Bulk Topological Invariants in Noninteracting Point Group Symmetric Insulators

TL;DR

This paper systematically explores bulk topological invariants in noninteracting insulators with point group symmetries, revealing how symmetry eigenvalues determine topological properties like Chern numbers and polarization in 2D and 3D materials.

Contribution

It provides a comprehensive survey of how point group symmetries influence quantized bulk invariants and introduces methods to compute these invariants from symmetry eigenvalues.

Findings

Chern number determined by symmetry eigenvalues at high-symmetry points

Chern number vanishes for dihedral point groups Dn

Quantized magnetoelectric polarization P3 occurs only with improper rotation symmetries

Abstract

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