Fig. 5: HOFA states and fragile Dirac semimetal phase in β′-PtO₂.

a Bulk bands incorporating the effects of SOC of the candidate Dirac semimetal β′-PtO₂ in SG 136 (\( P{4}_{2}/mnm1^{\prime} \))^60,61. Unlike in the Dirac semimetals α″-Cd₃As₂ and KMgBi examined in Fig. 4, the k_z = 0 plane of β′-PtO₂ is equivalent to a 2D TCI with mirror Chern number \( {C}_{{M}_{z}}=2 \) (ref. ⁶¹ and Supplementary Note 13). b The hinge spectrum of β′-PtO₂ exhibits two narrowly split HOFA states connecting the hinge projections of the bulk 3D Dirac points (white) to the projections of two surface TCI cones at k_z = 0 (blue). Fixing the system filling to that of the bulk Dirac points, we find that the lower HOFA state in energy in (b) is half-filled, and therefore exhibits a topological quadrupole moment, and the higher state is unoccupied, and is thus topologically trivial, as discussed in Supplementary Notes 9 and 13. c In the presence of a z-directed external electric field, the surface TCI cones in β′-PtO₂ become gapped (blue), allowing for the two HOFA states to meet at k_z = 0 in a quarter-filled Kramers pair of corner modes that is characteristic of the fragile phase introduced in this work (Figs. 2i–k, 3e, h, and Supplementary Notes 4, 9, and 13). The bulk band structure in (a) was obtained from first-principles, and then used to fit a tight-binding model whose hinge Green's functions in the absence and presence of an external electric field are shown in (b) and (c), respectively (Supplementary Note 13).