Ideal Intersecting Nodal Ring Phonons in a Body-Centered Cubic C$_{8}$

Carbon, a basic versatile element in our universe, exhibits rich varieties of allotropic phases, most of which possess promising nontrivial topological fermions. In this work, we identify a distinct topological phonon phase in a realistic carbon allotrope with a body-centered cubic structure, termed bcc-C$_{8}$. We show by symmetry arguments and effective model analysis that there are three intersecting phonon nodal rings perpendicular to each other in different planes. The intersecting phonon nodal rings are protected by time-reversal and inversion symmetries, which quantize the corresponding Berry phase into integer multiples of $pi$. Unlike the electron systems, the phonon nodal rings in bcc-C$_{8}$ are guaranteed to remain gapless due to the lack of spin-orbital coupling. The nearly flat drumhead surface states projected on semi-infinite (001) and (110) surfaces of bcc-C$_{8}$ are clearly visible. Our findings not only discover promising nodal ring phonons in a carbon allotrope, but also provide emergent avenues for exploring topological phonons beyond fermionic electrons in carbon-allotropic structures with attractive features.