Ising model on the Apollonian network with node-dependent interactions

This work considers an Ising model on the Apollonian network, where the exchange constant $J_{i,j}\sim 1/{\left(k_i{k}_j\right)}^{\mu }$ between two neighboring spins $\left(i,j\right)$ is a function of the degree $k$ of both spins. Using the exact geometrical construction rule for the network, the thermodynamical and magnetic properties are evaluated by iterating a system of discrete maps that allows for very precise results in the thermodynamic limit. The results can be compared to the predictions of a general framework for spin models on scale-free networks, where the node distribution $P\left(k\right)\sim k^{-\gamma }$, with node-dependent interacting constants. We observe that, by increasing $\mu$, the critical behavior of the model changes from a phase transition at $T=\infty$ for a uniform system $\left(\mu =0\right)$ to a $T=0$ phase transition when $\mu =1$: in the thermodynamic limit, the system shows no true critical behavior at a finite temperature for the whole $\mu \ge 0$ interval. The magnetization and magnetic susceptibility are found to present noncritical scaling properties.

Figure 1

Geometrical construction of the first three generations ($g=0$, 1, and 2) of the AN. Circles for corresponding Apollonian packing problem are not included. Nodes are numbered according to the scheme used in [16]

Figure 2

Dependence of $J_0$ on $\mu$, as a result from fixing the ground-state energy per spin $u_0=-3$.

Figure 3

(Color online) Plots of the specific heat $c$ for $g=40$ (symbols) and $g=50$ (solid line) for different values of $\mu$: 0 (black squares), 0.5 (red diamonds), 1.0 (green up-triangles), and 3.0 (blue down-triangles). The superposition of curves and symbols for $g=40$ and $g=50$ indicates that $c$ converged to its value in the thermodynamic limit.

Figure 4

(Color online) Dependence of the temperatures $T_d$ (symbols) and $T_m$ (solid lines) on the network generation $g$. They indicate, respectively, the temperature of numerical divergence of $1/\text{ln}\left({\lambda }_1/{\lambda }_2\right)$ and of the maximum of susceptibility $\chi$. (a) $\mu =0.0$ (squares), 0.5 (up-triangles), and 0.8 (down-triangles). (b) $\mu =1.0$ (circles) and 1.2 (diamonds).

Figure 5

(Color online) Behavior of the magnetization $m\left[T,h=0;N\left(g\right)\right]$ as function of the temperature $T$ for different values of $g$ when $\mu =0.5$, 1.0, and 1.5. (a) $\mu =0.5$, $g=20$, 30, and 40 indicated by dots, dashes, and solid lines. Four regions characterized by different behaviors are obtained: two of them are $g$ independent, the second of which has an exponential decay. The third region starts with a crossover to a second exponential regime. (b) $\mu =1.0$, same symbols as in (a). The first exponential region has disappeared. (c) $\mu =1.5$, $g=10$, 15, and 20 indicated by dots, dashes, and solid lines. The first region in (a) has disappeared. Two $T$ intervals separated by a $g$ dependent crossover temperature are observed. As $g$ increases, $m$ vanishes for any $T>0$.

Figure 6

(Color online) Main panels show scaling properties of the magnetic susceptibility $\chi \left[T,h=0;N\left(g\right)\right]$ as function of the temperature $T$ for different values of $g=40$ (solid), 45 (dashes), and 50 (dots), when (a) $\mu =0.5$, (b) 1.0, and (c) 1.5. The insets show the same curves in the original variables. (a) Scaling properties of $\chi$ as function of $T-T_m$, with exponent $\gamma /\nu =0.607$, are observed in region iii of the magnetization $m$ [see Fig. 5a]. (b) Scaling properties of $\chi$ as function of $T$, with exponent $\gamma /\nu =1.0$, valid in region iii, with exponential decrease in $m$, as discussed in Fig. 5b. (c) Scaling properties of $\chi$ as function of $T$, with exponent $\gamma /\nu =1.0$. Scaling is valid in the two regions shown in Fig. 5c. The horizontal axis is in logarithmic scale to evidence Curie’s law with $g$ dependent constant ${\mathcal{C}}_g$.