Fig. 3: HOFA states in magnetic, nonmagnetic, and fragile topological Dirac semimetals.

a The BZ, c bulk bands, and f hinge bands of a z-directed rod of a \( {\mathcal{T}} \)-broken 3D Dirac semimetal with HOFAs in magnetic SG \( P4/m^{\prime} mm \) (Eq. (5)). The Hamiltonian of each k_z slice describes either a trivial insulator or a QI, and the bulk Dirac points occur at the quantum critical points between the two phases. d HOFAs can also be realized in a closely related \( {\mathcal{T}} \)-symmetric Dirac semimetal in SG 123 \( P4/mmm1^{\prime} \) (Eq. (6)). g Here, the Hamiltonian of the k_z = 0 plane is in the 2D TCI phase¹⁰ shown in Fig. 2f, g, and therefore there is no gap in the hinge-projected surface states at \( \bar{\Gamma } \). e Upon breaking M_z and \( {\mathcal{I}} \) symmetries by adding Eqs. (3)–(6), which reduces the symmetry to SG 99 (\( P4mm1^{\prime} \)), h the face TCI cones gap as they did previously in Fig. 2i–k, resulting in a noncentrosymmetric Dirac semimetal without surface states²¹ whose only topological boundary modes are HOFA states. h Two sets of weakly split HOFAs meet in Kramers pairs at \( \bar{\Gamma } \); if the system filling is fixed to 1∕2 (i.e., to the filling of the Dirac points), then one set of HOFA states in h is half-filled and carries a topological quadrupole moment, and the other set is fully filled, and is topologically trivial. At \( \bar{\Gamma } \), this implies that the Kramers pairs of hinge states are three-quarters filled, and that they exhibit the same quadrupole moment (modulo e) as a QI (Supplementary Note 9). The Hamiltonian of the k_z = 0 plane in e, h exhibits the same fragile topology as the 2D model in Fig. 2i–k (Supplementary Note 4), implying that the hinge states at \( \bar{\Gamma } \) are equivalent to the fractionally charged corner modes of a 2D fragile TI. The model (Eqs. (6) and (3)) shown in e, h therefore represents a previously uncharacterized variant of topological semimetal that carries observable signatures of fragile topology. b The 3D semimetallic phases of Eqs. (5) and (6), whether or not Eq. (3) is additionally present, exhibit the same quadrupole-quantized nested Berry phase²³ θ₂(k_z).