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

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