Novel topological states of nodal points and nodal rings in 2D planar octagon TiB₄

Topological states of matter in two-dimensional (2D) materials have received increasing attention due to their potential applications in nanoscale spintronics. Here, we report the presence of unique topological electronic properties in a 2D planar octagon TiB₄ compound. Particularly, without considering the spin-orbit coupling (SOC), we found that the material showed a coexistence of novel quadratic node (QN), and two different types of nodal rings (NRs), namely type-I and type-II. The protection mechanism of fermions has been fully clarified in this study. Furthermore, these fermions showed clear edge states. It is worth noting that QN had a topological charge of 2 since it is different from linear nodes and exhibit clear Fermi arc edge states. Under lattice strain, we found that the system could further exhibit rich topological phase transition. When SOC was included, we determined that these crossing points open very tiny energy gaps, which were smaller than previously reported 3D and 2D examples. These results show that monolayer TiB₄ is an excellent nodal point and nodal ring semimetal, which also provides a feasible member for studying potential entanglements among multiple fermions.