Realization of the Haldane-Kane-Mele Model in a System of Localized Spins

We study a spin Hamiltonian for spin-orbit-coupled ferromagnets on the honeycomb lattice. At sufficiently low temperatures supporting the ordered phase, the effective Hamiltonian for magnons, the quanta of spin-wave excitations, is shown to be equivalent to the Haldane model for electrons, which indicates the nontrivial topology of the band and the existence of the associated edge state. At high temperatures comparable to the ferromagnetic-exchange strength, we take the Schwinger-boson representation of spins, in which the mean-field spinon band forms a bosonic counterpart of the Kane-Mele model. The nontrivial geometry of the spinon band can be inferred by detecting the spin Nernst effect. A feasible experimental realization of the spin Hamiltonian is proposed.

Figure 1

(a) The honeycomb lattice structure and the relative sign ${\nu }_{ij}$ of the DM interaction. (b) One-dimensional projection of the magnon bands [Eq. (5)], which are calculated with a ribbon geometry with zigzag terminations and 30 unit cells width. (c) The direction of the vector field $\mathbf{h}(\mathbf{k})$ [Eq. (4)]. (d) The Berry curvature of the upper band, ${\mathrm{\Omega }}^{+}$. For (b)–(d), the parameters $D=0.1J$ and $B=0$ are used. The shaded honeycomb in (c) is the first Brillouin zone. Two Dirac points, $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$ are denoted in (b) and (c). See the main text for detailed discussions.

Figure 2

(a) One-dimensional projection of the spinon bands [Eq. (10)] at the temperature $T=0.6J$ for the parameters $S=1/2$, $D=0.1J$, $B=0.1J$. A zoom of the middle sector is shown in the right. The band structure calculation corresponds to a ribbon geometry with zigzag terminations and 30 unit cells. (b),(c) The dependence of the mean fields $\eta ,{\zeta }_{\uparrow }$, and ${\zeta }_{\downarrow }$ on the temperature for the parameters $S=1/2$ and $D=0.1J$. See the main text for detailed discussions.

Figure 3

The spin Nernst conductivity ${\alpha }_{yx}^s$ as a function of temperature. The inset schematically shows a setup of an experiment and motions of spinons therein.