Classification of continuous phase transitions and stable phases. I. Six-dimensional order parameters.

It is well known that Landau's phenomenological description' of a continuous phase transition is very useful in classifying syminetry changes which can take place during such a transition. A good illustrative example is given by Lyubarskii. The procedure can be summarized as follows: (1) Find the space group 6 of the high-symmetry phase of a physical system (crystal, alloy, etc.) and the irreducible representation ("irrep") I' of the order parameter vi responsible for the transition. The irrep should comply with the Landau and Lifshitz conditions' for a continuous commensurate phase transition (referred to as an active irrepz). (2) Find a set of basic invariant polynomials in iI up to degree four. (3) Minimize the Landau potential consisting of this basic set of invariants and thus dettumine vis corresponding to the absolute minimum. (4) Find the subgroup H of 6 that leaves fixed the order-parameter components iso at the absolute minimum found in step (3). Step (3) is excessively laborious if carried out before step (4). Also, often one only needs to know the possible symmetries instead of the exact solution to {3). Birman proposed that step (4) should be carried out before step {3) is attempted. If a subgroup H af the space group 6 is ta be a symnltry group of the lower symmetry phase after a transition, the irrep I af 6 should subduce the identity representation of H at least once. (The number of times the identity representation occurs in the subduced representation is called the subduction frequency. ) This is called the subduction criterion. The group H is then called an "isotropy subgroup" of the representation I'. The chain subduction criterion further selects a minimal set of all "isotropy subgroups. " If two "isotropy subgroups" Hi and H2 (Hi &H2) have the same subduction frequency, then H2 is not eligible for any transition. That is, we need to consider only the largest "isotropy subgroup" in a chain of "isotropy subgroups" with the same subduction frequency. Normally there are many distinct chains at a given subduction frequency. The selection procedure needs to be carried out for all chains of "isotropy subgroups" at each subduction frequency, 6 starting from subduction frequency one down to the dimension of the order parameter. Good examples are given in Refs. 6 and 7. Since smaller "isotropy subgroups" do not appear in any physical context, the term isotropy subgroup is used in the literature for the largest "isotropy subgroup" in a given chain. The term maximal isotropy subgroup is used to label a member of the set of "high-level" isotropy subgroups of low index that are disjoint from each other and are supergroups of "lower level" ones of high index having higher subduction frequencies. Many authors overlooked the direction of the order parameter that is left fixed by the elements af an isotropy subgroup (called the H-invariant vector of the isotropy subgroup). Vinberg et al. made an extensive table, based on the abave selection, of possible phase transitions in crystals with space group Ot, (Pm 3m). In addition to listing isotropy subgroups, they included the invariant vector corresponding to each isotrapy subgroup. The minimization procedure is greatly facilitated if the full information of the isotropy subgroups and their invariant vectors is available in advance. In Kim's minimization method, instead of working with order-parameter components, one treats each independent invariant polynomial as a variable. Furthermore the dependence of the potential on the radial and directional parts of sI are separated. The directional minima of a low-degree Landau potential are obtained readily as a function of directions. ' Then the absolute minimum is the lowest of all directional minima in the space of directions. The method utilizes already known geometrical properties of the orbit space. " An orbit' is the set of all representation vectors (order-parameter vectors in our case) that are connected by group elements. All vectors on an orbit yield the same numerical value for a groupinvariant function and the isotropy subgroups leaving the vectors invariant are conjugate to each other. Thus a group-invariant function should be regarded as a function af the orbits rather than that of individual vectors. Conversely, an orbit can be completely specified by a set of I algebraically independent invariant polynomials. The number I is the dimension n of the representation vector