Phase Diagram and Quantum Order by Disorder in the Kitaev $K_1-K_2$ Honeycomb Magnet

We show that the topological Kitaev spin liquid on the honeycomb lattice is extremely fragile against the second-neighbor Kitaev coupling $K_2$, which has recently been shown to be the dominant perturbation away from the nearest-neighbor model in iridate ${\mathrm{Na}}_2{\mathrm{IrO}}_3$, and may also play a role in $\alpha \text{−}{\mathrm{RuCl}}_3$ and ${\mathrm{Li}}_2{\mathrm{IrO}}_3$. This coupling naturally explains the zigzag ordering (without introducing unrealistically large longer-range Heisenberg exchange terms) and the special entanglement between real and spin space observed recently in ${\mathrm{Na}}_2{\mathrm{IrO}}_3$. Moreover, the minimal $K_1-K_2$ model that we present here holds the unique property that the classical and quantum phase diagrams and their respective order-by-disorder mechanisms are qualitatively different due to the fundamentally different symmetries of the classical and quantum counterparts.

Popular Summary

The search for experimentally tangible scenarios of spin liquids as topologically ordered quantum states of matter is one of the most vibrant subfields of contemporary condensed matter research. Honeycomb iridates and related materials have originally been suggested as possible candidates for hosting a spin-liquid state. Intriguingly, no quantum paramagnetic ground state has been discovered so far in these materials, posing fundamental challenges to determining an accurate underlying microscopic spin model. In particular, a microscopic spin model is vital to predicting dynamical response functions from theory, which would help to quantify the proximity to the Kitaev spin-liquid regime and nurture hope for tracking traces of spin-liquid excitations at possibly higher frequencies above the magnetic ordering transition.

Here, we show that second-neighbor Kitaev coupling is an important ingredient to such a microscopic description for the strong spin-orbit transition-metal oxide ${\mathrm{Na}}_2{\mathrm{IrO}}_3$. We analyze the $K_1-K_2$ spin model—consisting of nearest-neighbor ($K_1$) and next-nearest-neighbor ($K_2$) Kitaev couplings—from a variety of methodological perspectives. As a coherent picture emerges from the investigation, the $K_1-K_2$ model naturally allows us to explain the onset of zigzag magnetic order that is also found experimentally. Furthermore, we find that the $K_1-K_2$ model is a suitable minimal description for resolving the substantially different nature of quantum and thermal fluctuations originating from such Kitaev couplings.

We expect that our findings will motivate future studies of similar compounds that are in close proximity to the Kitaev spin liquid.

Figure 1

The Kitaev $K_1-K_2$ model with three types of NN (solid) and NNN (dashed) Ising bonds. Here, ${\mathbf{t}}_1=a\mathbf{y}$ and ${\mathbf{t}}_2=[-(\sqrt{3}/2)\mathbf{x}+(1/2)\mathbf{y}]a$ are two primitive translations and $a$ is a lattice constant. We also show the vertical 2-leg ladders (shaded strips) discussed in the text, and the four-sublattice decomposition (A–D) related to the operations $H_{yzx}$ and $H_{xyz}$; see text.

Figure 2

The $T=0$ phase diagram of the model Eq. (1) as found by exact diagonalizations. Each of the magnetic regions (I–IV) hosts 12 degenerate quantum states. Here, we show two members [where spins point along the $\mathbf{z}$ axis, blue (red) circles denote spin-up (spin-down)] that are related to each other by flipping the spins in every second ladder (shaded strips) of Fig. 1. The Bragg peaks corresponding to $⟨S_i^z{S}_j^z⟩$ correlations are also shown in the extended Brillouin zone (assuming the same magnetic form factor in the two unit cell sublattices). The corresponding Bragg reflections for $⟨S_i^x{S}_j^x⟩$ and $⟨S_i^y{S}_j^y⟩$ are related to $⟨S_i^z{S}_j^z⟩$ by ${\tilde{\mathsf{C}}}_{6\mathrm{v}}$ spin-orbit rotations [35].

Figure 3

(a),(b) Exact low-energy spectra (measured from the ground state energy $E_0$) of the 24-site (a) and 32-site (b) clusters, defined, respectively [35], by the spanning vectors $(2{\mathbf{t}}_1-4{\mathbf{t}}_2,4{\mathbf{t}}_1-2{\mathbf{t}}_2)$ and $(2{\mathbf{t}}_1-4{\mathbf{t}}_2,4{\mathbf{t}}_1)$. A nonlinear $\mathbf{x}$ axis is used in order to highlight all regions of interest equally. The states are labeled by momenta $\mathbf{k}$ in the first BZ, parity (“e” for even, “o” for odd) under inversion through hexagon centers, and parity under global spin $\pi$ rotations around the $x$ axis (“Sze” for even, “Szo” for odd). The (red) numbers in (a) denote the multiplicity of the lowest five levels in regions I and II and the ground state degeneracy at $\psi =0$ and $\pi$. (c) ground state expectation value $⟨W_p⟩$ of Kitaev’s flux operators. (d) Square root of the “symmetrized” ground state spin structure factor $\widetilde{\mathcal{}\mathcal{S}}(\mathbf{Q})$ (see text), along with the spin length calculated from a self-consistent nonlinear spin-wave theory (NLSWT).

Figure 4

Real-space spin-spin correlation profiles evaluated at the ground state of the $N=32$ cluster, inside the first QSL phase ($\psi =0.01\pi$, left-hand column) and inside the magnetic phase I ($\psi =0.028\pi$, right-hand column). Different rows correspond to the three different channels $⟨S_i^{\alpha }{S}_j^{\alpha }⟩$, $\alpha =x$, $y$, and $z$. The reference site $i$ is indicated by the small black open circle. Positive (negative) correlations are shown by filled blue (filled red) circles, whose radius scales with the magnitude of the correlation. The difference between $\alpha =z$ and $\alpha =x$, $y$ stems from the fact that the 32-site cluster does not have the full point-group symmetry of the infinite lattice, and the momentum point ${\mathbf{M}}_z$ is not equivalent by symmetry to ${\mathbf{M}}_x$ and ${\mathbf{M}}_y$; see Ref. [35].

Figure 5

Top: The three types of virtual processes around the strong-coupling limit $r=0$ [35]. Bottom: $|J_W|/r^4$, $2|J_1|/r^4$, and $J_2/r^4$ versus $\psi$. The shaded strips denote the regions where $J_2$ competes with $J_1$ and $J_2>2|J_1|$.