Images, Landau expansions, and symmetry changes

The phase diagrams obtained from the fourth-degree Landau expansions associated with the $C_4$ and $C_{4v}$ images are compared. While the first image yields a Landau expansion with a four-dimensional parameter space (${\alpha }_1$,${\alpha }_2$,${\beta }_1$,${\beta }_2$), the second yields a potential which is contained in the three-dimensional subspace with ${\beta }_2$=0. Within this subspace the phase diagrams of both $C_4$ and $C_{4v}$ are identical. Two ordered phases (with the order parameter along either an axis or a diagonal, respectively) are separated by a first-order line. Analysis of symmetry changes involved lead, however, to different results. In the $C_{4v}$ case the two phases have distinct symmetries. On the other hand the $C_4$ image implies identical symmetry in the two ordered phases. Physical interpretation of the phase diagram is thus directly related to the actual image. Experimental consequences are discussed.