Edge states in topological magnon insulators

For magnons, the Dzyaloshinskii-Moriya interaction accounts for spin-orbit interaction and causes a nontrivial topology that allows for topological magnon insulators. In this theoretical investigation we present the bulk-boundary correspondence for magnonic kagome lattices by studying the edge magnons calculated by a Green function renormalization technique. Our analysis explains the sign of the transverse thermal conductivity of the magnon Hall effect in terms of topological edge modes and their propagation direction. The hybridization of topologically trivial with nontrivial edge modes enlarges the period in reciprocal space of the latter, which is explained by the topology of the involved modes.

Figure 1

Kagome lattice with lattice vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ in the $xy$ plane. Identical atoms A, B, and C are placed at the corners of the triangles. Dzyaloshinskii-Moriya vectors are aligned normal to the lattice plane and are represented by red dots: along $+z$ $\left(-z\right)$ for a counterclockwise (clockwise) chirality: A-B-C (C-B-A).

Figure 2

Edges of the kagome lattice. For two different terminations of the bulk system, (a) and (b), the semi-infinite lattice is divided into the thinnest principal layers possible if only nearest- and next-nearest-neighbor interactions are present. White dots show the basis consisting of six (a) and three (b) sites, respectively. The lattice constants $a_{\parallel }$ are indicated.

Figure 3

Topological phase diagram of the kagome lattice with regions characterized by sets $(C_1,C_2,C_3)$ of Chern numbers. The antiferromagnetic (AF) phase is colored blue. Red dots mark those variable sets $\{J_{\mathrm{NN}}/J_{\mathrm{N}},D/J_{\mathrm{N}}\}$ for which local densities of states are calculated. The sign of the transverse thermal conductivity ${\kappa }^{xy}$ of the magnon Hall effect is given in red; it is unique for the phases $(-1,0,1)$ and $(3,-2,-1)$.

Figure 4

Edge magnons in a kagome lattice. The spectral density of states of an edge site is shown as color scale [“local density of states” (LDOS), right] for the edge geometries of Fig. 2 [top row, (a)–(d)] and Fig. 2 [bottom row, (e)–(h)] in the entire edge Brillouin zone. Bulk magnons appear as extended regions separated by band gaps (“projected bulk band structure”), while edge magnons bridge these band gaps. The topological phases are characterized by their bulk-band-resolved Chern numbers $C_i$ $\left(i=1,2,3\right)$ given in each panel; the respective exchange parameters are given as well $\left(J_{\mathrm{N}}=1\mathrm{meV}\right)$. The topologically nontrivial edge magnon modes and their propagation direction are sketched in the central row.

Figure 5

Edge localization of magnons. The site-resolved spectral density (“LDOS”) is shown versus layer index (edge $=$ layer 1) for an edge magnon [blue; the respective $(\varepsilon ,\mathbf{k})$ is marked by the blue dot in the inset] and an edge resonance (green; cf. the green dot in the inset). $D/J_{\mathrm{N}}=1,J_{\mathrm{NN}}/J_{\mathrm{N}}=0$, and $J_{\mathrm{N}}=1\mathrm{meV}$, as in Fig. 4.

Figure 6

Hybridization of a topologically nontrivial with a trivial edge magnon. The local density of states (LDOS) for the edge geometry of Fig. 2 is shown as a color scale for several exchange parameters of the edge sites [(a)–(d); as indicated at the top of each panel]. $D/J_{\mathrm{N}}=0.1,J_{\mathrm{N}}=1\mathrm{meV}$, and $J_{\mathrm{NN}}=0$.

Figure 7

Topology and hybridization of a topologically nontrivial with a trivial edge magnon. Cylinder representation of a system with two bulk bands (black). The dashed line indicates the “seam” at $k_{\parallel }=0=2\pi /a_{\parallel }$. (a) Sketch of the band structure in Fig. 6. Trivial edge modes (green) circle around the cylinder; nontrivial edge modes (blue) connect the bulk bands. The trivial modes' energy is not located within the energy range of the nontrivial mode. (b) Sketch of the band structure in Fig. 6. A hybridization of a trivial and nontrivial edge mode generates a nontrivial mode circulating the cylinder (red).