Ferromagnetic model on the Apollonian packing

This work investigates the influence of geometrical features of the Apollonian packing (AP) on the behavior of magnetic models. The proposed model differs from previous investigations on the Apollonian network (AN), where the magnetic coupling constants depend only on the properties of the network structure defined by the packing, but not on quantitative aspects of its geometry. In opposition to the exact scale invariance observed in the AN, the circle's sizes in the AP are scaled by different factors when one goes from one generation to the next, requiring a different approach for the evaluation of the model's properties. If the nearest-neighbors coupling constants are defined by $J_{i,j}\sim 1/{(r_i+r_j)}^{\alpha }$, where $r_i$ indicates the radius of the circle $i$ containing the node $i$, the results for the correlation length $\xi$ indicate that the model's behavior depend on $\alpha$. In the thermodynamic limit, the uniform model ($\alpha =0$) is characterized by $\xi \to \infty$ for all $T>0$. Our results indicate that, on increasing $\alpha$, the system changes to an uncorrelated pattern, with finite $\xi$ at all $T>0$, at a value ${\alpha }_c\simeq 0.743$. For any fixed value of $\alpha$, no finite temperature singularity in the specific heat is observed, indicating that changes in the magnetic ordering occur only when $\alpha$ is changed. This is corroborated by the results for the magnetization and magnetic susceptibility.

Figure 1

(a)–(c) illustrate the first steps of the construction of the AN. The labels indicate how the TM's for each individual triangle of type $s=1$ are combined to generate a new TM corresponding to $s=2$ triangle. (d) indicates how to proceed to next step, when three $s=2$ triangles are combined to form the $s=3$ triangle.

Figure 2

Behavior of $f\left(T\right)$ [(a) and (b)] and $s\left(T\right)$ [(c) and (d)] for $g_{max}=5$ (red dashes) and 15 (black solid line), and $\alpha =0.5$ [(a) and (c)] and 0.8 [(b) and (d)]. To enable a comparison of the curves for different values of $g_{max}$, the values of $J_0$ were chosen in such a way that $f(T=0)=-3$, as observed for the uniform model with $J_0=1$. As a consequence, the high temperature limit, when $s\to ln2$, is observed in the same range of values of $T,\forall g_{max},\alpha$. The inset of all panels highlight the convergence of the results as $g_{max}$ increases, by displaying the corresponding differences from $g^{\prime }=5$ (red solid line), 7 (blue dashes), 10 (green dots), and 13 (violet dot-dashes), to $g_{max}=15$.

Figure 3

Behavior of $c\left(T\right)$ for $g_{max}=5$ (red dashes), 7 (blue dot-dashes), 10 (green dots), 13 (violet dot-dot-dashes), and 15 (black solid line), and (a) $\alpha =0.5$ and (b) 0.8. All curves follow the same Schottky profile. Because of the adequate definition of $J_0$, the smooth peaks always occur in the same temperature range. No indication of a singular behavior at a finite temperature has been detected. $g^{\prime }=5$ (red solid line), 7 (blue dashes), 10 (green dots), and 13 (violet dot-dashed line), to $g_{max}=15$.

Figure 4

Behavior of $1/ln(\eta /\epsilon )$ as a function of temperature for $g_{max}=5$ (red squares), 7 (blue circles), 10 (green up-triangles), 13 (violet down-triangles) and 15 (black diamonds). (a), (b), and (c) illustrate the different behavior of the divergence temperature $T_d$ for $\alpha =0.5,0.743,$ and 0.87.

Figure 5

Behavior of $T_d\left(g\right)$ as a function of $g_{max}$ for different $\alpha$. In (a), results for $\alpha =0.5,0.6,0.7$, and 0.8, are identified by black squares, red circles, green up-triangles, and blue down-triangles, respectively. With exception of $\alpha =0.8,T_d$ increases with $g_{max}$. In (b) $\alpha =0.7,0.743,0.75,0.8,$ and 0.87 are identified by black squares, red circles, green up-triangles, blue down-triangles, and violet diamonds. While $T_d$ increases when $\alpha =0.7$ and approaches a constant plateau when $\alpha =0.743$, it decreases with $g$ when $\alpha >0.743$.

Figure 6

Behavior of $m(T,h=0)$ as a function of $T$ for $g_{max}=5$ (red dashes), 7 (blue dot-dashes), 10 (green dots), 13 (violet dot-dot-dashes), and 15 (black solid line), (a), (b) and (c) correspond, respectively, to $\alpha =0.5,0.743$, and 1.3.

Figure 7

Behavior of $\chi (T,h=0)$ as a function of $T$ for $g_{max}=5$ (red dashes), 7 (blue dot-dashes), 10 (green dots), 13 (violet dot-dot-dashes), and 15 (black solid line), Panels (a), (b) and (c) correspond, respectively, to $\alpha =0.5,0.743$, and 1.3.