Critical Behavior of the Ising Model on Small-world Hanoi Networks

The addition of small-world bonds on hierarchical lattices changes a typical Ising model ferromagnetic phase transition to one of infinite order, referred to as the inverted-Berezinski-Kosterlitz-Thouless transition. We study this shift in phase behavior on Hanoi networks, which are one-dimensional Ising chains connected by small-world bonds that are self-similar and hierarchical in structure [1]. The phase behavior of the Ising model near T$_{c}$ on Hanoi networks is studied using an exact renormalization group and Monte Carlo techniques. We show that compared to the Migdal-Kadanoff hierarchical lattice, Hanoi networks possess characteristics in their thermodynamic densities that are more physical. These densities are studied in detail and the behavior of their critical exponents near T$_{c}$ is described. By introducing a continuous parameter which regulates the strength of small-world bonds in the Hanoi networks, we begin to uncover the essential small-world properties that dictate this change in phase behavior from second- to infinite-order. \\[4pt] [1] S. Boettcher and C.T. Brunson, Phys. Rev. E, \textbf{83}, 021103 (2011)