Topology and Geometry of Spin Origami

Kagome antiferromagnets are known to be highly frustrated and degenerate when they possess simple, isotropic interactions. We consider the entire class of these magnets when their interactions are spatially anisotropic. We do so by identifying a certain class of systems whose degenerate ground states can be mapped onto the folding motions of a generalized “spin origami” two-dimensional mechanical sheet. Some such anisotropic spin systems, including ${\mathrm{Cs}}_2{\mathrm{ZrCu}}_3{\mathrm{F}}_{12}$, map onto flat origami sheets, possessing extensive degeneracy similar to isotropic systems. Others, such as ${\mathrm{Cs}}_2{\mathrm{CeCu}}_3{\mathrm{F}}_{12}$, can be mapped onto sheets with nonzero Gaussian curvature, leading to more mechanically stable corrugated surfaces. Remarkably, even such distortions do not always lift the entire degeneracy, instead permitting a large but subextensive space of zero-energy modes. We show that for ${\mathrm{Cs}}_2{\mathrm{CeCu}}_3{\mathrm{F}}_{12}$, due to an additional point group symmetry associated with the structure, these modes are “Dirac” line nodes with a double degeneracy protected by a topological invariant. The existence of mechanical analogs thus serves to identify and explicate the robust degeneracy of the spin systems.

Figure 1

(a) Mapping from a spin configuration on the star of David to an origami where the spins in the former represent the edge vectors in the latter drawn in dotted lines. (b) A kagome “star of David” with nonuniform interactions which on the exterior bonds satisfy the star condition [Eq. (2)], necessary for a generic spin system to possess an origami analog. The expression of the interior angle ${\theta }_{ij}$ is given in Eq. (3).

Figure 2

(a) Right: The distorted kagome lattice structure of ${\mathrm{Cs}}_2{\mathrm{ZrCu}}_3{\mathrm{F}}_{12}$ with interactions that satisfy the star condition. Left: The origami analog of the $q=0$ state of (a) is a flat sheet ($\mathcal{G}=0$ at each vertex as shown) consisting of isosceles triangles. The dark blue and the light blue faces correspond, respectively, to the blue-black and red-blue triangles of the kagome lattice shown in the right. (b) Left: The distorted kagome lattice structure of ${\mathrm{Cs}}_2{\mathrm{CeCu}}_3{\mathrm{F}}_{12}$ with interactions obeying the star condition. Right: The spin origami for a $q=0$ state is a nonflattenable surface [with finite $\mathcal{G}$ defined in Eq. (5)] with coplanar pairs of triangles that form diamond shapes. The spin configurations for both are denoted by yellow arrows.

Figure 3

(a) Some of the lowest spin-wave frequencies of ${\mathrm{Cs}}_2{\mathrm{CeCu}}_3{\mathrm{F}}_{12}$ as plotted along the high-symmetry path in the BZ shown in the inset and corresponding to the ground state specified by $b_0=0.06$ (see Supplemental Material [28] for the definition of $b_0$). (b) A plot (in log scale) of the lowest frequency (${\omega }_0$) in the BZ reveals the Dirac line nodes.

Figure 4

(a),(b) Dirac line nodes (thick red lines) separating zones of different values of ${\eta }_{+}$ (yellow and blue correspond to $+$ and $-$, respectively) in the spin-wave dispersions of ${\mathrm{Cs}}_2{\mathrm{CeCu}}_3{\mathrm{F}}_{12}$. We study these here for two different ground states (defined by the parameter $b_0$) that represent two members of the one-dimensional family of origami configurations obtained for the periodic state (see Supplemental Material [28]). The insets in (a) and (b) are the plots of ${\eta }_{+}$ over a circle in the BZ shown on the dotted line. The locations of the lines are decided by the condition $\mathrm{Det}[\mathcal{R}(\mathbf{k})]=0$ and depend on $b_0$. (c) Under deformations that break the point group symmetry of ${\mathrm{Cs}}_2{\mathrm{CeCu}}_3{\mathrm{F}}_{12}$, each Dirac line splits into two Weyl lines which are characterized by $\eta$ in Eq. (8). The inset in (c) is the plot of $\eta$ over a circle in the BZ.