Stiefel–Whitney classes and topological phases in band theory
Ahn Junyeong1, 2, 3, Park Sungjoon1, 2, 3, Kim Dongwook4, Kim Youngkuk4, Yang Bohm-Jung1, 2, 3, †
       

A torus T2 wrapping the nodal line γ1 in the Brillouin zone. T2 is thin enough so that it does not intersect any other band degeneracies. Red lines (γ1 and γ6) are lines of touching between the conduction and valence bands. Orange (γ3 and γ4) and green (γ2 and γ5) lines are lines of touching between the first and the second topmost occupied bands and between the second and the third topmost occupied bands, respectively. Only the linking between γ1 and γ3 is protected as it generates the nontrivial second Stiefel–Whitney number on T2, whereas the linking between γ1 and γ5 is not protected [See Eq. (33)]. Figures are adopted from the supplemental materials in Ref. [7].