Ising models on the regularized Apollonian network

We investigate the critical properties of Ising models on a regularized Apollonian network (RAN), here defined as a kind of Apollonian network in which the connectivity asymmetry associated with its corners is removed. Different choices for the coupling constants between nearest neighbors are considered and two different order parameters are used to detect the critical behavior. While ordinary ferromagnetic and antiferromagnetic models on a RAN do not undergo a phase transition, some antiferrimagnetic models show an interesting infinite-order transition. All results are obtained by an exact analytical approach based on iterative partial tracing of the Boltzmann factor as an intermediate step for the calculation of the partition function and the order parameters.

Figure 1

Entropy versus the temperature $T$ (in units of the Boltzmann constant $k_B$) for the following values of $u$ (from bottom to top): $u=10,5,2,1.2,1$. The case $u=1$ is the regular antiferromagnet with nonvanishing zero-temperature entropy [the entropy is equal to $\ln \left(2\right)/3$]. Interestingly, whenever $u>1$, the zero-temperature entropy drops to zero. All curves behave regularly and do not show any sign of phase transition.

Figure 2

Specific heat versus the temperature $T$ (in units of the Boltzmann constant $k_B$) for the following values of $u$ (from right to left): $u=10,5,2,1.2,1$. The curves behaves regularly and do not show any sign of phase transition.

Figure 3

(a) Plot of the spontaneous magnetization $M$ (solid line) and the coordination $L$ (dashed line) versus the temperature $T$ (in units of the Boltzmann constant $k_B$) for the $u=5$ model. (b) Plot of $1/\ln \left(M\right)$ (solid line) and $1/\ln (-L)$ (dashed line) versus the temperature $T$ (in units of the Boltzmann constant $k_B$). In (a) it seems that the transition occurs at a temperature around 26, but this is a wrong perception. (b) shows in fact that the critical temperature is about $T_c=49.16$, which is the correct transition temperature that we found analytically.