Topological magnons in Kitaev magnets at high fields

We study the Kitaev-Heisenberg $\mathrm{\Gamma }\text{−}{\mathrm{\Gamma }}^{\prime }$ model that describes the magnetism in spin-orbit coupled honeycomb lattice Mott insulators. In strong magnetic fields perpendicular to the plane of the lattice ([111] direction) that bring the system into the fully polarized paramagnetic phase, we find that the spin-wave bands carry nontrivial Chern numbers over large regions of the phase diagram, implying the presence of chiral magnon edge states. In contrast to other topological magnon systems, the topological nontriviality of these systems results from the presence of anisotropic terms in the Hamiltonian that do not conserve the number of magnons. Since the effects of interactions are suppressed by the exchange scale divided by the applied field strength, the validity of the single-particle picture is tunable, making paramagnetic phases particularly suitable for the exploration of this physics. Using time-dependent density matrix renormalization group methods and interacting spin-wave theory, we demonstrate the presence of the chiral edge mode and its evolution with field.

Figure 1

(a) The phase diagram of the Kitaev-Heisenberg model as a function of $\vartheta$ and $h/S$ as extracted from the spin-wave spectrum. The region of ordered phases is determined from the gap closure in the spin-wave spectrum while at the boundary of this region the corresponding ordering wave vector is indicated. The true semiclassical phase diagram has small regions, away from the Kitaev points, separated from the paramagnetic phase by first-order transitions (shown in Ref. [47]) and above the threshold field shown here which are not captured using our technique [34, 35]. The rest of the phase diagram lies in the fully polarized phase. The entirety of the polarized paramagnetic region hosts topologically nontrivial magnons at the semiclassical level. Inset: Honeycomb lattice cluster with 24 sites. The different colored bonds correspond to the three types of coupling in the Kitaev model ${\mathsf{S}}_i^{\gamma }{\mathsf{S}}_j^{\gamma }$ for $\gamma =x$ (blue), $y$ (green), and $z$ (red) corresponding to the projections of the cubic axes onto the honeycomb plane. The [111] field direction, indicated on the figure, is perpendicular to the plane. The exact diagonalization results presented in this Rapid Communication were obtained for the Kitaev-Heisenberg model defined on this cluster. (b) shows the minimal gap between the spin-wave modes. The wave vector at which the gap is minimal is indicated by the color.

Figure 2

Dynamical correlation function $S(\mathit{k},\omega )$ computed along high-symmetry lines for $h=3$ ($S=1/2$) at the AFM Kitaev point $\vartheta =\pi /2$. The intensity scale is logarithmic from $5\times {10}^{-3}$ to 1. (a) Linear spin-wave theory, (b) nonlinear spin-wave theory, and (c) time-dependent DMRG, all with Gaussian broadening of the lines for purposes of presentation ${\sigma }_{\omega }=0.01$. The overlaid points in (c) show single magnon states obtained from 24-site ED with periodic boundary conditions.

Figure 3

Dynamical correlation functions $\langle {\mathsf{S}}^x(-k,x,-\omega ){\mathsf{S}}^x(k,x,\omega )\rangle$ computed using DMRG+tMPO (right column) on a slab [illustrated in (e)] of width $L_{\mathrm{x}}=5$ unit cells and length $L_{\mathrm{y}}=71$ and with periodic boundary conditions imposed along $y$. The crystal momentum along the translationally invariant direction is denoted by $k$ and $x$ is the line number [indicated in (e)]. The DMRG+tMPO energy resolution is $\mathrm{\Delta }\omega \approx 0.03$. For comparison, corresponding results are shown for linear spin-wave theory (left column) and $O\left(S^0\right)$ nonlinear spin-wave theory (middle column). All calculations were performed for $\vartheta =\pi /2$ and $h=3$. The plots in each row correspond to different line numbers (from top to bottom $x=1$, 5, and 9). The shading in each figure indicates the bulk single magnon bandwidths with the bulk band gap in between the shaded bands. The chiral modes at the two boundaries $x=1$ (top) and $x=9$ (bottom) and their edge character can been seen through the reduction in the band gap intensity in the middle of the slab $x=5$ (middle) and in (d), which shows the integrated intensity within the bulk band gap in different layers on the slab. The velocities of the edge modes are opposite at the two boundaries.

Figure 4

Dimensionless thermal Hall conductivity ${\kappa }_{xy}/T$ as a function of temperature at the ferromagnetic Kitaev point ($S=1/2$), $K=-1$, at fields $h=0.01$, 0.02, 0.05, 0.1, 0.2, 0.5, 1, 2, 3, 4 to be read in the arrow direction. The inset shows the dispersions of the bands at $h=0.1$. The color of the bands indicates the Berry curvature with the scale corresponding to the color bar.