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 ${C}_{n}$-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 ${C}_{n}$-invariant insulator is also determined, up to a multiple of $n$, by the ${C}_{n}$ eigenvalue of the Slater determinant of a noninteracting many-body system; and (iii) the Chern number vanishes in insulators with dihedral point groups ${D}_{n}$, 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 ${C}_{n}$, ${C}_{nh}$, and ${S}_{n}$ PGS can have nonzero 3D quantum Hall coefficient and (ii) only insulators with improper rotation symmetries can have quantized magnetoelectric polarization ${P}_{3}$ in the term ${P}_{3}mathbf{E}ifmmodecdotelsetextperiodcenteredfi{}mathbf{B}$, the axion term in the electrodynamics of the insulator (medium).