A model for incommensurate transitions proposed by Chen and Walker (Phys. Rev. B 43 (1991) 5634) has been extended to describe a sub-lattice model which possesses supersymmetry. If a zone boundary mode freezes, a superstructure with supersymmetry can appear and may transform into a superstructure without supersymmetry at lower temperature. Although both superstructures belong to the same space group, the transition is either second or first order one with accompanying a definite anomaly in specific heat and susceptibility.KEYWORDS: free energy model, structural transition, superstructure, supersymmetry §1. Introduction It is well known that a modulated structure can be stabilized if the nearest-neighbor interaction is competing with the second-neighbor interaction in an Ising spin model. 1) In general, the modulated structure has an incommensurate period against the basic structure, however, it becomes commensurate at lower temperature. From the numerical solution of the so-called ANNNI model,2) it has been demonstrated that high-order commensurate phases appear successively and the modulation wave number changes as if a devil's staircase.3) The symmetry property of the incommensurate phase has been successfully analyzed by the idea of 3+d dimensional superspace group. 4) Although the character of the commensurate phase is believed to be understood within the framework of the ordinal 3-dimensional space group, the symmetry of the superstructure must be considered carefully.For the ANNNI model or in some extended models, one Ising spin is located in a unit cell of the basic structure. On the other hand, one-degree of freedom of lattice distortion is assigned in the φ 4 -model describing the modulated structures.5) In any way, a simplified theoretical model with one-degree of freedom is easy to treat theoretically and is tutorial to get an essence of the incommensurate-commensurate phase transition. But a real crystal may consist of two or more sub-lattices, and may exhibit a complicated structural change at low temperature. For example, two distinct commensurate structures with the same periodicity can appear, and the transformation between them may take place.In studying a twofold commensurate structure, we have explained the additional extinction rule with introducing supersymmetry:6) an extra symmetry operation that does not belong to a three-dimensional space group. 7,8) In this report we discuss whether a structure which possesses supersymmetry is stable or not on the basis of a thermodynamic potential for a sub-lattice model.Since supersymmetry is additional symmetry between two sub-lattices within a unit cell, a more realistic model for the sub-lattice is needed in further discussions. A sub-lattice model which explains modulated structure has been proposed by Chen and Walker. 9) They considered A 2 BX 4 -type crystals which perform a successive transition from the normal phase (space group P mcn: Z=4) to the incommensurate phase, followed by some commensurate phases. In the incommensurate phase, atomic ...
