Bulk topological invariants in noninteracting point group symmetric insulators

We survey various quantized bulk physical observables in two- and three-dimensional topological band insulators invariant under translational symmetry and crystallographic point group symmetries (PGS). In two-dimensional insulators, we show that (i) the Chern number of a C₍-invariant insulator can be determined, up to a multiple of n, by evaluating the eigenvalues of symmetry operators at high-symmetry points in the Brillouin zone; (ii) the Chern number of a C₍-invariant insulator is also determined, up to a multiple of n, by the C₍ eigenvalue of the Slater determinant of a noninteracting many-body system; and (iii) the Chern number vanishes in insulators with dihedral point groups D₍, and the quantized electric polarization is a topological invariant for these insulators. In three-dimensional insulators, we show that (i) only insulators with point groups C₍, C₍₇, and S₍ PGS can have nonzero 3D quantum Hall coefficient and (ii) only insulators with improper rotation symmetries can have quantized magnetoelectric polarization P₃ in the term P₃E, the axion term in the electrodynamics of the insulator (medium).