Magnetic models on Apollonian networks

Thermodynamic and magnetic properties of Ising models defined on the triangular Apollonian network are investigated. This and other similar networks are inspired by the problem of covering a Euclidian domain with circles of maximal radii. Maps for the thermodynamic functions in two subsequent generations of the construction of the network are obtained by formulating the problem in terms of transfer matrices. Numerical iteration of this set of maps leads to very precise values for the thermodynamic properties of the model. Different choices for the coupling constants between only nearest neighbors along the lattice are taken into account. For both ferromagnetic and antiferromagnetic constants, long-range magnetic ordering is obtained. With exception of a size-dependent effective critical behavior of the correlation length, no evidence of asymptotic criticality was detected.

Figure 1

Third generation $(n=3)$ of construction of the AN. In (a), we show the optimal circles that define the network. In (b), sites represented by squares, circles, and diamonds are introduced in the first, second, and third steps of construction, respectively. Links represented by dotted, dashed, and solid lines correspond to $n=3$, $m=3,2$, and 1 in Eq. (1) or $n=1,2$, and 3 in Eq. (8), respectively.

Figure 2

Thermodynamic functions for the ferromagnetic model. Solid, dashed, and dotted lines indicate $\alpha =0,1$, and $\infty$, respectively.

Figure 3

Size-dependent critical temperature for distinct values of $\alpha$.

Figure 4

Thermodynamic functions for the antiferromagnetic model. Solid, dashed, and dotted lines indicate $\alpha =0,1$, and $\infty$, respectively. $\xi$ is drawn only for $\alpha =0$ and reciprocal temperature axis.

Figure 5

Thermodynamic functions for the model with alternating ferro- and antiferromagnetic coupling, starting with $J_0=1$. Solid, dashed, and dotted lines indicate $\alpha =0,1$, and $\infty$ respectively, and $J_0>0$. In (c), three curves for the magnetization when $J_0=-1$ and same values of $\alpha$ saturate at $m=1∕3$ are also drawn.