Symmetry-protected nodal points and nodal lines in magnetic materials

Abstract

Nodal-point and Nodal-line structures in the dispersion of electron energy bands are characterized by their high degeneracy in certain corners or lines in the Brillouin zone (BZ). These nodal structures can also exist in the dispersion of itinerant electrons in magnetically ordered materials whose symmetry groups are anti-unitary groups called the magnetic space groups. In the present work, we provide a complete list of magnetic space groups which can host symmetry-protected nodal-point/line band structures for spin-1/2 fermionic particles, where the degeneracies at the nodal points/lines are guaranteed by irreducible projective representations (IPReps) of the little co-groups. Our discussion is restricted to the magnetic space groups whose magnetic point group contains the space-time inversion operation $\tilde T=\mathcal IT$, the combined operation of spacial inversion $\mathcal I$ and time reversal $T$, such that the energy bands are at least doubly-degenerate at arbitrary points in the BZ. For these magnetic point groups we provide the invariants to label the classes of projective Reps, and for each class we calculate all the inequivalent IPReps. From the results we select out all the groups and the corresponding Rep classes which support high-dimensional ($d\geq$4) IPReps. We then list the magnetic space groups and their high symmetry points/lines whose little co-groups have high-dimensional ($d\geq$4) IPReps with the corresponding factor systems. Examples of candidate materials are discussed.