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We discuss the proximate phases of a three-dimensional system with Dirac-likedispersion. Using the cubic lattice with plaquette $\pi$-flux as a model, wefind, among others phases, a chiral topological insulator and singlettopological superconductor. While the former requires a special "chiral"symmetry, the latter is stable as long as time reversal and SU(2) spin rotationsymmetry are present. These phases are characterized by stable surface Diracfermion modes, and by an integer topological invariant in the bulk. The keyfeatures of these phases are readily understood in a two dimensional limit withan appropriate pairing of Dirac nodes between layers. This Dirac node-pairingpicture is also shown to apply to $Z_2$ topological insulators protected bytime-reversal symmetry (TRS). The nature of point-like topological defects inthese phases is also investigated, revealing an interesting duality relationbetween these topological phases and the Neel phase.

Chiral Topological Insulators, Superconductors and other competing orders in three dimensions