Ising model on a general Cayley tree with competing next-nearest-neighbour interactions - Ganikhodjaev, Nasir; Zakaria, Siti Fatimah (2011)
We study the Ising model on a general Cayley tree of arbitrary order and produce the phase diagram with competing interactions prolonged next-nearest-neighbour Jp and one-level k-tuple next-nearest-neighbour Jo . Vannimenus proved the existence of modulated phase in the phase diagram of Ising model with competing nearest-neighbour interaction J1 and prolonged next-nearest-neighbour interactions Jp, as found for similar models on periodic lattices. Later Mariz et al generalized this result for Ising model with Jo ≠ 0. For a given lattice model on a Cayley tree, i.e., Jp ≠ 0; Jo ≠ 0 with J1 = 0, we describe the general equation, phase diagram and clarify the role of nearest-neighbour interaction J1. In the presence of nearest-neighbour interaction J1, Vannimenus demonstrated that for arbitrary random initial data one can reach the same phase diagram. We show that in the case J1 = 0 the set of all possible initial data can reach different phase diagrams