  There has been a significant focus on real topological systems that enjoy
space-time inversion symmetry (PT ) and lack spin-orbit coupling. While the
theoretical classification of the real topology has been established, more
progress has yet to be made in the materials realization of such real
topological systems in three dimensions (3D). To address this crucial issue, by
selecting the carbon-based material candidates as targets, we perform
high-throughput computing to inspect the real topology in the phonon spectrums
of the 3D carbon allotropes in the Samara Carbon Allotrope Database (SACADA).
Among 1192 kinds of 3D carbon allotropes, we find 65 real topological systems
with a phononic real Chern insulating (PRCI) state, 2 real topological systems
with a phononic real nodal line (PRNL) state, 10 real topological systems with
a phononic real Dirac point (PRDP) state, and 8 real topological systems with a
phononic real triple-point pair (PRTPP) state. This extremely expands the
material candidates with real topology, especially for the gapless topological
phonons. We exhibit the PRCI, PRNL, PRTPP, and PRDP states of 27-SG. 166-pcu-h,
1081-SG. 194- 4 2T13-CA, 52-SG. 141-gis, and 132-SG. 191-3,4T157 as
illustrative examples, and explore the second-order boundary mode, i.e.,
phononic hinge mode. Among the four examples, the materials 1081-SG.
194-42T13-CA and 52-SG. 141-gis are so ideal that the PRNL and PRTPP in them
are well separated from other bands, and the phononic hinge mode can be clearly
observed. This study aims to broaden the understanding of 3D topological
phonons, and emphasizes the potential of 3D carbon allotropes as a valuable
framework for exploring the fascinating physics related to phononic hinge modes
and phononic real topology.
