Fully linear band crossings at high symmetry points in layers: classification and role of spin–orbit coupling and time reversal

Abstract

Symmetry imposed restrictions to the Hamiltonian are systematized and applied to all of 80 clusters of single/double ordinary/gray groups (320 groups in total), to single out linear (in all directions) band crossings and corresponding effective Hamiltonians in high-symmetry Brillouin zone points of layered materials. The resulting dispersion types are isotropic or anisotropic forms of: single cone (with double degenerate crossing point and non-degenerate branches, or four-fold degenerate crossing point with double degenerate conical branches), poppy-flower (four-fold degenerate crossing point with two pairs of non-degenerate mutually rotated conical branches), and fortune teller (with nodal lines). Further, we describe the nontrivial patterns of dispersions’ behavior in high symmetry points when symmetry is varied within a cluster. Namely, Clebsch–Gordan series of the products of spin representation with the integer ones are relevant when spin–orbit coupling is included, and clarify observed scenarios (gap closing, gap opening, cone preserving, cone splitting etc). Analogously, analysis of behavior of dispersions in transition from ordinary to gray group enlightens the role of time reversal symmetry. The results refine and expand data existing in literature, and interesting or even unexpected cases are singled out in discussion.