Color degeneracy of competing orders near topological defect cores in planar quadratic band touching systems

We study two-dimensional fermionic quadratic band touching (QBT) systems in presence of vortex and skyrmion of insulating and superconducting masses. A example of such systems is the Bernal bilayer graphene that eight zero-energy modes in the presence of a mass vortex with the U (1) symmetry. Inside the vortex core, additional ten masses that an SO (5) algebra can develop local expectation values by splitting the modes in five and ten different ways by lifting its SO (4) and SU (2) chiral, respectively. In particular, each SU (2) chiral symmetry can be by three distinct copies of chiral-triplet mass orders, giving rise to notion of the color or flavor degeneracy among the competing orders. By, a skyrmion of three anticommuting masses supports additional six in its core, and possesses an SU (2) isospin quantum number, besides the generalized U (1) charge. Consequently, charge 4e Kekule-density-waves can develop in the skyrmion core of N\\'eel layer, while a skyrmion of quantum spin Hall insulator in addition a mundane s-wave pairing. We also analyze the internal algebra of orders in the core of these defects on checkerboard or Kagome lattice supports only a single copy of QBT.