Possible proximity of the Mott insulating iridate Na${}_2$IrO${}_3$ to a topological phase: Phase diagram of the Heisenberg-Kitaev model in a magnetic field

Motivated by the recent experimental observation of a Mott insulating state for the layered iridate Na${}_2$IrO${}_3$, we discuss possible ordering states of the effective iridium moments in the presence of strong spin-orbit coupling and a magnetic field. For a field pointing in the $\langle 111\rangle$ direction—perpendicular to the hexagonal lattice formed by the iridium moments—we find that a combination of Heisenberg and Kitaev exchange interactions gives rise to a rich phase diagram with both symmetry breaking magnetically ordered phases as well as a topologically ordered phase that is stable over a small range of coupling parameters. Our numerical simulations further indicate two exotic critical points at the boundaries between these ordered phases.

Figure 1

Ground-state phase diagram of the Heisenberg-Kitaev model (1) in a $\langle 111\rangle$ magnetic field of strength $h$. Interpolating from the Heisenberg ($\alpha =0$) to Kitaev ($\alpha =1$) limit for small field strength, a sequence of three ordered phases is observed: a canted Néel state for $\alpha \lesssim 0.4$, a canted stripy Néel state illustrated in Fig. 2c for $0.4\lesssim \alpha \lesssim 0.8$, and a topologically ordered state for nonvanishing field around the Kitaev limit. All ordered phases are destroyed for sufficiently large magnetic field giving way to a polarized state.

Figure 2

(a) The honeycomb lattice spanned by unit vectors ${\mathrm{a⃗}}_1=(1,0)$ and ${\mathrm{a⃗}}_2=(1/2,\sqrt{3}/2)$. Illustration of magnetic states with (b) Neel order and (c) stripy Neel order.

Figure 3

Energy jump along the first-order transition between the canted Néel and stripy AFM. Scans of the ground-state energy across this transition for various values of the magnetic field strength are provided in the supplemental material.[19]

Figure 4

(Top panel) Ground-state phase diagram of the Kitaev model ($\alpha =1$) in the $h-\kappa$ plane, where $h$ is the strength of a magnetic field pointing in the $\langle 111\rangle$ direction and $\kappa$ is the strength of a time-reversal symmetry breaking three-site term. (Lower panels) Magnetization sweep and its derivative for various $\kappa$.

Figure 5

Phase transition for the Kitaev model ($\alpha =1$) in a $\langle 111\rangle$ magnetic field of strength $h$. (a) Second derivative of the ground-state energy $-d^{2}E/{\mathit{dh}}^2$, (b) magnetization $M(h)$ and its first derivative $\mathit{dM}(h)/\mathit{dh}$ for different system sizes. (c) Vortex number.

Figure 6

Constant field scan in the vicinity of the (critical) point separating the stripy AFM from the topological phase.