Electron in a tangled chain: Multifractality at the small-world critical point

We study a simple model of conducting polymers in which a single electron propagates through a randomly tangled chain. The model has the geometry of a small-world network, with a small density $p$ of crossings of the chain acting as shortcuts for the electron. We use numerical diagonalization and simple analytical arguments to discuss the density of states, inverse participation ratios, and wave functions. We suggest that there is a critical point at $p=0$ and demonstrate finite-size scaling of the energy and wave functions at the lower band edge. The wave functions are multifractal. The critical exponent of the correlation length is consistent with criticality due to the small-world effect, as distinct from the previously discussed, dimensionality-driven Anderson transition.

Figure 1

(Color online) Density of states (DOS), estimated from full-spectrum numerical diagonalization for (a) $R=32$ realizations, $L=1024$ sites, and different densities of shortcuts per site $p$; (b) $R=16$, $p=0.1$, and different values of $L$; and (c) different $R$ with $L=1024$ and $p=0.1$. All the plots have been obtained by drawing histograms using 100 uniformly spaced values of the energy from $ϵ=-3t$ to $ϵ=+3t$.

Figure 2

(Color online) Energy dependence of the IPR (left-hand axis) and the exponent ${\alpha }_1$ characterizing its power-law dependence on system size $L$ (right-hand axis). The data have been obtained by exact numerical diagonalization for $R=16$ realizations with $p=0.1$ shortcuts/site. The IPR vs energy curve corresponds to $L=1024$ sites. It has been obtained by splitting the energy interval from $ϵ=-3t$ to $ϵ=+3t$ into 20 subintervals and averaging the IPR over each subinterval. The 16 384 individual values of the IPR in our ensemble are also shown. Similar curves have been obtained for different values of $L$, and fits of the averaged IPR to a power law in each energy interval have been used to determine ${\alpha }_1$. Two examples of such fits are shown in the inset.

Figure 3

Numerically determined single-particle wave functions for a particular realization of our tight-binding model, with $L=460$ sites. Each wave function has been represented on a circle, whose radius is its energy, measured from $-3t$. The innermost wave function in (a) is the ground state. The three excited states with energy below $-2t$ have been plotted separately for clarity: (b), (c), and (d), in order of increasing energy. The other wave functions in (a) have energies in $(-2t,0)$. Finally, the straight lines in the innermost circles indicate which sites are joined by “shortcuts.”

Figure 4

Dependence on the system size $L$ of (a) the energy and (b) the IPR $(n=1)$ and next two higher moments of the density distribution $(n=2,3)$ at the lower band edge obtained by Lanczos diagonalization and averaged over 2000 random realizations. The density of shortcuts is $p=5\times {10}^{-4}$. The dotted lines indicate (a) $⟨{ϵ}_0⟩=-2t$ and $-\sqrt{5}t$ and (b) the analytical results for $p=0$.

Figure 5

Finite-size scaling of the ground-state energy, obtained as in Fig. 4. The main plot shows the collapse of the data for $p⩽{10}^{-4}$ on the curve given by Eq. (8) for values of the scaling variable from $L∕\xi =0.001$ to $L∕\xi =20$. The two dotted lines indicate ${ϵ}_0=-2t$ and ${ϵ}_0=-\sqrt{5}t$. The inset shows data for two higher values of $p$ for comparison.

Figure 6

Finite-size scaling of the IPR $(n=1)$ and next two higher moments of the local-density distribution $(n=2,3)$ at the lower band edge, obtained as in Fig. 4. (a) shows the collapse of the data for $p⩽{10}^{-4}$ on the curve given by Eq. (10) for values of the scaling variable from $L∕\xi =0.1$ to $L∕\xi =5$. The solid lines correspond to $\alpha =0.48$, 0.40, and 0.33 and fit the $n=1$, 2, and 3 data, respectively. The dotted lines correspond to $\alpha =0$ and $\alpha =1$. In (b), data for two higher values of $p$ over a wider range of values of $L∕\xi$ are displayed for comparison.