We study hexagonal spin-channel ("triplet") density waves with commensurate
M-point propagation vectors. We first show that the three Q=M components of
the singlet charge density and charge-current density waves can be mapped to
multi-component Q=0 nonzero angular momentum order in three dimensions (3D)
with cubic crystal symmetry. This one-to-one correspondence is exploited to
define a symmetry classification for triplet M-point density waves using the
standard classification of spin-orbit coupled electronic liquid crystal phases
of a cubic crystal. Through this classification we naturally identify a set of
non-coplanar spin density and spin-current density waves: the chiral spin
density wave and its time-reversal invariant analog. These can be thought of as
3DL=2 and L=4 spin-orbit coupled isotropic β-phase orders. In
contrast, uniaxial spin density waves are shown to correspond to
α-phases. The non-coplanar triple-M spin-current density wave realizes
a novel 2D semimetal state with three flavors of four-component spin-momentum
locked Dirac cones, protected by a crystal symmetry akin to non-symmorphic
symmetry, and sits at the boundary between a trivial and topological insulator.
In addition, we point out that a special class of classical spin states,
defined as classical spin states respecting all lattice symmetries up to global
spin rotation, are naturally obtained from the symmetry classification of
electronic triplet density waves. These symmetric classical spin states are the
classical long-range ordered limits of chiral spin liquids.Comment: 14 + 4 pages, 5 + 0 figures; published versio