Z. Phys. B CondensedMatter 57, 49-58 (1984) Condensed Zeitschrift Matter for Physik B 9 Springer-Verlag 1984 The Mean Field Theory of the Three-Dimensional ANNNI Model W. Selke Institut ftir Festk6rperforschung der Kernforschungsanlage Jiilich, Federal Republic of Germany P.M. Duxbury Department of Theoretical Physics, Oxford University, Oxford, United Kingdom Received May 9, 1984; revised version July 3, 1984 The mean field equations of the simple cubic or tetragonal ANNNI model are studied on finite lattices. Structure combination branching processes are found which allow us to considerably refine previous mean field calculations on the model. L Introduction The axial next-nearest neighbor Ising (or ANNNI) model [-1, 2] is one of the simplest statistical mechanical models to exhibit complex spatially mo- dulated phases. It is composed of spin-~ Ising vari- ables, Si= _+1, situated on a regular d-dimensional lattice formed of (d- 1) dimensional layers of coordi- nation number q• normal to, say, the z-axis. Within the layers each spin is coupled only by nearest- neighbour ferromagnetic interactions, do>0. How- ever, along the z-axis, spins are coupled by compet- ing nearest, J1 >0, and next-nearest neighbour, J2 = -~cJ~<0, interactions. The parameter ~c thus con- trols the degree of competition which leads to the complex behaviour of the model. In this article we shall study the ANNNI model on a simple cubic or tetragonal lattice, q• For rea- sons of simplicity Jo=J~ is assumed. The model is known to form a low temperature ferromagnetic phase for ~c< 89 and a (2) phase for ~c> 89The wedge in the (~:, T) phase diagram between these two pha- ses is, at low temperatures, filled by distinct com- mensurate phases. In particular, it has been shown, using systematic low temperature series expansion, that a countably infinite sequence of phases of type (2k-13), k=l, 2, 3, ..., spring directly from the mul- tiphase point (to= 89 T=0) [1, 3, 4]. To explore the modulated region of the phase diagram in between the discrete commensurate phases and the structures with supposedly continuously varying wave-vector close to the transition to the paramagnetic phase [2] approximate methods have been invoked, especially Monte Carlo techniques [-5, 6] and various mean field theories [7-14]. Results of such calculations suggest a variety of interesting features [2], but do not give a completely coherent picture of the modu- lated region, in particular on the mechanism by which more complicated commensurate phases evolve. Only recently, we presented a mean field ap- proach on finite lattices to systematically study this problem [15]. We found results consistent with the ones of the low temperature series expansion, and gave evidence for new structure combination branching processes which generate further commen- surate phases of type ((2 ~ 3)m(2 ~+~3)") or ((231)m(2Y+l)"), l,m, and n are integers, which be- come stable at non-zero temperatures in the vicinity of the multiphase point for ~c> 89 (We use a short hand notation which neglects the order of the ele- ments (2z3), (2~§ etc. The order follows from the structure combination branching process.) Based on this observation we can study, within mean field theory, structures of much larger periods than con- sidered previously, and we can hence considerably refine and/or check previous, partly contradictory, mean field calculations [7-14]. In the next section various approximate mean field theories are sketched and our method is outlined. The resulting global mean field phase diagram is presented. In Sect. III details for ~> 89 i.e. on the (2) side of the phase diagram, are given and corn-