Pseudo Dirac Nodal Sphere and its Topological Phase Transitions in a Semimetallic Carbon Network

Because the global symmetry requirement is hard to achieved, topological nodal sphere is generally a pseudo phase, in which multiple nodal rings form the spherical backbone while the other points on the sphere are approximately degenerate. The physical understanding of nodal sphere, however, remains unclear. Here, we present a tight-binding (TB) model in a tetragonal crystal to simulate the evolution from a nodal sphere to a nodal surface. To provide clear picture, by performing first-principle calculations, a tetragonal carbon network, in which carbon nanotubes (CNTs) interconnected by graphene nanoribbons, is used to identify the pseudo nodal sphere phase. By applying strains, many topological phase transitions and novel semimetallic phases are achieved. The tensile strains lead to a transition from a nodal sphere to a nodal point, and eventually a trivial insulator; while a compressive strain leads to a transition to a nodal tube (uniaxial strains) or a nodal crossbar (biaxial strain), eventually to nodal surfaces, which is in good agreement with our TB modeling. Our results suggest that nodal sphere and nodal surface share the same origin, and their morphologies depend on the inter-CNTs hopping.