spectrum f(α) of individual eigenfunctions. We also present numerical results concerning critical surface properties.