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 IST.ISTis an anti-unitary symmetry which is local in momentum space and satisfies I2 ST= 1 such as PT in two dimensions(2 D)and three dimensions(3 D) without spin–orbit coupling and C2 T in 2 D with or without spin–orbit coupling, where P, T, C2 indicate the inversion, time-reversal, and two-fold rotation symmetries, respectively. Under IST, 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 w1 and w2, respectively,are the corresponding invariants in 1 D and 2 D, which are equivalent to the quantized Berry phase and the Z2 monopole charge, respectively. We first describe the topological phases characterized by the first Stiefel–Whitney number, including1 D topological insulators with quantized charge polarization, 2 D Dirac semimetals, and 3 D nodal line semimetals. Next we review how the second Stiefel–Whitney class characterizes the 3 D nodal line semimetals carrying a Z2 monopole charge.In particular, we explain how the second Stiefel–Whitney number w2, the Z2 monopole charge, and the linking number between nodal lines are related. Finally, we review the properties of 2 D and 3 D topological insulators characterized by the nontrivial second Stiefel Whitney class.

Magnetic wallpaper Dirac fermions and topological magnetic Dirac insulators

Topological crystalline insulators(TCIs)can host anomalous surface states which inherits the characteristics of crystalline symmetry that protects the bulk topology.Especially,the diversity of magnetic crystalline symmetries indicates the potential for novel magnetic TCIs with distinct surface characteristics.Here,we propose a topological magnetic Dirac insulator(TMDI),whose two-dimensional surface hosts fourfold-degenerate Dirac fermions protected by either the p'_(c)4mm or p40 g0 m magnetic wallpaper group.The bulk topology of TMDIs is protected by diagonal mirror symmetries,which give chiral dispersion of surface Dirac fermions and mirror-protected hinge modes.We propose candidate materials for TMDIs including Nd_(4)Te_(8)Cl_(4)O_(2)0 and DyB_(4) based on first-principles calculations,and construct a general scheme for searching TMDIs using the space group of paramagnetic parent states.Our theoretical discovery of TMDIs will facilitate future research on magnetic TCIs and illustrate a distinct way to achieve anomalous surface states in magnetic crystals.