Stiefel-Whitney classes and topological phases in band theory

We review the recent progress in the study of topological phases in systems with space-time inversion symmetry I-ST. I-ST is an anti-unitary symmetry which is local in momentum space and satisfies I-ST(2) = 1 such as PT in two dimensions (2D) and three dimensions (3D) without spin-orbit coupling and C2T in 2D with or without spin-orbit coupling, where P, T, C-2 indicate the inversion, time-reversal, and two-fold rotation symmetries, respectively. Under I-ST, the Hamiltonian and the periodic part of the Bloch wave function can be constrained to be real-valued, which makes the Berry curvature and the Chern number vanish. In this class of systems, gapped band structures of real wave functions can be topologically distinguished by the Stiefel-Whitney numbers instead. The first and second Stiefel-Whitney numbers w(1) and w(2), respectively, are the corresponding invariants in 1D and 2D, which are equivalent to the quantized Berry phase and the Z(2) monopole charge, respectively. We first describe the topological phases characterized by the first Stiefel-Whitney number, including 1D topological insulators with quantized charge polarization, 2D Dirac semimetals, and 3D nodal line semimetals. Next we review how the second Stiefel-Whitney class characterizes the 3D nodal line semimetals carrying a Z(2) monopole charge. In particular, we explain how the second Stiefel-Whitney number w(2,) the Z(2) monopole charge, and the linking number between nodal lines are related. Finally, we review the properties of 2D and 3D topological insulators characterized by the nontrivial second Stiefel Whitney class.