Dirac loops in trigonally connected 3D lattices

We consider different generalizations of the honeycomb lattice to three dimensional structures. We address the family of the hyper-honeycomb lattice, which is made up of alternating layers of 2D honeycomb nano-ribbons, with each layer rotated by $\pi/2$ with the layer below. When the orbitals of the lattice sites are symmetric with respect to the planes of the trigonal links, these structures can produce a Dirac loop, a closed line of Dirac nodes in momentum space. For orbitals that break that symmetry, such as the carbon $p$-wave orbitals, hyper-honeycomb lattices do not possess the loop structure. We also consider a new structure, the screw hyper-honeycomb, in which the successive layers of parallel units are rotated by $2\pi/3$. This structure has a Dirac loop if reflection symmetry in the unit cell is imposed, regardless the symmetry of the onsite orbitals. We discuss the implementation of those systems in optical lattices.