Tsallis scaling in the long-range Ising chain with competitive interactions

Abstract

A numerically efficient transfer matrix approach is used to investigate the validity of the Tsallis scaling hypothesis in the long-range Ising spin chain with competitive interactions. In this model, the interaction between two spins i and j placed r lattice steps apart is J i, j = (−1) ζ(i, j) J 0 /r α , where ζ(i, j) is either 0 or 1. This procedure has succeeded to show the validity of the scaling hypothesis for the well investigated ferromagnetic version of the model, i.e., ζ(i, j) = 0, ∀i, j, ∀α ≥ 0. Results are reported for some models of a set, which is defined by requiring ζ(i, j) to be a periodic sequence of 0 s and 1 s. As expected from symmetry arguments, we find that the hypothesis is not valid when ζ(i, j) = 1, ∀i, j and α < 1. However, it is verified, with high degree of numerical accuracy, when α < 1, for sequences in which the occurrence of ζ(i, j) = 0 is more frequent than that of ζ(i, j) = 1.