The decoupling point of the axial next-nearest-neighbour Ising model and marginal crossover

The axial next-nearest-neighbour Ising (or ANNNI) model and the four-state asymmetric clock model both exhibit a decoupling multicritical point where they decompose into two uncoupled Ising models at criticality. The corresponding scaling exponents for various operators are given generally, the leading modulation or chiral crossover exponent being phi = gamma - nu , where gamma and nu are the pure Ising susceptibility and correlation exponents. Implications for the ANNNI model phase diagrams are discussed. Crossover scaling forms appropriate to logarithmic and marginal behaviour, with non-universal exponents, are presented.