Topological Insulators and Semimetals with Point Group Symmetries*

In this work, we study the theory of topological phases in systems with point group symmetries (PGSs) in one-, two- and three-dimension. The systems we study in general do not require time-reversal symmetry, and hence may be realized in both non-magnetic and magnetic materials. We show that a point group symmetry introduces new quantum numbers which reveal themselves in the entanglement spectrum as mid-gap states. PGSs also define a series of topological semimetals, in which the band touching points are protected by certain symmetries. We apply our theory to 3D ferromagnetic semimetal HgCr$_2$Se$_4$ which possesses a double-vortex band crossing protected by $C_4$ rotation symmetry.