Perturbation expansions of the axial next-nearest-neighbour Ising and asymmetric four-state clock models

The axial next-nearest-neighbour Ising (ANNNI) and four-state asymmetric clock (4 ASC) models are investigated by perturbing about points at which they reduce the simple nearest-neighbour Ising models. These limits are (a) for the ANNNI model-(i) small nearest-neighbour interaction, J 1 , in the x direction and (ii) small next-nearest-neighbour interaction, J 2 , in the x direction, and (b) for the 4 ASC model small asymmetry, Delta . In the limit J 1 to 0, the phase boundary is located to first order in J 1 and found to be Ising-like. In the limit J 2 to 0, the behaviour to second order in J 2 is more complicated but is identical to that found to second order in Delta for the 4 ASC model. These results are discussed in the light of other recent work.