Phase transitions in an Ising model on a Euclidean network.

A one-dimensional network on which there are long-range bonds at lattice distances l>1 with the probability P(l) proportional to l(-delta) has been taken under consideration. We investigate the critical behavior of the Ising model on such a network where spins interact with these extra neighbors apart from their nearest neighbors for 0<or=delta<2. It is observed that there is a finite temperature phase transition in the entire range. For 0<or=delta<1, finite-size scaling behavior of various quantities are consistent with mean-field exponents while for 1<or=delta<or=2, the exponents depend on delta. The results are discussed in the context of earlier observations on the topology of the underlying network.