Large-disorder renormalization group study of the Anderson model of localization

We describe a large-disorder renormalization group (LDRG) method for the Anderson model of localization in one dimension which decimates eigenstates based on the size of their wave functions rather than their energy. We show that our LDRG scheme flows to infinite disorder, and thus becomes asymptotically exact. We use it to obtain the disorder-averaged inverse participation ratio (IPR) and density of states (DOS) for the entire spectrum. A modified scheme is formulated for higher dimensions, which is found to be less efficient, but capable of improvement.

Figure 1

The fraction of the total system size, $f_{\mathrm{cl}}$, that is decimated as a function of cluster size $N_{\mathrm{cl}}$ during the LDRG flow for the one-dimensional Anderson model with initial system size $L={10}^5$. LDRG data are averaged over 100 runs.

Figure 2

(a) IPR (in blue, left $y$ axis) and DOS (in red, right $y$ axis) for the Anderson model in one dimension with $x=2.5$ $(w=10)$ from exact diagonalization (solid lines) and from LDRG with initial system size $L={10}^5$, using different values of the cutoff $m_0$. (b) The error in the IPR (as defined in the text) obtained using LDRG with initial system size $L={10}^5$ as a function of disorder, $x=w/4$, for the one-dimensional Anderson model. Lines are a guide to the eye. LDRG data are averaged over 100 runs. A similar measure can be defined for the DOS and lies below $0.1\%$ for the values of $w$ shown here.

Figure 3

(a) Evolution of the distribution of the logarithms of bond values $m_{ij}$ as the LDRG progresses for the one-dimensional Anderson model with $x=2.5$ $(w=10)$ and $m_0=0.2$. Initial system size is $L={10}^5$. $N_{\mathrm{cl}}$ is the size of the clusters decimated just before the distribution was measured. $N_{\mathrm{cl}}=0$ is thus the initial distribution. LDRG data are averaged over 100 runs. The vertical line marks ${\Gamma }_0$. The straight lines are fits to $ln\left[R\right(\Gamma \left)\right]=b\Gamma +ln\left(a\right)$. (b) The inset shows that $\lambda =\frac{a}{b}{e}^{{\Gamma }_{0}b}$ is approximately constant. (c) The evolution of the fitting parameter $a$ as a function of cluster size $N_{\mathrm{cl}}$ for $w=10,14,18,22$ and cutoff $m_0=0.2$.

Figure 4

IPR for Anderson model in two dimensions with $x=6.25$ $(w=50)$ from exact diagonalization (solid line) and from LDRG with different values of the cutoff $m_0$. LDRG data are averaged over 100 runs of systems with $100\times 100$ sites.