Comment on "Irrelevant variables, Landau expansions, and cubic anisotropy."

The relationship of the XY model, which arises from a C4 " image, to the phase transition in the rare-earth molybdate Tb2(Mo04)3 (TMO) is discussed. We point out that this model does not describe the TMO transition, but that a Landau expansion arising from a C4 image is the appropriate free-energy model. Recently, Galam' considered phase transitions occurring in the XY model with cubic anisotropy. He emphasized, in a quite thorough discussion, that the extension of the Lan-dau expansion to higher orders may significantly affect the resulting phase diagram. For the XY model, $s terms generate a new symmetry breaking and the generic phase appears as a possible lower-symmetry phase. ' Here we use the term "generic phase" to characterize the phase which has maximum symmetry breaking, i.e. , the symmetry group mapped onto the identity matrix by the homomorphism. Higher-order terms beyond those of eighth order are "ir-relevant" in that they do not introduce additional phase symmetries. Within the Landau theory of phase transitions the free-energy expansion is constructed as an invariant function of the representation basis. It is usually assumed that the transition is driven by a single, multicomponent order parameter — the vector of an irreducible representation (irrep). The cubic XY model occurs in the description of many crystal systems (see Ref. 2 for an example of the XI'model in surface transitions). It is defined by the representation of a space group whose complete set of distinct representation matrices (image) is isomorphic to the point group in two dimensions C4 ". Thus the set of symmetry transformations of the crystal map homomorphically onto a set of two-dimensional representation matrices equivalent to the usual vector representation of C4 " , namely, mrs-(It f), C4,-(f)), etc. The Landau expansion for the C4 " image is then an invariant polynomial expansion in terms of the order-parameter components. For C4 " the expansion can be constructed3 as a polynomial expansion of the two basic in-variants Ii-r2 and I 2r4c s(os). To eighth degree the free energy takes the form F X u If'+vtI2+ wiItI2+yil/I2+ y2IJ Here the terms with coefficients ~i, wi, and yl are anisotro-pic terms of fourth, sixth, and eighth degree, respectively. This form is equivalent to that used by Galam' and defines the XY model. As mentioned, the XY model is the polynomial expansion which is invariant under the order-parameter symmetry (image group) C4 ". The C4 " …