Ising models on the regularized Apollonian network

Abstract

We investigate the critical properties of Ising models on a Regularized Apollonian Network (RAN), here defined as a kind of Apollonian Network (AN) in which the connectivity asymmetry associated to 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 behaviour. While ordinary ferromagnetic and anti-ferromagnetic models on RAN do not undergo a phase transition, some anti-ferrimagnetic 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 intermediate steps for the calculation of the partition function and the order parameters.PACS numbers: 89.75. Hc, 89.20.Hh, 89.75.Da Many real world networks exhibit complex topological properties as the small word effect, related to a very short minimal path between nodes, and the scale-free property, related to the power-law nature of the connectivity distribution. These properties have important implications in the real phenomena as virus spreading in computers, sharing of technological information and diffusion of epidemic diseases, to name just a few.In this context, the Apollonian network [1] is a particularly useful theoretical tool, since it is scale-free, displays small-world effect, can be embedded in a Euclidean lattice and shows space-filling as well as matching graph properties. Therefore, in spite of its deterministic nature, it shares the most relevant characteristics of real world networks.Phase transitions has been detected for a number of different physical models on Apollonian Networks. For example, the ideal gas undergoes to Bose-Einstein condensation [2-6] and epidemics exhibits a transition between an absorbing state and an active state [7,8]. In particular, in [6] it has been adopted an analytical strategy which has some similarities with that in this paper.In this work we focus on the infinite order transition exhibited by some Ising models on the Apollonian network. Previous studies of similar Ising models [9, 10] and Potts model [11] have not detected critical properties, due to the fact that infinite order transitions are elusive. On the contrary, a second order phase transition, as a function of the noise parameter, has been detected for a majority vote model [12].Ising models on different hierarchical fractal have been studied [13] and in the case of diamond fractal they have been exactly solved by exact renormalization [14,15]. These models, differently to present model, show a second order phase transition. Indeed, the standard Ising behavior is second order transition in plane models [16,17], nevertheless, it may be very intricate, with many phases [18], when the interactions are made more complicated.We start by regularizing the standard Apollonian Network in order to remove the connectivity asymmetry associated to its corners which consistently simplify the analytical computation of the thermodynamics o...

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