Critical properties of the Anderson localization transition and the high-dimensional limit

In this paper we present a thorough study of transport, spectral, and wave-function properties at the Anderson localization critical point in spatial dimensions $d=3$, 4, 5, 6. Our aim is to analyze the dimensional dependence and to assess the role of the $d\to \infty$ limit provided by Bethe lattices and treelike structures. Our results strongly suggest that the upper critical dimension of Anderson localization is infinite. Furthermore, we find that $d_U=\infty$ is a much better starting point compared to $d_L=2$ to describe even three-dimensional systems. We find that critical properties and finite-size scaling behavior approach by increasing $d$ those found for Bethe lattices: the critical state becomes an insulator characterized by Poisson statistics and corrections to the thermodynamics limit become logarithmic in the number $N$ of lattice sites. In the conclusion, we present physical consequences of our results, propose connections with the nonergodic delocalized phase suggested for the Anderson model on infinite-dimensional lattices, and discuss perspectives for future research studies.

Figure 1

Sketch of the quasi-one-dimensional bar along the $x$ direction of cross section $L^{(d-1)}$.

Figure 2

$\overline{ln\text{Im}G\left(x\right)}$ as a function of $x$ in 6 dimensions, for $L=6$ and for several values of the disorder, showing that ${\xi }_{1d}$ can be measured from Eq. (6) by linear fitting of the data at large enough $x$.

Figure 3

Left panel: ${\lambda }_{1d}$ as a function of the disorder $W$ in $4d$ for several system sizes $L$ from 2 to 18. The vertical dashed line spots the position of the critical point, $W_c\simeq 34.5$. Top-right panel: Finite-size scaling of the same data for $L$ from 10 to 18, showing data collapse for $\nu \simeq 1.11$. Bottom-right panel: $\psi f_1=({\lambda }_{1d}-f_{\infty })/L^y$ as a function of the scaling variable $(W-W_c)L^{1/\nu }$ for different sizes $L$ from 2 to 7, showing data collapse for the same value as before of $W_c$ and $\nu$ and for $y\simeq -1$.

Figure 4

Left panel: ${\lambda }_{1d}$ as a function of the disorder $W$ in $5d$ for several system sizes $L$ from 2 to 9. The vertical dashed line spots the position of the critical point, $W_c\simeq 57.5$. Top-right panel: Finite-size scaling of the same data for $L$ from 6 to 9, showing data collapse for $\nu \simeq 0.96$. Bottom-right panel: $\psi f_1=({\lambda }_{1d}-f_{\infty })/L^y$ as a function of the scaling variable $(W-W_c)L^{1/\nu }$ for different sizes $L$ from 2 to 6, showing data collapse for the same value as before of $W_c$ and $\nu$ and for $y\simeq -1.2$.

Figure 5

Left panel: ${\lambda }_{1d}$ as a function of the disorder $W$ in $6d$ for several system sizes $L$ from 2 to 6. The vertical dashed line spots the position of the critical point, $W_c\simeq 83.5$. Top-right panel: Finite-size scaling of the same data for $L$ equal to 4, 5, and 6, showing data collapse for $\nu \simeq 0.84$. Bottom-right panel: $\psi f_1=({\lambda }_{1d}-f_{\infty })/L^y$ as a function of the scaling variable $(W-W_c)L^{1/\nu }$ for different sizes $L$ from 2 to 5, showing data collapse for the same value as before of $W_c$ and $\nu$ and for $y\simeq -1.4$.

Figure 6

$\overline{r}$ (top left) and $q^{\mathrm{typ}}$ (bottom left) as a function of the disorder $W$ in $3d$ for several system sizes $L$ from 4 to 30. The horizontal dashed lines correspond to the reference GOE and Poisson asymptotic values. The vertical dashed line spots the position of the AL transition, $W_c\simeq 16.35$. Finite-size scaling of the same data (top-right and bottom-right panels) showing data collapse obtained for $\nu \simeq 1.57$. Finite-size corrections to Eq. (12) are observed at small sizes (open symbols), and can be described by Eq. (10) with $y\simeq -1$.

Figure 7

$\overline{r}$ (top left) and $q^{\mathrm{typ}}$ (bottom left) as a function of the disorder $W$ in $4d$ for several system sizes $L$ from 3 to 13. The horizontal dashed lines correspond to the reference GOE and Poisson asymptotic values. The vertical dashed line spots the position of the AL transition, $W_c\simeq 34.5$. Finite-size scaling of the same data (top-right and bottom-right panels) showing data collapse obtained for $\nu \simeq 1.11$. Finite-size corrections to Eq. (12) are observed at small sizes (open symbols), and can be described by Eq. (10) with $y\simeq -1$.

Figure 8

$\overline{r}$ (top left) and $q^{\mathrm{typ}}$ (bottom left) as a function of the disorder $W$ in $5d$ for several system sizes $L$ from 3 to 8. The horizontal dashed lines correspond to the reference GOE and Poisson asymptotic values. The vertical dashed line spots the position of the AL transition, $W_c\simeq 57.5$. Finite-size scaling of the same data (top-right and bottom-right panels) showing data collapse obtained for $\nu \simeq 0.96$. Finite-size corrections to Eq. (12) are observed at small sizes (open symbols), and can be described by Eq. (10) with $y\simeq -1.2$.

Figure 9

Left panel: $\overline{r}$ as a function of the disorder $W$ for several system sizes $L$ from 2 to 6 in $6d$. The horizontal dashed lines correspond to the reference GOE and Poisson asymptotic values. The vertical dashed line spots the position of the AL transition, $W_c\simeq 83.5$. Top-right panel: Finite-size scaling of the same data for the largest system sizes only, $L=4$, 5, and 6, showing data collapse for $\nu \simeq 0.84$. Bottom-right panel: $\psi g_1=(\overline{r}-g_{\infty })/L^y$ as a function of the scaling variable $(W-W_c)L^{1/\nu }$ for different sizes $L$ from 2 to 5, showing a reasonably good data collapse for the same value as before of $W_c$ and $\nu$, and for $y\simeq -1.4$.

Figure 10

Left panel: Flowing fractal exponent $\beta$ describing the scaling of the typical value of the IPR with the system size for $d=4$. The vertical dashed black line corresponds to the critical disorder $W_c\simeq 34.5$. Right panel: Finite-size scaling of the same data showing a reasonably good data collapse obtained for $\nu \simeq 1.11$. Strong finite-size corrections to the one-parameter scaling are observed at small sizes (open symbols), and can be described by Eq. (10) with $y\simeq -1$.

Figure 11

Sketch of the SDRG decimation procedure for a site (top), and a bond (bottom) transformation. Dotted blue lines represent preexisting hopping amplitudes before decimation. Solid blue lines represent new or renormalized bonds. The on-site energies of all the neighbors of the decimated sites (blue circles) are renormalized as well.

Figure 12

Quasi-$1d$ dimensionless localization length, ${\lambda }_{1d}$, obtained using the SDRG procedure for different values of $k_{\max }$, at the AL critical point, $W_c\simeq 83.5$, in 6 dimensions, and for $L=3$ (blue circles) and $L=6$ (red squares). The horizontal blue (resp. red) solid and dashed lines correspond to the average value of ${\lambda }_{1d}$ and its fluctuations computed using the TM method for $L=3$ (resp. $L=6$), showing that for $k_{\max }\gtrsim 240$ the approximate SDRG results converge, within our numerical accuracy, to the exact values.

Figure 13

Bottom-left and bottom-right panels: The stationary distributions $R_{{\tau }_{☆}}\left(t\right)$ and $Q_{{\tau }_{☆}}\left(\epsilon \right)$ at the AL critical points in dimensions from 3 to 6. The system size is $L=33$ in $3d, L=14$ in $4d, L=8$ in $5d$, and $L=6$ in $6d$, in such a way that the total number of sites is approximately the same, $N\sim 4\times {10}^4$, in all dimensions. The stationary state is reached for a RG time ${\tau }_{☆}$ such that the number of sites left in the system are approximately 1/8 of the initial ones in $3d$, 1/16 in $4d$, 1/26 in $5d$, and 1/40 in $6d$. The value of $k_{\max }$ is set to 360 in all dimensions. Top panel: The same data as in the bottom-left panel plotted in a log-log scale, showing the power law behavior of $R_{{\tau }_{☆}}\left(t\right)\sim t^{-\gamma }$ with $\gamma \simeq 2$ (black dashed line), for $t$ smaller than a cutoff of $O\left(1\right)$.

Figure 14

Numerical values of the inverse of the critical exponent $\nu$ as a function of $1/d$ in dimensions from 3 to 6 (blue circles), showing a smooth behavior interpolating from $\nu \to \infty$ in $d=2$ to $\nu =1/2$ in $d\to \infty$ [46]. The turquoise dashed line shows the predictions of the self-consistent theory of [27], with $d_U=4$. The dashed-dotted magenta line corresponds to the lower bound $\nu \ge 2/d$ provided by the Harris criterion [59]. The red solid line shows the dimensional dependence of $\nu$ obtained from a perturbative analysis of the $\mathrm{NL}\sigma \mathrm{M}$ to five-loop in $\epsilon =d-2$ [10], Eq. (20). The straight dotted blue line is a linear fit corresponding to the first correction in $1/d$ from which we find $1/\nu \simeq 2-4.75/d$. The prediction of the semiclassical approach of Ref. [25], ${\nu }^{-1}=2-4/d$, corresponds to the straight line interpolating from ${\nu }^{-1}=2$ in $d\to \infty$ to ${\nu }^{-1}=1$ in $d=2$.

Figure 15

Dimensional dependence of ${\overline{r}}_c$ (top-left panel), $q_c^{\mathrm{typ}}$ (bottom-left panel), ${\left({\lambda }_{1d}\right)}_c$ (top-right panel), and ${\beta }_c$ at the AL critical point as a function of $1/d$. The dashed horizontal red (resp. black) lines correspond to the reference GOE (resp. Poisson values).

Figure 16

Dimensional dependence of the critical value of the disorder strength, $W_c$, divided by $8dln\left(2d\right)$, which corresponds to the exact asymptotic behavior on treelike lattices in the large-connectivity limit [60]. The straight dotted blue line is a fit that takes into account corrections in $1/d$, from which we find $W_c/8dln\left(2d\right)\simeq 1-1.81/d$.

Figure 17

Rescaled singularity spectrum $f\left(\alpha \right)/d$ as a function of $\alpha /d$ at the AL critical point in dimensions from 3 to 5. The dashed black straight line $f\left(\alpha \right)=\alpha /2$ for $\alpha \in [0,2]$ corresponds to the prediction of [61] in the $d\to \infty$ limit.