Critical behavior of the ideal-gas Bose-Einstein condensation in the Apollonian network

We show that the ideal Boson gas displays a finite-temperature Bose-Einstein condensation transition in the complex Apollonian network exhibiting scale-free, small-world, and hierarchical properties. The single-particle tight-binding Hamiltonian with properly rescaled hopping amplitudes has a fractal-like energy spectrum. The energy spectrum is analytically demonstrated to be generated by a nonlinear mapping transformation. A finite-size scaling analysis over several orders of magnitudes of network sizes is shown to provide precise estimates for the exponents characterizing the condensed fraction, correlation size, and specific heat. The critical exponents, as well as the power-law behavior of the density of states at the bottom of the band, are similar to those of the ideal Boson gas in lattices with spectral dimension $d_s=2ln\left(3\right)/ln(9/5)\simeq 3.74$.

Figure 1

(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) Integrated density of states (IDOS) within the tight-binding approximation for a single particle in an Apollonian network with $g=15$ generations. Degenerescences and gaps are signaled by vertical and horizontal segments, respectively. The inset shows the power-law behavior near the band bottom IDOS $\propto {(E-E_0)}^{1+\sigma }$ [$1+\sigma =\ln \left(3\right)/\ln (9/5)\simeq 1.87$] modulated by a fractal-like structure.

Figure 2

(a) Condensed fraction $N_0/N_p$ as a function of temperature for networks with distinct number of generations, $g=11$ up to $g=15$. Here, we used a particle density $N_p/N=1/2$. The inset shows the data in a universal finite-size scaling form. The straight lines are consistent with the expected scaling behavior below and above the transition (see text); (b) Condensed fraction $N_0/N_p$ as a function of temperature in a network with $g=15$ generations and different particle densities: $N_p/N=2$ (solid line), $N_p/N=1$ (dotted line), $N_p/N=1/2$ (dashed line), and $N_p/N=1/4$ (dotted-dashed line). The inset shows the density dependence of the transition temperature $T_c$. The dashed lines correspond to the high density $T_c\propto N_p/N$ and low density $T_c\propto {(N_p/N)}^{1/(1+\sigma )}$ behavior.

Figure 3

The specific heat per particle (in units of $k_B$) as a function of temperature for a fixed particle density $N_p/N=1/2$ in Apollonian networks with distinct number of generations. The low-temperature behavior $C_v\propto T^{1+\sigma }$ (shown as a straight dashed line) is consistent with the low-energy overall power-law behavior of the IDOS. Near the transition, the specific heat develops a peak rounded by finite-size effects. The data point toward a saturation of the peak height in the thermodynamic limit.

Figure 4

(a) Auxiliary scaling functions $f(N,N^{\prime },T)$ versus temperature using $N=7174456$ ($g=15$) and distinct values of $N^{\prime }$ ($g=11,12,13$, and 14). Here, the particle density of $N/N_p=1/2$ was used. The crossing point signals the BEC transition at $k_B{T}_c/t=0.1136$. It is also shown the finite-size scaling of (b) the condensed fraction ${\rho }_0$, (c) its logarithmic derivative $\frac{d}{dT}\ln {\rho }_0(N,T)$, and (d) the specific heat singularity $C_v(N\to \infty )-C_v\left(N\right)$ at the transition temperature. The power-law scaling correspond $\zeta /\tilde{\nu }=1/\tilde{\nu }=(d_s-2)/d_d\simeq 0.46$ and $\alpha /\tilde{\nu }=-(4-d_s)/d_s\simeq -0.07$.