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 $P_3\mathbf{E}·\mathbf{B}$, the axion term in the electrodynamics of the insulator (medium).

Figure 1

Schematic of the loop integrals used in the monodromy proof of the relationship between the Chern number and the eigenvalues at high-symmetry points within the Brillouin zone in $C_n$-invariant topological insulators with $n=2,4,3,6$ in (a), (b),(c), and (d), respectively.

Figure 2

Primitive cells of four 2D Braivis lattices: (a) rectangle, (b) centered-rectangle, (c) square, and (d) triangle. The basis vectors of the real lattice ${\mathbf{a}}_{1,2}$ are plotted as blue arrows and the basis vectors of the reciprocal lattice ${\mathbf{b}}_{1,2}$ are plotted as red arrows. All possible $C_2^{\prime }$ axes are plotted as dashed lines in (a)–(c). In (d), dashed lines and dotted lines represent two sets of $C_2^{\prime }$ axes. $D_3$ PGS includes either one of the two sets and $D_6$ PGS contains both sets.

Figure 3

The phase of $\det \left({\mathcal{B}}_{C_3}\right)$ at six corners of the BZ. ${\lambda }_{1,2,3}$ denote the paths of integration considered in the text.