Investigation of the Maier–Saupe–Zwanzig Model in the Apollonian Network

We analyze the thermodynamic behavior of a discrete version of the Maier–Saupe model for the nematic phase transitions in liquid crystals in the Apollonian network. This simple model, which we call Maier–Saupe–Zwanzig (MSZ) model, has been investigated in a variety of situations and geometric substrates. In terms of a single energy parameter, it has been shown to account for the well-known transition between uniaxially ordered and disordered phases in nematic liquid crystals. We consider a special Apollonian lattice, and use an exact transfer matrix approach, to investigate the occurrence of these phase transitions. We obtain numerical precise results for the free energy and its derivatives, as well as for the correlation lengths, which come from the iterations of recursion relations for the transfer matrix elements of successive generations of the lattice. Results are compared with similar findings on a diamond hierarchical lattice. In contrast with the diamond lattice, and in agreement with similar analyses for the Ising model, we find some special thermodynamic features, including peculiar behavior of the correlations functions, but no singular behavior of the free energy.