Fig. 2: TCI, fragile, and QI phases in 2D insulators with p4m symmetry.

a \( {\mathcal{T}} \)-symmetric layer group \( p4/mmm1^{\prime} \) reduces to type-III magnetic layer group⁶² \( p4/m^{\prime} mm \) under the application of a magnetic potential with no net magnetic moment in each unit cell (Supplementary Note 3); this can be achieved by placing spin-1∕2 magnetic moments (red arrows in a) at the 4d Wyckoff position with orientations related by C_4z and M_x,y. b BZ and c bulk bands for a model (Eq. (1)) that respects \( p4/mmm1^{\prime} \) (Table 1), which has M_z and \( {\mathcal{T}} \) symmetries, as well as the symmetries of \( p4/m^{\prime} mm \) in a; this model is constructed from s and \( {d}_{{x}^{2}-{y}^{2}} \) orbitals at the 1a Wyckoff position. Eq. (1) can be tuned between a trivial and a mirror TCI phase (f), distinguished by their (d, g) ribbon edge spectra and (e, h) x-directed Wilson loops plotted as functions of k_y (Eq. (2)). i Relaxing M_z while preserving C_2z and \( {\mathcal{T}} \) by introducing Eq. (3) to Eq. (1), (k) we realize a four-band model with the same Wilson loop winding as the 2D TCI phase in f–h, but (j) without topological edge states. The Wilson loop in k can either be trivialized by the addition of more orbitals to the model (Supplementary Note 4), or gapped by the magnetic potential depicted in a. l Upon gapping (m) the surface and (n) Wilson bands with magnetism that breaks M_z while preserving magnetic wallpaper group^28,62 p4m, which we accomplish by adding the potential in Eq. (4), the Wannier centers of the topological phases in f–k localize at the 1b position (Supplementary Note 3), realizing an insulator topologically equivalent to the QI introduced in ref. ²³ (Supplementary Note 2).