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.
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Received 28 July 1998
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Binder, K., Frisch, H. Surface effects on phase transitions of modulated phases and at Lifshitz points: A mean field theory of the ANNNI model. Eur. Phys. J. B 10, 71–90 (1999). https://doi.org/10.1007/s100510050831
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DOI: https://doi.org/10.1007/s100510050831