Regular phases of quasi-one-dimensional spin systems: Classification and imprints on diffraction

Referring to the propulsive field of low-dimensional magnetism, the paper discusses the foundation and principles of exploitation of the spin line groups, thus establishing a base for full implementation of the symmetry in further studies of frustrated quasi-one-dimensional magnetics. The classification of the symmetry-allowed magnetic phases reveals full diversity of the complex helimagnets, tilted along a singled-out direction, as well as within the cross sections perpendicular to it. The neutron diffraction amplitudes are exhaustively calculated for all the possible spin arrangements, providing experimentally verifiable fingerprints of the particular order of the spin structures. A recent prediction of the gate-voltage-dependent phases of ${}^{13}\text{C}$ carbon nanotubes serves as an illustration of the introduced concepts and their applications in the ground state determination.

Figure 1

Conformation classes ${\mathit{Y}}^{\left(F\right)}$ with examples of regular spin arrangements. Besides class number $F$ a letter of the spin set is given. Connected (in different colors) are the atoms generated from ${\mathit{r}}_0$ (a bit larger) by the group generators; mirror planes and horizontal axis are additionally depicted.

Figure 2

Spin sets for the line groups. For the spin sets ${\mathcal{S}}_{\mathrm{j}}$ and ${\mathcal{S}}_{\mathrm{k}}$ positive and negative cones viewed from above are outer and interior circles.

Figure 3

Spin arrangements of the generic orbit of the first family line group ${\mathit{T}}_{56/9}\left(f\right){\mathit{C}}_4$, labeled by the ordinals in Table 2.

Figure 4

Geometrical factors of the diffraction amplitudes for ${}^{13}\text{C}$ carbon nanotube $(24,4)$: left and right panels correspond to the ground state of the first (or the third) and the second regime. Three pairs of the principal nuclear layer lines (black) are enumerated as 1, $\overline{1}$, 2, $\overline{2}$, and 3, $\overline{3}$, and the satellite spin lines (red and blue) of the first one (bolded) are shown at the corresponding heights $u_z$. The amplitudes within layer lines are indicated by the curves above the lines; grid lines emphasize maximal intensity of the first-principles nuclear layer line.