Topologically nontrivial magnons at an interface of two kagome ferromagnets

Magnon band structures of topological magnon insulators exhibit a nontrivial topology due to the Dzyaloshinskii-Moriya interaction, which manifests itself by topologically protected edge magnons. Bringing two topological magnon insulators into contact can lead to nontrivial unidirectional magnons located at their common interface. We study theoretically interfaces of semi-infinite kagome ferromagnets in various topological phases, with a focus on the formation and the confinement of nontrivial interface magnons. We analyze generic magnon dispersions with respect to the number of band gaps and the respective winding numbers. Eventually, we prove that interfaces of topologically identical phases can host nontrivial interface magnons as well.

Figure 1

Sketch of facing semi-infinite kagome phases $\mathsf{A}$ (left, green) and $\mathsf{B}$ (right, blue), which share a common interface (dashed line) along the $y$ direction. Each phase is divided into the thinnest principal layers possible (green and blue layers, respectively) if only nearest and next-nearest interactions are present in the Hamiltonian; four atomic layers form one principal layer from which four are numbered (on the top; $-2,...,+1$). The six white dots in $\mathsf{B}$ indicate the principle-layer basis.

Figure 2

Topological phase diagram of ferromagnetic kagome lattices with respect to the ratios $\frac{J_{\mathrm{NN}}}{J_{\mathrm{N}}}$ and $\frac{D}{J_{\mathrm{N}}}$. The tinted topological phases are identified by their Chern numbers $C=(C_1,C_2,C_3)$ and winding numbers $\nu =({\nu }_1,{\nu }_2)$.

Figure 3

Bulk-boundary correspondence and energy-resolved winding numbers. (a) and (c) display energy-resolved winding numbers ${\nu }^{\mathsf{A}}\left(\varepsilon \right)$ and ${\nu }^{\mathsf{B}}\left(\varepsilon \right)$ (black lines) for an $\mathsf{A}$/vacuum and a $\mathsf{B}$/vacuum interface, respectively. These are superimposed on the spectral density of the rightmost principle layer (color scale, with white indicating zero spectral weight and dark orange indicating maximum weight). (b) shows the same quantities as (a) and (c) but for an $\mathsf{A}/\mathsf{B}$ interface. Phase $\mathsf{A}$ with Chern triple $(-1,2,-1)$: $J_{\mathrm{N}}=23/20D=23/10J_{\mathrm{NN}}=0.46\mathrm{meV}$; phase $\mathsf{B}$ with Chern triple $(-1,0,1)$: $J_{\mathrm{N}}=2D=1\mathrm{meV}, J_{\mathrm{NN}}=0\mathrm{meV}$.

Figure 4

Localization of topological magnons at an $\mathsf{A}/\mathsf{B}$ interface. The site-resolved spectral density is shown vs layer index for two interface magnons (blue and green histogram). The respective $(\varepsilon ,k_y)$ is marked by the blue (green) dot in the inset, which reproduces Fig. 3. Parameters are as in Fig. 3.

Figure 5

Bulk-boundary correspondence applied to magnon interface modes. (a)–(c) show three selected interface band structures that differ with respect to the number of interface band gaps and winding numbers. On the left-hand side one-dimensional magnon dispersions $\varepsilon \left(k_y\right)$ are sketched, with bulk bands in gray and winding numbers given within the band gaps. The right-hand panels display the evolution of the spectral density per principle layer (top; labeled $-3,...,+3$) across the interface (layer 0). The Chern triples $C$ of phases $\mathsf{A}$ and $\mathsf{B}$ as well as the corresponding winding numbers are indicated. Exchange parameters of $\mathsf{A}$: (a) $J_{\mathrm{N}}=2D=1\mathrm{meV}, J_{\mathrm{NN}}=0\mathrm{meV}$, (b) $J_{\mathrm{N}}=23/20D=23/10J_{\mathrm{NN}}=0.46\mathrm{meV}$, and (c) $J_{\mathrm{N}}=20/7D=1\mathrm{meV}, J_{\mathrm{NN}}=0\mathrm{meV}$; $\mathsf{B}$: (a) $J_{\mathrm{N}}=13/7J_{\mathrm{NN}}=13/11D=0.65\mathrm{meV}$, (b) $J_{\mathrm{N}}=2D=1\mathrm{meV}, J_{\mathrm{NN}}=0\mathrm{meV}$, and (c) $J_{\mathrm{N}}=13/6D=1.3\mathrm{meV}, J_{\mathrm{NN}}=0\mathrm{meV}$.

Figure 6

Bulk-boundary correspondence of magnon interface modes. (a)–(f) display generic interface band structures with different combinations of interface band gaps and winding numbers. Bulk bands are sketched in gray; winding numbers are given within the gaps.