Multi-Qhexagonal spin density waves and dynamically generated spin-orbit coupling: Time-reversal invariant analog of the chiral spin density wave

Abstract

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 multicomponent 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 noncoplanar 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 4 spin-orbit coupled isotropic β-phase orders. In contrast, uniaxial spin density waves are shown to correspond to α phases. The noncoplanar 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 nonsymmorphic 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.

Engineering chiral density waves and topological band structures by multiple-Qsuperpositions of collinear up-up-down-down orders

“…In Eqs. (16) and (17), (η,η ) = (1,2) for 2Q and (η,η ) = (1,2), (1,3), and (2,3) for 3Q. Figure 3 shows F (4) in Eqs.…”

“…Skyrmion crystals are one of the most prominent examples, where the relativistic spin-orbit coupling plays an important role [7][8][9][10]. Other examples have been extensively discussed for 3d-and 4f -electron compounds, on the basis of the Kondo-type spin-charge coupling on several lattice structures: triangular [11][12][13][14][15][16], honeycomb [16,17], kagome [3,18,19], square [20,21], cubic [22], face-centered-cubic [23], and pyrochlore lattices [24]. In particular, the noncoplanar magnetic orders with ferroic ordering of the scalar chirality have attracted much interest, as the spatially uniform scalar chirality generates a coherent spin Berry phase for itinerant electrons and may lead to a quantized anomalous Hall effect [3,11,12,23].…”

Symmetry analysis of translational symmetry broken density waves: Application to hexagonal lattices in two dimensions

“…This study provides a stepping stone for a classification of M-point spin-triplet orders. A full symmetry classification of hexagonal spin-triplet orders will be presented elsewhere [52].…”

“…Whether the nondegenerate or degenerate level is higher in energy (i.e., the relative ordering ofˆ A andˆ E ) is determined by the sign of s . This can be related to a Ginzburg-Landau expansion of the free energy in terms of the order parameter s , which allows for a cubic term ∼c 3 s [52,53]. As a result, the sign of α determines the sign of s .…”

“…Our analysis shows that this scenario is a generic feature of hexagonal lattice M-point order. Indeed, the present analysis can be generalized to explicitly take the electron spin into account, leading to a unified framework of M-point order [52].…”