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

Abstract

For topological materials with a coexistence of Weyl nodes and nodal rings, their unique surface-state configuration and connection still need to be studied and discussed. In this Letter, we predict a ferromagnetic (FM) material, Cs2MoCl6, with a coexistence of Weyl and nodal ring fermions in its spinful FM electronic band structure, which is unusual since FM materials are very rare in nature and nodal ring band crossings will usually open a gap when spin-orbit coupling is taken into consideration. We find that the surface states of Cs2MoCl6 show different properties along different directions, i.e., the surface states are in a drumhead shape showing the nodal ring property on the (001) surface and in a 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 changing abruptly as a function of the energy. We implement both first-principles calculation and an analytical model to understand the unique surface-state connection for systems with a coexistence of Weyl nodes and nodal rings (or nodal lines). This result is universal and irrespective of the presence or absence of time-reversal symmetry (T).