Mean-field theory of the three-dimensional anisotropic Ising model as a four-dimensional mapping

Abstract

The mean-field theory of systems which are inhomogeneous in one direction, such as modulated structures, can be formulated as an area- (volume-) preserving mapping. As a specific example we have investigated the anisotropic Ising model with nearest- and next-nearest-neighbor interactions (ANNNI model) by studying the corresponding four-dimensional mapping. The mapping generates spin structures characterized by fixed points, one-dimensional orbits, two-dimensional surfaces, and chaotic trajectories. From the eigenvalues of the linearized mapping in the disordered phase the transition temperature and the critical wave vector can be calculated analytically. We have developed numerical methods to locate the various orbits and determine their physical stability away from the critical line. The orbits of the thermodynamically stable phases are embedded in regimes with positive Lyapunov exponents and they are unstable under numerical iteration. The one-dimensional trajectories describe incommensurate phases which may be stable between the commensurate phases associated with high-order limit cycles. At low temperatures and near the low-order commensurate phases there seem to be no incommensurate trajectories. As the commensurate phase of period 4 is approached, the incommensurate phase takes the form of a soliton lattice. The soliton energy, soliton-soliton interaction, and pinning energy are calculated, and the critical wave vector for a pinned soliton lattice is estimated. The transition from the modulated phase to the ferromagnetic phase is found to be of first order. Our findings are in agreement with the conclusions reached in a previous study of an approximate two-dimensional mapping representation of the model.