Two-dimensional (2D) carbon nitride materials play an important role in energy-harvesting, energy-storage and environmental applications. Recently, a new carbon nitride, 2D polyaniline (C3N) was proposed [PNAS 113 (2016) 7414-7419]. Based on the structure model of this C3N monolayer, we propose two new carbon nitride monolayers, named dumbbell (DB) C4N-I and C4N-II. Using first-principles calculations, we systematically study the structure, stability, and band structure of these two materials. In contrast to other carbon nitride monolayers, the orbital hybridization of the C/N atoms in the DB C4N monolayers is sp3. Remarkably, the band structures of the two DB C4N monolayers have a Dirac cone at the K point and their Fermi velocities are comparable to that of graphene. This makes them promising materials for applications in high-speed electronic devices. Using a tight-binding model, we explain the origin of the Dirac cone. ; Comment: 21 pages, 5 figures