Unique surface-state connection between Weyl and nodal ring fermions in ferromagnetic material Cs₂MoCl₆

 Surface states connection for Weyl/Dirac/nodal line/ring semimetals have been widely studied, yet the one for systems with the coexistence of Weyl points and nodal line/ring lacks detailed research. We offer a ferromagnetic (FM) material as a candidate to study, and build k・p model for further analysis. In the case of topological materials with coexistence of Weyl nodes and nodal rings, the surface-state configuration and connection are unique yet have never been studied and discussed before.

 In our presentation, we predict a FM material, Cs₂MoCl₆, with coexistence of Weyl and node-ring fermions in its spinful FM electronic band structure. It is unusual since FM materials are very rare in nature and node-ring band crossings will usually open a gap when spin-orbit coupling (SOC) is taken into consideration. We found that the surface states of Cs₂MoCl₆ show different properties along different directions, e.g, the surface states are in the drumhead shape showing the node-ring property on the (001) surface and in the helicoid shape showing the Weyl property on the (010) surface. Interestingly, both the drumhead surface states and the helicoid surface states will cross the projected points of the Weyl and nodal ring along different directions. In particular, helicoid surface states on the (010) surface will meet the nodal ring tangentially, with their shapes change abruptly as a function of the energy. We implement both first-principle calculation and an analytical model to understand the unique surface-state connection for systems with the coexistence of Weyl nodes and nodal rings (or nodal lines). This result is universal and irrespective of the presence/absence of and time-reversal symmetry.