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

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 $ϵ$ 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 $\left(d=3\right)$, with a finite density at $ϵ=0$ and a power-law decay $1/{ϵ}^2$ at large $\left|ϵ\right|$. For the renormalized hoppings in the localized phase, we find: $\text{ln}{V}_L\simeq -\frac{L}{{\xi }_{loc}}+L^{\omega }u$, 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 $1+\left(d-1\right)$, with $\omega \left(d=2\right)=1/3$ and $\omega \left(d=3\right)\simeq 0.24$. At criticality $\left(d=3\right)$, 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 $\overline{\text{ln}{V}_L}\simeq -{\rho }_{typ}\text{ln}L$, in agreement with previous numerical measures of ${\alpha }_{typ}=d+{\rho }_{typ}\simeq 4$ for the singularity spectrum $f\left(\alpha \right)$ of individual eigenfunctions. We also present numerical results concerning critical surface properties.

Figure 1

(Color online) Renormalization procedure in dimension $d=2$. The initial state is the tight-binding Anderson model on a square lattice of size $L^2$, with periodic boundary conditions in the two directions. Sites are then iteratively eliminated using the RG rules of Eqs. (9, 10) until there remains only the four sites corresponding to the large disks, i.e., there are four renormalized on-site energies and four renormalized hoppings at distance $L/2$ per sample.

Figure 2

(Color online) Renormalization procedure in dimension $d=3$. The initial state is the tight-binding Anderson model on a cubic lattice of size $L^3$, with periodic boundary conditions in the three directions (here for clarity, the sites of the initial model have not be drawn in contrast to Fig. 1 concerning the case $d=2$ which is more explicit). Sites are then iteratively eliminated using the RG rules of Eqs. (9, 10) until there remains only the eight sites corresponding to the large disks, i.e., there are eight renormalized on-site energies and twelve renormalized hoppings at distance $L/2$ per sample.

Figure 3

(Color online) Renormalization procedure for the PRBM model with the ring geometry. Sites are iteratively eliminated using the RG rules of Eqs. (9, 10) until there remains only the two sites $L/2$ and $L$ corresponding to the large disks, i.e., there are two renormalized on-site energies and one renormalized hoppings per sample.

Figure 4

(Color online) Statistics of renormalized on-site energies $ϵ$ for the Anderson model on the square lattice in dimension $d=2$ for sizes $12\le L\le 120$ (disorder strength $W=40$): the histograms of $\text{ln}\left|ϵ\right|$ are identical (apart for the cutoff imposed by the statistics over the samples) The left and right slopes of value unity corresponds to Eqs. (41, 43).

Figure 5

(Color online) Statistics of renormalized on-site energies $ϵ$ for the Anderson model on the cubic lattice in dimension $d=3$ for sizes $4\le L\le 20$ (a) Histograms of $\text{ln}\left|ϵ\right|$ in the localized phase $\left(W=40\right)$ (b) Histograms of $\text{ln}\left|ϵ\right|$ at criticality $\left(W_c=16.5\right)$

Figure 6

(Color online) Statistics of renormalized on-site energies $ϵ$ in the PRBM model of parameter $b=0.1$ (a) The histograms of $\text{ln}\left|ϵ\right|$ at criticality $a=1$ for various sizes $L=100,200,400,600,800$ are identical (apart for the cutoff imposed by the statistics over the samples). (b) Comparison of the stationary distributions in the localized phase $\left(a=1.4\right)$, at criticality $\left(a=1\right)$ and in the delocalized phase $\left(a=0.6\right)$. The left and right slopes of value unity correspond to Eqs. (41, 43).

Figure 7

Statistics of the typical renormalized hopping in $d=2$ where only the localized phase exists (the data shown correspond to the disorder strength $W=40$) (a) Typical exponential decay: $\overline{\text{ln}{V}_L}$ is linear in $L$ and the slope represents the inverse of the localization length ${\xi }_{loc}$ [Eq. (48)]. (b) The fluctuation term $\Delta \left(\text{ln}{V}_L\right)$ grows as $L^{\omega }$ [Eq. (48)] with $\omega \left(d=2\right)\simeq 0.33$.

Figure 8

(Color online) Histograms of the logarithm of the renormalized hopping $\text{ln}{V}_L$ for various lengths $L$ in the localized phase (for the disorder strength $W=40$ (a) in dimension $d=2$ (b) in dimension $d=3$

Figure 9

Statistics of the typical renormalized hopping in $d=3$ in the localized phase $\left(W=40\right)$ (a) $\overline{\text{ln}{V}_L}$ as a function of $L$: the slope represents the inverse of the localization length ${\xi }_{loc}$ [Eq. (48)] (b) The fluctuation term $\Delta \left(\text{ln}{V}_L\right)$ grows as $L^{\omega }$ [Eq. (48)] with $\omega \left(d=3\right)\simeq 0.24$.

Figure 10

Statistics of the typical renormalized hopping in $d=3$ at criticality $\left(W=16.5\right)$ (a) $\overline{\text{ln}{V}_L}$ as a function of $\text{ln}L$ for bulk sites (see Fig. 2): the measured slope ${\rho }_{typ}\simeq 1.05$ represents the typical exponent of Eq. (56). (b) $\overline{\text{ln}{V}_L^{surf}}$ as a function of $\text{ln}L$ for surface sites (see Fig. 11): the measured slope ${\rho }_{typ}^{surf}\simeq 1.6$ represents the typical exponent of Eq. (60).

Figure 11

(Color online) Renormalization procedure in dimension $d=3$ to measure the renormalized hopping $V_L^{surf}$ between two boundary sites. The initial state is the tight-binding Anderson model on a cubic lattice of size $L^3$, with periodic boundary conditions in two directions $x$ and $y$ and free boundary conditions in the third direction $z$. Sites are then iteratively eliminated using the RG rules of Eqs. (9, 10) until there remains only the two surface sites corresponding to the large disks.