Dynamic systems corresponding to some lattice models on the cayley tree of arbitrary order

Abstract

We study the Ising model on a general Cayley tree of arbitrary order k and produce the phase diagram with competing next-nearest-neighbour interaction; prolonged Jp and one-level k-tuple next-nearest-neighbour J0. Previously, Vannimenus proved for k = 2 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, are similar to other models on periodic lattices. Later Mariz et al generalized this result of k = 2 for Ising model with one-level interacton J0 ≠ 0. Here, for a given lattice model on a Cayley tree of arbitrary order, i.e., Jp ≠ 0; J0 ≠ 0 with J1 = 0 we produce the general equations, describe 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 condition one can reach the same result of phase diagram. Here however, two cases have been covered for analysis which are when k is even and k is odd. An analytical analysis has been done to check the stability of paramagnetic phase of the resultant phase diagram. An asymptote of transition line is achieved through the analysis. Also, a duality condition coming to the attention based on the stability analysis for the recurrence system. With this, we show that in the case J1 = 0 the set of all possible initial conditions can reach different phase diagrams.