Theory-Guided Design and Machine-Learning Identification of a Nested Nodal-Surface Semimetal in the Carbon Allotrope bct-C12

Extending the Dirac physics of two-dimensional graphene into three-dimensional (3D) carbon allotropes with higher-dimensional band degeneracies remains a central challenge in topological materials science. Here, we propose a general symmetry-engineering principle that systematically transforms graphene’s Dirac cone into a 3D nodal surface via controlled layering and registry shift, and employ this principle to guide a machine-learning-accelerated inverse design. By integrating a crystal diffusion variational autoencoder (CDVAE) with CrystalFormer, we identify a body-centered tetragonal carbon structure (previously reported as bct-C12) which we also name Netsene. First-principles calculations reveal that this known material hosts a previously unrecognized nested nodal-surface system near the Fermi level, protected by non-symmorphic symmetries, alongside Dirac-like linear crossings with Fermi velocities comparable to graphene. Its non-trivial bulk topology, verified by Wilson-loop and Berry-phase calculations, is expected to manifest in drumhead surface states, including a nearly flat band. Netsene thus provides a structurally robust, bulk platform that combines high Fermi velocities comparable to graphene, topological nodal surfaces, and a high density of states that may foster correlation effects worthy of future study, demonstrating the power of theory-guided, machine-learning-accelerated identification for engineering exotic topological quantum phases.