Application of Morse theory to the symmetry breaking in the Landau theory of second order phase transition

Abstract

We treat here the case of all irreps (irreducible representations) on the reals of the 32 point groups. For each point group these irreps are irreps with wave vecor k = 0 of the corresponding space groups. Landau model of second order phase transition can be applied to those irreps with no third degree invariants : one has to look for minima of a bounded below fourth degree polynomial which is not minimum at the origin, and determine the little groups (= isotropy groups) of these minima ; they are the subgroups into which the symmetry can be broken in the transition. By an efficient strategy we reduce the study of the 153 equivalence classes of irreps to few cases (6). Moreover we do not need to study the minima of invariant polynomials, we simply apply Morse theory to find the possible little groups of minima.