Bose-Einstein condensation in the Apollonian complex network

We demonstrate that a topology-induced Bose-Einstein condensation (BEC) takes place in a complex network. As a model topology, we consider the deterministic Apollonian network which exhibits scale-free, small-world, and hierarchical properties. Within a tight-binding approach for noninteracting bosons, we report that the BEC transition temperature and the gap between the ground and first excited states follow the same finite-size scaling law. An anomalous density dependence of the transition temperature is reported and linked to the structure of gaps and degeneracies of the energy spectrum. The specific heat is shown to be discontinuous at the transition, with low-temperature modulations as a consequence of the fragmented density of states.

Figure 1

(Color online) (a) Apollonian network with $g=3$. $P_1$, $P_2$, and $P_3$ are the sites of the original triangle at the zeroth generation. (b) IDOS within the tight-binding approximation for a single particle in an Apollonian network with $g=9$ generations. The vertical segments correspond to the degenerated energy levels while the horizontal segments are associated with the energy gaps. Observe in the inset the power-law scaling $\Delta E_0\propto N^{1/3}$, where $N$ is the total number of sites in a finite network.

Figure 2

(a) The fugacity as a function of temperature for Apollonian networks with distinct number of generations for $N_p/N=1/2$. Note the existence of a Bose-Einstein condensation at low temperature $\left(z\simeq 1\right)$, which is rounded by finite-size effects. (b) The fugacity as a function of temperature for an Apollonian network with $g=9$ generations and different numbers of particles.

Figure 3

(a) Fraction of particles $N_0/N_p$ in the ground state as a function of temperature for networks with distinct number of generations: $g=6$ (solid line), $g=7$ (dotted line), $g=8$ (dashed line), and $g=9$ (dotted-dashed line). Here, we used the ratio $N_p/N=1/2$. The inset shows the dependence of transition temperature, $T_c$, on the total number of sites $N$ of the network. The dashed line corresponds to $T_c\propto N^{1/3}$. (b) Fraction of particles $N_0/N_p$ in the ground state as a function of temperature for an Apollonian network with $g=9$ generations and different number of particles: $N_p/N=1/2$ (solid line), $N_p/N=1/4$ (dotted line), $N_p/N=1/8$ (dashed line), and $N_p/N=1/16$ (dotted-dashed line). The inset show the dependence of the transition temperature $T_c$ with the number of particles $N_p/N$. The dashed line corresponds to $N_p/N\propto \text{exp}\left(-{ϵ}^{\prime }/k_B{T}_c\right)$.

Figure 4

The normalized specific heat as a function of temperature for noninteracting bosons in Apollonian networks: (a) considering networks with distinct generations and a fixed particle density; (b) considering a network with $g=9$ generations but different particle densities $N_p/N$. A discontinuity in the specific heat at the Bose-Einstein transition temperature develops as the thermodynamic limit is approached. The oscillations in the low-temperature behavior reflects the fragmented structure of the density of states.