Unified theory of spiral magnetism in the harmonic-honeycomb iridates $\alpha ,\beta$, and $\gamma {\mathrm{Li}}_2{\mathrm{IrO}}_3$

A family of insulating iridates with chemical formula ${\mathrm{Li}}_2{\mathrm{IrO}}_3$ has recently been discovered, featuring three distinct crystal structures $\alpha ,\beta ,\gamma$ (honeycomb, hyperhoneycomb, stripyhoneycomb). Measurements on the three-dimensional polytypes, $\beta$- and $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$, found that they magnetically order into remarkably similar spiral phases, exhibiting a noncoplanar counter-rotating spiral magnetic order with equivalent $q=0.57$ wave vectors. We examine magnetic Hamiltonians for this family and show that the same triplet of nearest-neighbor Kitaev-Heisenberg-Ising ($KJI$) interactions reproduces this spiral order on both $\beta$- and $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ structures. We analyze the origin of this phenomenon by studying the model on a one-dimensional zigzag chain, a structural unit common to the three polytypes. The zigzag-chain solution transparently shows how the Kitaev interaction stabilizes the counter-rotating spiral, which is shown to persist on restoring the interchain coupling. Our minimal model makes a concrete prediction for the magnetic order in $\alpha \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$.

Figure 1

Lattices of Ir in $\alpha$-, $\beta$-, $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$, with parent orthorhombic $a,b,c$ axes. Experiments on the 3D lattices, $\beta$- and $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$, found strikingly similar spiral orders.

Figure 2

Phase diagrams on $\alpha$-, $\beta$-, $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$. In the vicinity of the spiral phase (shaded blue) which contains the experimentally observed magnetic order, the semiclassical phase diagrams appear remarkably similar across the $\alpha$-, $\beta$-, $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ lattices. (a) The nearest-neighbor $KJ{I}_c$ model ($J_2=0$) is sufficient for capturing the observed spiral, and exhibits this cross-lattice similarity. (b) (Left) the spiral from the 1D zigzag chain model persists to the full lattices; (right) taking $J_2\to 0$ requires large $\left|K\right|/J$; see parameters below. For the 2D $\alpha$ polytype, shading indicates the equivalent spiral $\mathit{q}$ along $\mathit{a}$ as described in the text.

Figure 3

Zigzag chain and spiral. As evident in this 1D minimal model for the ${\mathrm{Li}}_2{\mathrm{IrO}}_3$ lattices (top left), the counter-rotating coplanar spiral order can be stabilized by Kitaev interactions within the coplanar ansatz Eq. (3) (bottom left; here with $K<0$, $J=\left|K\right|/3$). For each lattice, restoring the interchain couplings preserves the counter-rotating $S^a,S^c$ spiral (top right), while also introducing noncoplanar $S^b$ components (overlayed in blue, bottom right). Together they form the experimentally observed order.

Figure 4

Visualization of geometry of Kitaev and Ising exchanges. The two neighboring Ir sites (purple spheres), with surrounding oxygens (vertices of octahedra), are shown. The oxygen octahedra of neighboring Ir sites are edge sharing in these structures. The axes for the anisotropic interaction terms are then determined as follows [see the discussion following Eq. (1) of the main text for details]. The Ising interaction axis $\vec{r}$ is the vector connecting the two Ir sites. The Kitaev interaction axis $\gamma$ is perpendicular to the plane which contains $r$ and the shared octahedra edge. For both interaction terms, the coupling axis for the quadratic spin interaction is defined as an axis with no orientation; here it is shown as an arrow (with an arbitrary direction of the arrow head) for ease of visualization.

Figure 5

Here we observe that for $J=\left|K\right|/20$, finite $I_c<0$ is required regardless of the sign or magnitude of $J_2$.