Quantum order by disorder in the Kitaev model on a triangular lattice

We identify and discuss the ground state of a quantum magnet on a triangular lattice with bond-dependent Ising-type spin couplings, that is, a triangular analog of the Kitaev honeycomb model. The classical ground-state manifold of the model is spanned by decoupled Ising-type chains, and its accidental degeneracy is due to the frustrated nature of the anisotropic spin couplings. We show how this subextensive degeneracy is lifted by a quantum order-by-disorder mechanism and study the quantum selection of the ground state by treating short-wavelength fluctuations within the linked cluster expansion and by using the complementary spin-wave theory. We find that quantum fluctuations couple next-nearest-neighbor chains through an emergent four-spin interaction, while nearest-neighbor chains remain decoupled. The remaining discrete degeneracy of the ground state is shown to be protected by a hidden symmetry of the model.

Figure 1

(a) Ising-type spin couplings on the three nonequivalent bonds of the triangular lattice of model (1). The lattice lies in the $(1,1,1)$ plane of the spin-quantization axes. On a $\left(\gamma \right)$ bond, the one perpendicular to the $\gamma (=x,y,z)$ spin-quantization axis, only the $\gamma$ components of neighboring spins are coupled. (b) Four-sublattice structure of the triangular lattice used for the unitary transformations discussed in the text. (c) Sketch of the fourth-order perturbation process leading to the coupling, via quantum fluctuations, of the four spins sitting around a diamond [see Eq. (3)]. In the virtual states, the locations of the misaligned spins (spin flips are performed in pairs at one of the four bonds of the diamond at each step: top left $\Rightarrow$ bottom right $\Rightarrow$ bottom left $\Rightarrow$ top right) are shown by filled circles and the wavy lines mark the broken $\left(z\right)$ bonds.

Figure 2

Color map of the second-order quantum energy correction (2) for an arbitrary direction $\mathbf{m}$ of the ordered moment. The minimum energy is achieved for $\mathbf{m}$ pointing along one of the spin-quantization axes.

Figure 3

Color map of the spin-wave excitation spectrum of model (1) in (a) the FM state Eq. (4) and (b) the stripy AF state Eq. (5), for $K_{\gamma }=1$. Only one of the lower branches is shown in (b). Hexagons mark the crystallographic Brillouin zone. The right panels report the corresponding magnetic structures.