Abundant surface-semimetal phases in three-dimensional obstructed atomic insulators

Three-dimensional obstructed atomic insulators (OAIs) are characterized by the appearance of floating surface states (FSSs) at specific surfaces. Benefiting from this feature, our study here shows the presence of abundant surface-semimetal phases in 3D OAIs. The symmetries of obstructed Wannier charge centers ensure the degeneracy of such FSSs at high-symmetry points or invariant lines in the surface Brillouin zone. Utilizing topological quantum chemistry theory, we identify a carbon allotrope with a body-centered tetragonal structure, named $\mathrm{bct}\text{\ensuremath{-}}{\mathrm{C}}_{20}$, as an ideal candidate for realizing different kinds of surface-semimetal phases. For the (001) surface of $\mathrm{bct}\text{\ensuremath{-}}{\mathrm{C}}_{20}$, there are four in-gap FSSs, and these four FSSs form two kinds of surface Dirac cones, i.e., topological Dirac cones with linear dispersion and symmetry-enforced quadratic Dirac cones. The band topology of a surface Dirac cone is captured by the effective surface Hamiltonian and the emergence of hinge states. Moreover, the existence of the surface-nodal-line state is also discussed. This work reports an approach to obtain $d$-dimensional semimetal phases from the surface states of $(d+1)$-dimensional systems, which is of great significance for the studies in revealing topological states and their practical applications in high-dimensional crystals.