Possible ground states of D-wave condensates in isotropic space through geometric invariant theory

“The problem of classifying spontaneous symmetry breaking (SSB) patterns in theories where the ground state is determined as a minimum of a potential invariant under the action of a compact group G of transformations is relevant both in elementary particle physics and in solid state physics. “Even if trivial in principle, the concrete determination of minima of G-invariant potentials is, generally, a difficult task, owing to the degeneracies of the extremal points. A geometric approach, based on the analysis of local properties of the G-spaces, has been devised to exploit the invariance properties of the potential. For many years, the study of the lattice of the G-space isotropy subgroups, complemented by the famous Michel conjecture, was used to determine the residual symmetry after SSB. Independently, in 1971, during the first years of the development of the G-space approach, Yu. Gufan [Fiz. Tverdogo Tela 13 (1971), no. 1, 225–231] proposed the use of a fundamental system of polynomial invariants (integrity bases) to write the most general form of Landau non-equilibrium potential. But it was in 1981, when counterexamples to Michel’s conjecture began to be discovered, that a newrigorous method, fully exploiting geometric invariant theory, was proposed [M. Abud and G. Sartori, Phys. Lett. B 104 (1981), no. 2, 147–152; MR0627570 (83d:81059); Ann. Physics 150 (1983), no. 2, 307–372; MR0724117 (85i:58083)]. It was demonstrated that the range of a set of basic polynomial invariants yields an isomorphic image S of the orbit space. A clear method (hereafter called the P-matrix or orbit space approach), founded on a sounder mathematical analysis, was discovered to construct S. “In solid state physics, the analytical precision of the approach seemed not to be essential to treat second-order phase transitions. Actually, the P-matrix approach appeared to be complicated in real cases, for high cardinality or high-degree integrity bases. Some results were obtained with numerical techniques, artificially truncating the non-equilibrium Landau potential in order that just the lower-degree invariants appeared in the expansion. “In the following years, it was also proved that the P-matrix approach allows one to get, at least in principle, a model-independent classification of the theoretically admissible SSB schemes. “This communication aims at demonstrating that the geometric method may be successfully employed in a real problem: the study of the possible ground states of a D-wave condensate. “A complete and rigorous determination of the possible ground states for D-wave pairing Bose condensates is presented. Using an orbit space approach to the problem, we find 15 allowed phases (besides the unbroken one), with different symmetries, that we thoroughly