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

TL;DR

This study predicts a ferromagnetic material, Cs2MoCl6, exhibiting unique surface-state connections between Weyl and nodal ring fermions, with distinct surface states on different crystal surfaces, analyzed through first-principles calculations and models.

Contribution

It reveals the coexistence and surface-state connection of Weyl and nodal ring fermions in a ferromagnetic material, a rare phenomenon in topological materials.

Findings

Surface states show drumhead and helicoid shapes on different surfaces.

Surface states cross projected points of Weyl and nodal rings along different directions.

Helicoid surface states meet the nodal ring tangentially and change shape with energy.

Abstract

For 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 this paper, we predict a ferromagnetic (FM) material, Cs2MoCl6, with coexistence of Weyl and nodering fermions in its spinful FM electronic band structure, which 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 find that the surface states of Cs2MoCl6 show different properties along different directions, i.e, 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 (T).