D Orbital Topological Insulator and Semimetal in the Antifluorite Cu₂S Family: Contrasting Spin Helicities, Nodal Box, and Hybrid Surface States

We reveal a class of three-dimensional d orbital topological materials in the antifluorite Cu₂S family. Derived from the unique properties of low-energy t_2g states, their phases are solely determined by the sign of the spin-orbit coupling (SOC): topological insulator (TI) for negative SOC and topological semimetal for positive SOC, both having Dirac cone surface states but with contrasting helicities. With broken inversion symmetry, the semimetal becomes one with a nodal box consisting of butterfly-shaped nodal lines that are robust against SOC. Further breaking the tetrahedral symmetry by strain leads to an ideal Weyl semimetal with four pairs of Weyl points. Interestingly, the Fermi arcs coexist with a surface Dirac cone on the (010) surface, as required by a Z2 invariant.