Free-electron gas in the Apollonian network: Multifractal energy spectrum and its thermodynamic fingerprints

We study the free-electron gas in an Apollonian network within the tight-binding framework. The scale-free and small-world character of the underlying lattice is known to result in a quite structured energy spectrum with deltalike singularities, gaps, and minibands. After an exact numerical diagonalization of the corresponding adjacency matrix of the network with a finite number of generations, we employ a scaling analysis of the moments of the density of states to characterize its multifractality and report the associated singularity spectrum. The fractal nature of the energy spectrum is also shown to be reflected in the thermodynamic behavior by logarithmic modulations on the temperature dependence of the specific heat. The absence of chiral symmetry of the Apollonian network leads to distinct thermodynamic behaviors due to electrons and holes thermal excitations.

Figure 1

(Color online) Second generation of the Apollonian network. $P_1$, $P_2$, and $P_3$ represent the corner sites. Different symbols refer to sites added at each new generation (central circle: $n=0$; squares: $n=1$; and triangles: $n=2$). The number of sites added in each generation is given by $3^n$.

Figure 2

Integrated density of states for an Apollonian network with $n=8$ generations. The vertical segments signal degenerated energy levels while the horizontal segments are associated with energy gaps. The largest degeneracy is at $E=0$ which supports $1∕3$ of the one-particle states. Minibands are present in the upper half of the spectrum.

Figure 3

Density of states representing the fraction of states located in small energy windows $\Delta E∕t={10}^{-2}$ for a $n=8$ Apollonian network. The first half of the energy spectrum is mainly composed of a discrete set of degenerated levels separated by gaps, while in the second half the minibands are visible. The presence of many scales of energy gaps and degeneracies suggests a fractal scaling. The inset is an amplification of the DOS showing the minibands above the band center.

Figure 4

Multifractal singularity spectrum of the density of states for one particle in Apollonian networks with $n=5$, 6, 7, and 8 generations. For computing the multifractal exponents, the energy spectrum was partitioned in $N$ boxes, where $N$ is the total number of sites for each generation. Finite-size corrections are mainly affecting the extremal segments of the singularity spectrum. The slight increase of the range of values for $\alpha$ for larger sizes confirms the multiscaling nature of the density of states in the thermodynamic limit.

Figure 5

Specific heat per electron as a function of temperature for the case of a half-filled band $N_e∕N=1∕2$. Apollonian networks with $n=5$, 6, 7, and 8 generations were considered. Finite-size corrections are only significant at very small temperatures. The high temperature decay $C_v\propto 1∕T^2$ is typical of systems with a bounded energy spectrum. The anomaly observed as a strong deviation from the usual power-law low-temperature behavior reflects the existence of energy gaps near the Fermi energy.

Figure 6

Specific heat per electron as a function of temperature in an Apollonian network with $n=8$ generations and band fillings $N_e∕N=1∕4$, $1∕8$, and $1∕16$. The oscillations in the specific heat are associated with the increasing number of scales of energy gaps and degeneracies as one approaches the bottom of the band. In the limit of $N_e∕N\to 0$ the specific heat starts to develop log-periodic oscillations, as expected for classical systems with a fractal energy spectrum.

Figure 7

Specific heat per hole $(N_h=N-N_e)$ as a function of temperature in an Apollonian network with $n=8$ generations and band fillings $N_e∕N=3∕4$, $7∕8$, and $15∕16$, complementary to those depicted in Fig. 6. The absence of electron-hole symmetry is clearly evident. The presence of minibands at the upper half of the energy spectrum smoothes the oscillations in the specific heat. The specific heat per hole for large band fillings becomes smaller than the corresponding specific heat per electron at low fillings due to the lower degree of degeneracy at the top of the energy spectrum.