Strong-disorder approach for the Anderson localization transition

We propose a strong-disorder renormalization-group approach to study the Anderson localization transition in disordered tight-binding models in any dimension. Our approach shifts the focus from the lower to the upper critical dimension, thus emphasizing the strong-coupling/strong-disorder nature of the transition. By studying the two-point conductance, we (i) show that our approach is in excellent agreement with exact numerical results, (ii) confirm that the upper critical dimension for the Anderson transition is $d_c^{+}=\infty$, (iii) find that the scaling function shows a previously reported ‘mirror symmetry’ in the critical region, and (iv) demonstrate that the range of conductances for which this symmetry holds increases with the system dimensionality. Our results open an efficient avenue to explore the critical properties of the Anderson transition using the strong-coupling high-dimension limit as a starting point.

Figure 1

(a) The typical two-point conductance $g$ of the ER random graph with $〈k〉=3.0$ and $N=3^{L_{\mathrm{ER}}}, L_{\mathrm{ER}}=4,\cdots ,7$ for several disorder parameters $W$ using the exact SDRG procedure (colorful solid lines) and our modified algorithm using $k_{\max }=20$ (colorful symbols). We average over as many disorder realizations as needed to reach $5\%$ of precision. Inset: the weighted conductance $g_w$ as a function of $L_{\mathrm{ER}}$. The legend corresponds to the inset, not the main panel. (b) The inverse localization length ${\xi }^{\prime }$ (Lyapunov exponent) near the localization transition.

Figure 2

Same as in Fig. 1 for a 3D cubic lattice of sizes $L=8,10,12,15$, and 20 with periodic boundary conditions. The inset of panel (b) shows the CPU time (in arbitrary units) for systems of $N=L^3$ sites in the critical regime $W=17$ when using the methods of exact diagonalization, exact SDRG, and modified SDRG.

Figure 3

The inverse of the critical exponent $\nu$ as a function of $1/d$. For comparison, we show the numerical results by Ueoka and Slevin [5] (red triangles) and García-García and Cuevas [37] (blue diamonds). Error bars are about the size of the symbols. Analytical predictions of other well-known theories are also shown in order to stress their limited range of reliability (see main text).

Figure 4

The inverse of the mirror symmetry range $G$ as a function of $1/d$. The dashed line is the best fit for $G^{-1}=aexp(b/d)$ where $a=2.1\times {10}^{-6}$ and $b=31.86$.