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

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 H-1 and for interactions J(S) in the surface plane which differ from the interactions J(0) in the bulk, while only in the z-direction perpendicular to the surface competing nearest neighbor ferromagnetic exchange (J(1)) and next nearest neighbor antiferromagnetic exchange (J(2)) occurs. We show that for kappa equivalent to -J(2)/J(1) < kappa(L) = 1/4 and temperatures in between the critical point of the bulk (T-cb(kappa)) and the disorder line (Td(kappa) the decay of the profile is exponential with two competing lengths xi+, xi- with xi+ proportional to [T/T-cb(kappa)- 1](-1/2) while xi- stays finite at T-cb The amplitudes of these exponentials exp(-na/xi+/-) (a is the lattice spacing) are obtained from boundary conditions that follow from the molecular field equations. For kappa < kappa(L) but T > T-d(kappa), as well as at the Lifshitz point (kappa = kappa(L) = 1/4) and in the modulated region (kappa > kappa(L)), we obtain a modulated profile m(n+1) = A cos (naq + psi)e(-na/xi), where again the amplitude A and the phase Psi 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 partial derivative(2)m/partial derivative z(2) and partial derivative(4)m/partial derivative z(4) here) is supplemented by two boundary conditions, which both contain terms up to order partial derivative(2)m/partial derivative z(2). It is shown that the solution of the continuum theory reproduces the lattice model only when both the leading correlation length (xi(+) or xi, respectively) and the second characteristic length (xi- or the wavelength of the modulation lambda = 2 pi/q, respectively) are very large. We obtain for J(s) > J(sc) (kappa) a surface transition, with a two-dimensional ferromagnetic order occurring at a transition T-cs(kappa) 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 T-cs(kappa(L)) - T-cb(kappa(L)) proportional to (J(s) - J(sc))(1/phi L) With phi(L) = 1/4 while for kappa not equal kappa(L) the crossover exponent is phi = 1/2. We also consider the "ordinary transition" (J(s) < J(sc)(kappa)) and obtain the critical exponents and associated critical amplitudes (the latter are often singular when kappa --> kappa(L)). At the Lifshitz point, the exponents of the surface layer and surface susceptibilities take the values y(11)(L) = -1/4, gamma(1)(L) = 1/2, gamma(s)(L) = 5/4, while from scaling relations the surface "gap exponent" is found to be Delta(1)(L) = 3/4 and the surface order parameter exponents are beta(1)(L) = 1, beta(s)(L) = 1/4. Open questions and possible applications are discussed briefly.