Localization properties of a tight-binding electronic model on the Apollonian network

An investigation on the properties of electronic states of a tight-binding Hamiltonian on the Apollonian network is presented. This structure, which is defined based on the Apollonian packing problem, has been explored both as a complex network and as a substrate on the top of which physical models can be defined. The Schrödinger equation of the model, which includes only nearest-neighbor interactions, is written in a matrix formulation. In the uniform case, the resulting Hamiltonian is proportional to the adjacency matrix of the Apollonian network. The characterization of the electronic eigenstates is based on the properties of the spectrum, which is characterized by a very large degeneracy. The $2\pi /3$ rotation symmetry of the network and large number of equivalent sites are reflected in all eigenstates, which are classified according to their parity. Extended and localized states are identified by evaluating the participation rate. Results for other two nonuniform models on the Apollonian network are also presented. In one case, interaction is considered to be dependent of the node degree, while in the other one random on-site energies are considered.

Figure 1

Generations $n=0$, 1, and 2 of AN. The adopted site numbering is the one introduced in Ref. 7.

Figure 2

(Color online) Illustration of parity properties for three eigenstates at generation $n=2$. Green (light gray) circles indicate AN nodes, while red (dark gray) stars indicate corresponding wave-function amplitude. (a) Even state corresponding to the largest value $E$ in class $C_1^2$. (b) and (c) Odd states corresponding to a twofold degenerate eigenvalue in class $C_2^2$.

Figure 3

(Color online) Dependence of ${\xi }_i/N$ with respect to $i/N$ for $n=5$ (black dots), 6 [red (dark gray) dashes], and 7 [green (light gray) solid line]. The states are labeled by increasing values of $E$ and, within each degenerate level, by increasing values of $\xi$.

Figure 4

(Color online) Absolute value of wave amplitudes for three states when $n=7$ indicated by red (dark gray) stars. (a) and (b) correspond to states with $E=0$ and, respectively, to smallest and largest values of $\xi$. In (c) we show the state with largest value of $\xi$ in the class $C_2^7$ and $E=\sqrt{3}$. The increasing value of $\xi$ stays in close correspondence with the trend to change from localized to extended character.

Figure 5

(Color online) Participation ratio $\xi$ for some specific states, as a function of $N$, for generations $n=4$, 5, 6, and 7. Red (dark gray) circles and black squares correspond, respectively, to states with large and low values of $\xi$. (a) $E=0$ and (b) $E=-\sqrt{3}$ correspond to states of class $C_3^n$. (c) and (d) illustrate localized and extended states belonging classes $C_1^n$ and $C_2^n$, respectively.

Figure 6

(Color online) (a) Dependence on $N$ of the average participation ratio $⟨\xi ⟩$ for generations $n=5$, 6, 7, and 8. All states with energy $E=0$ have been cast into ten groups according to their values of $\xi$. The values of the different slopes, drawn in the inset, indicate that most states are extended. (b) The participation ratio probability distribution $P\left(\xi /N\right)$ of states with $E=0$, for $n=6$ [red (dark gray) squares], 7 [green (light gray) circles], and 8 (black triangles), is largely concentrated on large values of $\xi /N$ for all $n$. This is in accordance with (a).

Figure 7

(Color online) Dependence of ${\xi }_i/N$ with respect to $i/N$ for $n=5$ (black dots), 6 [red (dark gray) dashes], and 7 [green (light gray) solid line] when (a) $\beta =1$ and (b) $\beta =-1$. The states are labeled by increasing values of $E$. In (c), we show the dependence of $⟨\alpha ⟩$ with respect to $\beta$, with a clear indication of a sudden change in the nature of the states for $\beta =3$.