Heisenberg-Kitaev model on the hyperhoneycomb lattice

Motivated by recent experiments on $\beta$-Li${}_2$IrO${}_3$, we study the phase diagram of the Heisenberg-Kitaev model on a three-dimensional lattice of tricoordinated Ir${}^{4+}$, dubbed the hyperhoneycomb lattice. The lattice geometry of this material, along with Ir${}^{4+}$ ions carrying $J_{\mathrm{eff}}=1/2$ moments, suggests that the Heisenberg-Kitaev model may effectively capture the low-energy spin-physics of the system in the strong-coupling limit. Using a combination of semiclassical analysis, exact solution, and slave-fermion mean-field theory, we find, in addition to the spin liquid, four different magnetically ordered phases depending on the parameter regime. All four magnetic phases—the Néel, the polarized ferromagnet, the skew-stripy, and the skew-zig-zag—have collinear spin ordering. The three-dimensional $Z_2$ spin liquid, which extends over an extended parameter regime around the exactly solvable Kitaev point, has a gapless Majorana mode with a deformed Fermi circle (codimensions, $d_c=2$). We discuss the effect of the magnetic field and finite temperature on different phases that may be relevant for future experiments.

Figure 1

The tricoordinated orthorhombic lattice. The orthorhombic unit cell is outlined in gray. The primitive unit cell contains four Ir atoms colored yellow and labeled from 1 to 4. The ten blue sites show the smallest closed loop on this lattice. These sites are labeled from $a$ to $j$. All the other Ir atoms are colored gray. The primitive vectors for the four-site unit cell are given by ${\mathbf{a}}_i$. For the Kitaev interactions, the red bonds refer to $S^x{S}^x$, the green to $S^y{S}^y$, and the blue to $S^z{S}^z$ interactions, respectively. The orientation of the global $x,y,z$ coordinates are shown in the bottom right.

Figure 2

The Néel phase. This is the classical ground state for $K=0$.

Figure 3

The equivalent of the four-sublattice rotation defined by Khaliullin [29] and later by Chaloupka et al. [27] for the hyperhoneycomb lattice. The spins at the sites denoted by blue circles are left unrotated, the spins at the sites denoted by red triangles are rotated by 180${}^{\circ }$ about the $z$ axis, the spins at the sites denoted by yellow hexagons are rotated by 180${}^{\circ }$ about the $y$ axis, and the spins at the sites denoted by green squares are rotated by 180${}^{\circ }$ about the $x$ axis.

Figure 4

The skew-stripy phase with ordering in $S^z$. This is the exact solution to the model at the point $K=2J$. The antiferromagnetic chains run along the $x-y$ bonds (shaded in yellow) which form almost skew lines. The ferromagnetic $z$ bonds form a stripy order (shaded in red and blue).

Figure 5

The spectrum of the dispersing Majorana fermion in the pure Kitaev model on the hyperhoneycomb lattice along paths of high symmetry in the first Brillouin zone (the first Brillouin zone and the paths are shown in Appendix pp1).

Figure 6

The green curve indicates the Fermi surface at $J=0$. This occurs on the boundary of the first Brillouin zone. The red curve indicates the Fermi surface at $K/J=8(\alpha =0.2)$ as computed within mean-field theory. The two Fermi surfaces almost coincide with minute differences. (The Fermi surfaces of neighboring cells have been appended to aid visualization.)

Figure 7

The classical phase diagram within Luttinger-Tisza approximation for arbitrary $J$ and $K$. Diagonal dotted lines indicate the four-sublattice rotation mapping from $(J,K)\to (-J,K-2J)$. Black dots indicate the exactly solvable ferromagnetic point in both the rotated and the unrotated bases. Red dots indicate the antiferromagnetic Heisenberg point in both the rotated and the unrotated bases. Four magnetic phases have been found; see main text for details.

Figure 8

The skew-zig-zag phase with $S^z$ ordering. The ferromagnetic chains run along the $x-y$ bonds in a zig-zag fashion (indicated in blue and red), while the $z$ bonds are ferromagnetic (indicated in yellow).

Figure 9

The spin-wave dispersion for various values of $K/J$ within the skew-stripy phase. We have chosen the ordering in $S^z$ as an example. $K/J=2(\alpha =0.5)$ maps to the pure ferromagnetic model in the rotated basis (see text); $K/J=1.1(\alpha =0.35)$ is near the classical boundary of the Néel and the skew-stripy order; and, $K/J=0.6$ is a general point within the skew-stripy phase.

Figure 10

The zero-point energy corrections from spin-wave theory for the different skew-stripy states for $K=3J$. The different stripy states can be labeled by $(\theta ,\varphi )$ (see text). We find that the $x$-, $y$-, and the $z$-skew-stripy states have lower energies (but not same, as this figure may deceptively suggest, due to lack of resolution; see Fig. 11 for the difference). Due to the three $C_2$, two inversion, and time-reversal symmetries of our Hamiltonian, energy correction for other $(\theta ,\varphi )$ not explicitly shown is related to the plotted octant by mirror operations ${\sigma }_{yz}$, ${\sigma }_{xz}$, and ${\sigma }_{xy}$ in $(\theta ,\varphi )$ space.

Figure 11

The zero-point energy difference between the $S^z$-ordered and $S^x/S^y$-ordered skew-stripy phases as a function of $K/J$. Negative values indicating $S^z$-ordered phase have lower energy. At the exactly solvable point, $K/J=2(\alpha =0.5)$, SU(2) symmetry is restored and hence the energy difference is zero. Away from that point, the $S^z$-ordered phase has lower energy and hence is selected by this quantum-order-by-disorder mechanism.

Figure 12

(a) The spinon band structure at the exactly solvable point $K/J=\infty$ ($\alpha =1$). (b) The spinon band structure at the point $K/J=8$ ($\alpha =0.8$). At this point, magnetic order has not yet stabilized. (c) The spinon band structure at the point $K/J=4.9$ ($\alpha =0.71$). At this point, magnetic order is present.

Figure 13

The magnitude of the mean-field order parameters, plotted as a function of $\alpha$, where $J=(1-\alpha )$ and $K=2\alpha$.

Figure 14

The Brillouin zone. The high-symmetry paths are: $\Gamma \to Y\to T\to Z\to \Gamma \to X\to A_1\to Y$, $T\to X_1$, $X\to A\to Z$, and $\Gamma \to L$. The following are the position of the high-symmetry points: $\Gamma =(0,0,0)$, $Y=(0,0,-\frac{\pi }{2})$, $T=(-\frac{\pi }{6},-\frac{\pi }{6},-\frac{\pi }{2})$, $Z=(-\frac{\pi }{6},-\frac{\pi }{6},0)$, $X=(\frac{29\pi }{72},-\frac{29\pi }{72},0)$, $A_1=(\frac{11\pi }{72},-\frac{11\pi }{72},-\frac{\pi }{2})$, $X_1=(-\frac{19\pi }{72},-\frac{5\pi }{72},-\frac{\pi }{2})$, $A=(\frac{13\pi }{72},-\frac{37\pi }{72},0)$, and $L=(\frac{\pi }{6},-\frac{\pi }{3},-\frac{\pi }{4})$.

Figure 15

The four loops as shown are (1) b-c-d-e-f-g-h-i-j-a; (2) m-n-o-p-d-c-b-a-k-l; (3) m-n-q-r-h-i-j-a-k-l; (4) q-r-h-g-f-e-d-p-o-n. Sublattices 1, 2, 3, and 4 are colored green, red, orange, and yellow, respectively, to aid visualization of the four-site unit cell.