Exact solutions and infinite-order phase transitions for a general class of Ising models on the regularized Apollonian network.

The regularized Apollonian network (RAN) is defined starting from a tetrahedral structure with four nodes all connected. At any successive generations, new nodes are added and connected with the surrounding three nodes. As a result, a power-law cumulative distribution of connectivity P (k) ∝ 1/k η with η = ln(3)/ ln(2) 1.585 is obtained. We consider a very general class of Ising models on this network, whose exact solutions for both finite and infinite (thermodynamic limit) size are achieved by using an approach based on recursive partial tracing of the Boltzmann factor as an intermediate step for the calculation of the partition function. Afterwards, we focus on some relevant choices for the coupling constants between connected spins, and we show that ordinary ferromagnets and anti- ferromagnets (all equal couplings) do not undergo a phase transition. In contrast, some anti-ferrimagnets show an infinite-order transition, which is detected by the spontaneous magnetization M (and also by the coordination L, a second-order parameter), which, at transition, goes as exp[−b/(T c − T )] for T < T c , vanishing for T > T c .