Multi-$Q$ hexagonal spin density waves and dynamically generated spin-orbit coupling: time-reversal invariant analog of the chiral spin density wave

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 $3D$ $L=2$ and $L=4$ spin-orbit coupled isotropic $\beta$-phase orders. In contrast, uniaxial spin density waves are shown to correspond to $\alpha$-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.