Statistics of renormalized on-site energies and renormalized hoppings for Anderson localization models in dimensions d=2 and d=3

Abstract

For Anderson localization models, there exists an exact real-space renormalization procedure at fixed energy which preserves the Green functions of the remaining sites [H. Aoki, J. Phys. C13, 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 $ε$ and of the renormalized hoppings $V$ as a function of the linear size $L$. We find that the renormalized on-site energies $ε$ remain finite in the localized phase in $d=2,3$ and at criticality ($d=3$), with a finite density at $ε=0$ and a power-law decay $1/ε^2$ at large $| ε|$. For the renormalized hoppings in the localized phase, we find: ${\rm ln} V_L \simeq -\frac{L}{ξ_{loc}}+L^ωu$, where $ξ_{loc}$ is the localization length and $u$ a random variable of order one. The exponent $ω$ 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 $ω(d=3) \simeq 0.24$. 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 $ρ_{typ}\simeq 1$ for the exponent governing the typical decay $\overline{{\rm ln} V_L} \simeq -ρ_{typ} {\rm ln}L$, in agreement with previous numerical measures of $α_{typ} =d+ρ_{typ} \simeq 4$ for the singularity spectrum $f(α)$ of individual eigenfunctions. We also present numerical results concerning critical surface properties.