Eur. Phys. J. B 10, 71-90 (1999)
https://doi.org/10.1007/s100510050831

Surface effects on phase transitions of modulated phases and at Lifshitz points: A mean field theory of the ANNNI model

Institut für Physik, Johannes Gutenberg-Universität Mainz, 55099 Mainz, Staudinger Weg 7, Germany

Corresponding author: ^a binder@chaplin.physik.uni-mainz.de

Received: 28 July 1998
Published online: 15 July 1999

Abstract

The semi-infinite axial next nearest neighbor Ising (ANNNI) model in the disordered phase is treated within the molecular field approximation, as a prototype case for surface effects in systems undergoing transitions to both ferromagnetic and modulated phases. As a first step, a discrete set of layerwise mean field equations for the local order parameter m_n in the nth layer parallel to the free surface is derived and solved, allowing for a surface field H₁ and for interactions J_S in the surface plane which differ from the interactions J₀ in the bulk, while only in the z-direction perpendicular to the surface competing nearest neighbor ferromagnetic exchange (J₁) and next nearest neighbor antiferromagnetic exchange occurs. We show that for and temperatures in between the critical point of the bulk and the disorder line the decay of the profile is exponential with two competing lengths with while stays finite at T_cb. The amplitudes of these exponentials (a is the lattice spacing) are obtained from boundary conditions that follow from the molecular field equations. For but , as well as at the Lifshitz point and in the modulated region , we obtain a modulated profile , where again the amplitude A and the phase Ψ can be found from the boundary conditions. As a further step, replacing differences by differentials we derive a continuum description, where the familiar differential equation in the bulk (which contains both terms of order and here) is supplemented by two boundary conditions, which both contain terms up to order . It is shown that the solution of the continuum theory reproduces the lattice model only when both the leading correlation length ( or ξ, respectively) and the second characteristic length ( or the wavelength of the modulation , respectively) are very large. We obtain for a surface transition, with a two-dimensional ferromagnetic order occurring at a transition exceeding the transition of the bulk, and calculate the associated critical exponents within mean field theory. In particular, we show that at the Lifshitz point with while for the crossover exponent is . We also consider the "ordinary transition" and obtain the critical exponents and associated critical amplitudes (the latter are often singular when ). At the Lifshitz point, the exponents of the surface layer and surface susceptibilities take the values , while from scaling relations the surface "gap exponent" is found to be and the surface order parameter exponents are . Open questions and possible applications are discussed briefly.