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

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 mn in the nth layer parallel to the free surface is derived and solved, allowing for a surface field H1 and for interactions JS 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 (J 2 ) occurs. We show that for $$\kappa \equiv - {J_2}/{J_1} < {\kappa _L} = 1/4$$ and temperatures in between the critical point of the bulk $$({T_{cb}}(\kappa ))$$ and the disorder line $$({T_{d}}(\kappa ))$$ the decay of the profile is exponential with two competing lengths $$\xi + ,\xi \_$$ with $$\xi + \propto {[T/{T_{cb}}(\kappa ) - 1]^{ - 1/2}}$$ while $$\xi \_$$ stays finite at Tcb. The amplitudes of these exponentials $$\exp ( - na/\xi \pm )$$ (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 $${m_{n + 1}} = A\cos (naq + \psi )$$ , 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 $${\partial ^2}m/\partial {z^2}$$ and $${\partial ^2}m/\partial {z^4}$$ here) is supplemented by two boundary conditions, which both contain terms up to order $${\partial ^2}m/\partial {z^4}$$ . 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 $${J_s} > {J_{cs}}(\kappa )$$ 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 $${T_{cs}}({\kappa _L}) \propto {({J_s} - {J_{sc}})^{1/\phi L}}$$ with $$\kappa \ne {\kappa _L}$$ while for $$\kappa \ne {\kappa _L}$$ 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 $$\kappa \to {\kappa _L}$$ ). At the Lifshitz point, the exponents of the surface layer and surface susceptibilities take the values $$\gamma _{11}^L = - 1/4,\gamma _1^L = 1/2,\gamma _s^L = 5/4$$ , 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.