Stiefel–Whitney classes and topological phases in band theory

Ahn Junyeong^{1, 2, 3}, Park Sungjoon^{1, 2, 3}, Kim Dongwook⁴, Kim Youngkuk⁴, Yang Bohm-Jung^{1, 2, 3, †}

Topological phase transition from a normal insulator to a 3D weak Stiefel–Whitney insulator. The Hamiltonian in Eq. (45) is used with r = 0.5 and m = 0.9. (a) Shape of the nodal lines formed by touching between a conduction and a valence band (red) and between valence bands (orange). As M increases from −1 to 2.5, a pair of monopole nodal lines are created near (0,0,0), and then they are annihilated near (0,0,π). (b)–(d) Wilson loop operators are calculated along the k_x direction at each value of k_y and k_z. (b) Normal insulator at M = −1. (c) Nodal line semimetal with monopole nodal lines at M = 1. (d) 3D weak Stiefel–Whitney insulator at M = 3.