How the site degree influences quantum probability on inhomogeneous substrates

We investigate the effect of the node degree and energy $E$ on the electronic wave function for regular and irregular structures, namely, regular lattices, disordered percolation clusters, and complex networks. We evaluate the dependency of the quantum probability for each site on its degree. For a class of biregular structures formed by two disjoint subsets of sites sharing the same degree, the probability $P_k\left(E\right)$ of finding the electron on any site with $k$ neighbors is independent of $E\ne 0$, a consequence of an exact analytical result that we prove for any bipartite lattice. For more general nonbipartite structures, $P_k\left(E\right)$ may depend on $E$ as illustrated by an exact evaluation of a one-dimensional semiregular lattice: $P_k\left(E\right)$ is large for small values of $E$ when $k$ is also small, and its maximum values shift towards large values of $\left|E\right|$ with increasing $k$. Numerical evaluations of $P_k\left(E\right)$ for two different types of percolation clusters and the Apollonian network suggest that this observed feature might be generally valid.

Figure 1

Examples of biregular (a–c) and semiregular (d) decorated linear chains.

Figure 2

Probability of finding the particle on the set of sites with different degree according to Eqs. (A6) ($k=2$, squares) and (7) ($k=4$ circles) as a function of the wave function energy.

Figure 3

Probability of finding the particle on the set of sites with different degree as a function of the wave function energy for the Apollonian network with $g=8$. With exception of panel (h), for which $k=257$, panels (a)–(g) and (i) correspond, respectively, to degrees $k=3\times 2^m$, with $m=0,1,...,7$.

Figure 4

Probability of finding the particle on sites with different degree as a function of the wave function energy for percolation clusters obtained by the Gaussian (a, c) and usual models (b, d) when $p=p_{c,G}=0.56244$. The used samples correspond to $L=32$ (a, b) and $L=64$ (c, d). Black squares, red (dark gray) circles, dark yellow (gray) triangles, and cyan (light gray) diamonds indicate data points for node degree $k=1,2,3,$ and 4, respectively.

Figure 5

Probability of finding the particle on sites with different degree as a function of the wave function energy for percolation clusters obtained by the Gaussian (a, c) and usual models (b, d) when $p=p_{c,G}=0.8$. The used samples correspond to $L=32$ (a, b) and $L=64$ (c, d). Black squares, red (dark gray) circles, dark yellow (gray) triangles, and cyan (light gray) diamonds indicate data points for node degree $k=1,2,3,$ and 4, respectively.

Figure 6

Dependency of $\mathrm{\Delta }{P}_{G,U}\left(k\right)=P_{k,G}\left(E\right)-P_{k,U}\left(E\right)$ as a function of $E$ for decreasing probabilities $p=0.56244,0.6,0.7,0.8,$ and 0.9, when $L=64$. Since $\mathrm{\Delta }{P}_{G,U}\left(k\right)$ decreases as $p$ increases, the vertical axis is conveniently rescaled in each panel. For different $k$, the relative positions of $\mathrm{\Delta }{P}_{G,U}\left(k\right)$ also change with $p$. Black squares, red (dark gray) circles, dark yellow (gray) triangles, and cyan (light gray) diamonds indicate data points for node degree $k=1,2,3,$ and 4, respectively.

Figure 7

Distribution of site occupancy probability ${\mathrm{\Pi }}_k({\rho }_E\left(\vec{r}\right))$ as a function of ${log}_{10}\left[{\rho }_E\left(\vec{r}\right)\right]$ for systems with $L=64$. Points in panels (a)–(d) correspond, respectively, to number of sites with $k=1–4$ in histograms with bin width 1, for $p=0.8$. Black squares, red (dark gray) circles, and cyan (light gray) diamonds indicate data points for $E=-0.01,-1.50,$ and $-3.65$, respectively.

Figure 8

Distribution of site occupancy probability ${\mathrm{\Pi }}_k\left[{\rho }_E\left(\vec{r}\right)\right]$ as a function of ${log}_{10}\left[{\rho }_E\left(\vec{r}\right)\right]$ for systems with $L=64$. Points in panels (a)–(d) correspond, respectively, to number of sites with $k=1–4$ in histograms with bin width 1, for $E\simeq -1.5$. Black squares, red (dark gray) circles, and cyan (light gray) diamonds indicate data points for $p=0.9,0.8,$ and $p_c$, respectively.

Figure 9

Color plots of ${log}_{10}\left[{\rho }_E\left(\vec{r}\right)\right]$ as a function of position for a percolation cluster at $p=p_c$, for $E=-0.2,-1.5,$ and $-3.065$, in panels (a)–(c). White region corresponds to sites with $k=0$. Sites with $k=1–4$ are indicated, respectively, by circles, vertical bars, horizontal bars, and open squares. Sites with ${log}_{10}\left[{\rho }_E\left(\vec{r}\right)\right]<-6$ remain white, otherwise they are shaded according to a color (gray tone) code for the following intervals: (i) yellow (very light gray), $[-6,-4.7)$; (ii) cyan (light gray), $[-4.7,-3.3)$; (iii) green (gray), $[-3.3,-2.0)$; and (iv) pink (dark gray), $[-2.0,-0.7]$. Only the state close to the band edge ($E=-3.065$) displays strong localization, with peaks centered on $k=4$ sites. The less localized states with $E=-0.2$ and $-1.5$ are spread over sites with different values of $k$.