  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 bct-C20, as an ideal candidate for realizing
different kinds of surface-semimetal phases. For the (001)surface of bct-C20,
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.
