Landau's theory of crystalline phase transitions in a superspace formulation

Abstract

We assume that the reader is familiar with the superspace group approach and with Landau’s theory for continuous structural phase transitions [1,2]. This Landau theory assumes that the high-temperature crystal phase has a space group G0 as symmetry group and that there is a group-subgroup relation between the symmetry groups above and below the phase transition. Normally such a subgroup is a space group also. This type of crystal state can be destabilized by the occurrence of a so called Lifshitz term, leading to a phase which is incommensurate and lacks 3-dimensional lattice symmetry. The superspace approach has been developed precisely for recovering the space group symmetry of such phases by adding an internal space (see [3] and references therein). The problem is that those space groups have a dimension higher than three. In order to retain the group-subgroup relation (essential in a Landau theory) the high temperature phase is embedded as well (trivially) in a superspace of the same dimension (as that needed for the incommensurate phase) and has accordingly as symmetry group the direct product G0×E, with E the Euclidean group of the d-dimensional internal space. A preliminary paper on the subject has already appeared [4], and a more detailed one will be published elsewhere [5].