Kitaev-Heisenberg models for iridates on the triangular, hyperkagome, kagome, fcc, and pyrochlore lattices

The Kitaev-Heisenberg (KH) model has been proposed to capture magnetic interactions in iridate Mott insulators on the honeycomb lattice. We show that analogous interactions arise in many other geometries built from edge-sharing IrO${}_6$ octahedra, including the pyrochlore and hyperkagome lattices relevant to Ir${}_2$O${}_4$ and Na${}_4$Ir${}_3$O${}_8$, respectively. The Kitaev spin liquid exact solution does not generalize to these lattices. However, a different, exactly soluble point of the honeycomb lattice KH model, obtained by a four-sublattice transformation to a ferromagnet, generalizes to all of these lattices and even to certain additional further neighbor Heisenberg couplings. A Klein four-group $\cong$${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ structure is associated with this mapping (hence Klein duality). A finite lattice admits the duality if a simple geometrical condition is met. This duality predicts fluctuation-free ordered states on these different 2D and 3D lattices, which are analogues of the honeycomb lattice KH stripy order. This result is used in conjunction with a semiclassical Luttinger-Tisza approximation to obtain phase diagrams for KH models on the different lattices. We also discuss a Majorana fermion based mean-field theory at the Kitaev point, which is exact on the honeycomb lattice, for the KH models on the different lattices. We attribute the rich behavior of these models to the interplay of geometric frustration and frustration induced by spin-orbit coupling.

Figure 1

Phase diagrams using the Klein duality as well as the Luttinger-Tisza approximation (LTA). Phase diagrams are shown for the Kitaev-Heisenberg Hamiltonian, $H_{\mathrm{KH}}=\eta \left(1-\left|\alpha \right|\right){\vec{S}}_i·{\vec{S}}_j-2\eta \alpha S_i^{{\gamma }_{ij}}{S}_j^{{\gamma }_{ij}}$ with $\eta =\pm 1$, on the various lattices. Note that the parameter space is a ring: for the $\alpha$ parameter segments shown here, the endpoints should be identified, i.e., writing the parameter as $(\eta ,\alpha )$, identify the points $(-1,-1)\cong (+1,+1)$ and also identify the points $(-1,+1)\cong (+1,-1)$. Rich phase diagrams are found. 2D and 3D stripy magnetic phases (blue), found on all lattices, are exact and fluctuation-free by the Klein duality at the FM SU(2) point (yellow star) $\eta =+1$, $\alpha =1/2$. On the kagome and hyperkagome lattices, outside the FM SU(2) points, the FM and stripy orders are given non-unit-length spins by the LTA (gray shading), suggesting frustration. Extensive degeneracy of LTA ordering wave vectors hints at a non-spin-ordered “quantum” phase, labeled “$Q$,” where the Hamiltonians hosting $Q$ phases have been solved, exactly [9] for the honeycomb at $\alpha =\pm 1$ and numerically [35, 36] for the kagome at $\eta =+1$, $\alpha =0$, they have turned out to host nonmagnetic phases. The hyperkagome hosts $Q$ points at the Heisenberg antiferromagnet and its Klein dual, as well as at the pure Kitaev Hamiltonians, because any single Kitaev bond type fragments the hyperkagome into disjoint clusters (see Fig. 4). The ${120}^{\circ }$ triangular lattice and Neel and zigzag honeycomb orders are found with normalized spins. Apparent ordering with LTA non-normalized spins is found at incommensurate wave vectors on the triangular lattice and in the cluster-ferrimagnet and cluster-antiferromagnet (AF) regimes on the hyperkagome. Kagome, fcc, and pyrochlore lattices also host frustrated regimes with no definitive ordering within the LTA, as described in the text (gray).

Figure 2

Edge-sharing IrO${}_6$ octahedra generating Kitaev exchange. Iridium ions (spheres) are each coordinated by six oxygen ions forming vertices of octahedra. Octahedra of neighboring Ir ions share edges. Dotted (purple) lines show the iridium-oxygen-iridium hopping paths, which form a square with ${90}^{\circ }$ angles. As described in the text, these superexchange paths generate an Ising interaction between the iridium effective spins, which couple a spin component $x$, $y$, or $z$ depending on the orientation of the shared octahedra edge (shown in red, green, and blue). Ir lattices hosting this Kitaev exchange must arise from a regular tiling of these edge-sharing octahedra.

Figure 3

Kitaev-Heisenberg lattices. Iridium ions arranged in these lattices may generate the Kitaev spin exchange, coupling component $x,y,z$ on bonds colored red, green, and blue, respectively. A blue colored bond connecting two Ir sites implies that the respective IrO${}^6$ octahedra share a blue edge as in Fig. 2. We also list examples of possible relevant iridium compounds that form these lattices.

Figure 4

Hyperkagome unit cell and decomposition into Kitaev clusters. A symmetric depiction of the hyperkagome structure, highlighting the four disjointed three-site clusters, which split up the unit cell for a given Kitaev bond label (red, green, and blue). The fact that the lattice fragments into these disjointed clusters for a given Kitaev bond type has substantial repercussions, as described in the text. The four clusters of a given type appear in two parallel-orientation pairs that are perpendicular to each other (with different shading). The unit cell is composed of the 12 sites participating in the four drawn triangular faces, as well as all 24 drawn bonds. We label the 12 sites by a letter A, B, and C as in Ref. [45] so that A spins lie on midpoints of type-A (here blue) clusters, etc, and by a number 1–4 (chosen to not repeat within a triangle face or cluster). The camera angle, i.e., the vector pointing into the page, is just slightly off (up-right) of a Cartesian (here, also Bravais) axis; for Na${}_4$Ir${}_3$O${}_8$, it is the vector from an iridium ion to a neighboring coordinating oxygen.

Figure 5

Action of the Klein duality on the Kitaev-Heisenberg Hamiltonian. The $\eta ,\alpha$ parameters on the top line (oriented left to right) of the Hamiltonian (1) map according to the blue lines to $\eta ,\alpha$ parameters on the bottom line (oriented right to left) in the Klein-dual Hamiltonian. Thick blue lines map the points shown exactly, thin blue lines are a qualitative sketch. Note that the right and left edges of the figure are identified, forming a ring. Both pure Kitaev Hamiltonians ($\alpha =\pm 1$) are self-dual, mapping to themselves. The points at $\alpha =1/2$ and $\eta =+1$, $\eta =-1$ are dual to the SU(2) symmetric FM-Heisenberg and AF-Heisenberg points (yellow and brown stars), respectively, with the FM dual point hosting the exactly soluble stripy phases on all lattices in Fig. 3.

Figure 6

3D-stripy orders on the fcc, pyrochlore, and hyperkagome lattices. Black (white) spheres represent up (down) spins. Bonds are colored red, green, and blue according to the Kitaev label. The ordering pattern is of alternating planes, here normal to $\widehat{z}$; $z$-type bonds (blue), i.e., those within the planes, connect spins of the same orientation. On the pyrochlore and hyperkagome lattices, planes are broken up into uniformly oriented chains, with chains in spin-up planes oriented perpendicularly to chains on spin-down planes. On the hyperkagome lattice, the chains are further broken into the linear three site clusters shown in Fig. 4. These stripy orders are exact at $\eta =+1,\alpha =1/2$, being Klein duals of the ferromagnet.

Figure 7

Magentic order of the cluster-ferrimagnet state on the hyperkagome. Black (white) spheres are up (down) spins; bonds are colored according to Kitaev label. Here shown for $\widehat{z}$ ordering, notice that the $z$-type clusters (blue), lying in planes normal to $\widehat{z}$, all have (up, down, up) spin configurations. This configuration has nonzero net magnetization. Note that the cluster-ferrimagnet order is Klein dual to the cluster-AF order.

Figure 8

Magentic order of the cluster-AF state on the hyperkagome lattice. Black (white) spheres are up (down) spins; bonds are colored according to Kitaev label. Here shown for $\widehat{z}$ ordering, notice that the $z$-type clusters (blue), lying in planes normal to $\widehat{z}$, have (up, down, up) spin configurations on even-numbered planes, and the opposite (down, up, down) spin configurations on odd-numbered planes. This configuration has zero net magnetization. Note that the cluster-AF order is Klein dual to the cluster-ferrimagnet order.