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 $\epsilon$ and of the renormalized hoppings $V$ as a function of the linear size $L$. We find that the renormalized on-site energies $\epsilon$ remain finite in the localized phase in $d=2,3$ and at criticality ($d=3$), with a finite density at $\epsilon =0$ and a power-law decay $1/{\epsilon }^2$ at large $|\epsilon |$. For the renormalized hoppings in the localized phase, we find: $\ln V_L\simeq -\frac{L}{{\xi }_{loc}}+L^{\omega }u$, where ${\xi }_{loc}$ is the localization length and $u$ a random variable of order one. The exponent $\omega$ is the droplet exponent characterizing the strong disorder phase of the directed polymer in a random medium of dimension $1+(d-1)$, with $\omega (d=2)=1/3$ and $\omega (d=3)\simeq 0.24$. 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 ${\rho }_{typ}\simeq 1$ for the exponent governing the typical decay $\overline{\ln V_L}\simeq -{\rho }_{typ}\ln L$, in agreement with previous numerical measures of ${\alpha }_{typ}=d+{\rho }_{typ}\simeq 4$ for the singularity spectrum $f(\alpha )$ of individual eigenfunctions. We also present numerical results concerning critical surface properties.

16 pages, 17 figures

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