  The study of topological phases with unconventional bulk-boundary
correspondences and nontrivial real Chern number has garnered significant
attention in the topological states of matter. Using the first-principle
calculations and theoretical analysis, we perform a high-throughput material
screening of the 3D obstructed atomic insulators (OAIs) and 3D real Chern
insulators (RCIs) based on the Samara Carbon Allotrope Database (SACADA).
Results show that 422 out of 703 3D carbon allotropes are 3D OAIs with multiple
bulk-boundary correspondences, including 2D obstructed surface states (OSSs)
and 1D hinge states, which are in one dimension and two dimensions lower than
the 3D bulk, respectively. The 2D OSSs in these OAIs can be modified when
subjected to appropriate boundaries, which benefits the investigation of
surface engineering and the development of efficient topological catalysts.
These 422 OAIs, which have 2D and 1D boundary states, are excellent platforms
for multi-dimensional topological boundaries research. Remarkably, 138 of 422
OAIs are also 3D RCIs, which show a nontrivial real topology in the protection
of spacetime inversion symmetry. Our work not only provides a comprehensive
list of 3D carbon-based OAIs and RCIs, but also guides their application in
various aspects based on multiple bulk-boundary correspondences and real
topological phases.
