Surface critical behaviour near the uniaxial Lifshitz point of the axial next-nearest-neighbour Ising model

The semi-infinite axial next-nearest-neighbour Ising (ANNNI) model in the d isordered phase is treated within a molecular-field approximation, and the singularities of various response functions characterizing the critical beh aviour at the surface are obtained. In previous work (Binder K and Frisch H L 1999 Eur. Phys. J. B 10 71) the axis where a nearest-neighbour ferromagn etic (J(1)) and next-nearest-neighbour antiferromagnetic (J(2)) exchange co mpete was chosen perpendicular to the surface plane. In the present work we consider an orientation of this axis parallel to the surface, allowing als o for different values of these exchange interactions (j(1), j(2)) in the s urface plane. We derive the conditions for the occurrence of a surface tran sition, where the surface plane (at a temperature where the hulk is still d isordered) orders either ferromagnetically or into a modulated structure. A t the ordinary transition of the surface we obtain the mean-field values of the surface critical exponents, including the Lifshitz point of the bulk, where gamma(1)(L) = 1/2, y(11)(L) = -1/2, eta(perpendicular to)(L) = 0, eta (\\)(L) = 3. These exponents differ from their counterparts for the case wh ere the axis of competing interactions is oriented perpendicular to the sur face, and thus it is shown that for a uniaxial Lifshitz point the surface c ritical behaviour depends on the surface orientation.