Topological magnon insulator in insulating ferromagnet

In the ferromagnetic insulator with the Dzyaloshinskii-Moriya interaction, we theoretically predict and numerically verify a topological magnon insulator, where the charge-free magnon is topologically protected for transporting along the edge/surface while it is insulating in the bulk. The chiral edge states form a connected loop as a $4\pi$- or $8\pi$-period Möbius strip in the spin-wave vector space, showing the nontrivial topology of magnonic bands. Using the nonequilibrium Green's function method, we explicitly demonstrate that the one-way chiral edge transport is indeed topologically protected from defects or disorders. Moreover, we show that the topological edge state mainly localizes around edges and leaks into the bulk with oscillatory decay. Although the chiral edge magnons and energy current prefer to travel along one edge from the hot region to the cold one, the anomalous transports are identified in the opposite edge, which reversely flow from the cold region to the hot one. Our findings could be validated within wide energy ranges in various magnonic crystals, such as Lu${}_2$V${}_2$O${}_7$.

Figure 1

(a) Pyrochlore crystallographic structure of the sublattice of magnetic atoms V of Lu${}_2$V${}_2$O${}_7$. (b) Two tetrahedrons in the pyrochlore lattice, where DM vectors on bonds 1-3, 1-2, and 2-4 are indicated by orange arrows. (c) Schematic magnetic flux due to DM interaction in the [111] plane of the pyrochlore lattice, i.e., a kagome lattice. The coupling of two sites along the arrows is $(J+iD)S$ while the opposite direction corresponds to $(J-iD)S$. (d) The quasi-one-dimensional kagome-lattice strip. The area enclosed by the dotted line can be regarded as a center which is connected to two semi-infinite leads in equilibrium at temperatures $T_L$ (left) and $T_R$ (right). The two big arrows schematically depict the magnitudes and directions of energy flows along the lattice edges when $T_L>T_R$. The width of the strip example is $W=5$, which is defined as the number of atoms in the left column of each unit cell.

Figure 2

(a), (b), and (c) The Berry curvature of the three bands at zero DM interaction. (d), (e), and (f) The Berry curvature of the three bands at nonzero DM interaction ($D=0.2$). For all the insets (a)–(f), the horizontal and vertical axes correspond to wave vector $k_x$ and $k_y$, respectively; the unit is $2\pi /a$, where $a$ is the lattice spacing. (g) The Chern numbers of the three energy bands for the two-dimensional periodic kagome lattice. The dotted, solid, and dashed lines correspond to Chern numbers of the first, the second, and the third bands, respectively.

Figure 3

The dispersion relations of the periodic kagome strip lattices with different width sizes. The insets (a), (b), (c), and (d) correspond to $W=2$, $W=5$, $W=20$, and $W=80$, respectively.

Figure 4

The energy differences vs the width of the periodic kagome strip at the anticrossing points. The solid, dashed, and dotted lines correspond to energy differences at anticrossing points A, B, and C, respectively in Fig. 3d.

Figure 5

The dispersion relation of chiral magnonic edge modes with nonzero DM interactions. (a) and (b) are dispersion relations in the range of $k_x\in [0,8\pi /a]$ for $D/J=0.1$ and 0.4, respectively. (c) A conventional cylinder strip with two boundaries, of which each has a period of $2\pi$. (d) A Möbius strip which only has one boundary of $4\pi$ period. (e) A looped Möbius strip which only has one boundary of $8\pi$ period.

Figure 6

The local energy current and density of state for edge magnon transport at equilibrium. (a) The uniform kagome lattice. (b) The lattice with a defect at the upmost site of the left fifth column. The red arrows, the blue dots, and the small black dots correspond to the local energy current, the local density of magnon, and atom sites, respectively. The color of the arrows and dots indicates the magnitude of the local current and density of states, respectively. Parameters are $\varepsilon =1.5$, $T_L=T_R=1.0$, $D/J=0.1$, and $a=1$.

Figure 7

(a) Local energy current vs the coordinate along the $y$ direction. The solid and dashed lines correspond to the local currents in two different columns in one unit cell. (b) Local density of states of magnons vs the coordinate along the $y$ direction. The solid, dashed, and dotted lines correspond to the local density of states in three different columns in one unit cell. The energy of magnon is $\varepsilon =1.5$ in the lower bulk band gap.

Figure 8

The local energy current and density of state for edge magnon transport at nonequilibrium. The red arrows, the blue dots, and the small black dots correspond to the local energy current, the local density of magnon, and atom sites, respectively. The color of the arrows and dots indicate the magnitude of the local current and density of states, respectively. The uniform kagome lattice are with (a) $T_L=1.2,T_R=0.8$ and (b) $T_L=0.8,T_R=1.2$. The lattice with a defect at the upmost site of the left fifth column are with (c) $T_L=1.2,T_R=0.8$ and (d) $T_L=0.8,T_R=1.2$. Other parameters are $\varepsilon =1.5$, $D/J=0.1$, $a=1$, $W=80$.

Figure 9

Two tetrahedrons in the pyrochlore lattice of the atom vanadium of the ferromagnet Lu${}_2$V${}_2$O${}_7$. On a single tetrahedron, all the DM vectors on bonds 1-2, 2-4, 4-1, 1-3, 2-3, and 4-3 are shown by the arrows.

Figure 10

(a) The current density vs energy of magnon for uniform and edge-defect kagome lattices with the parameters of Lu${}_2$V${}_2$O${}_7$. The solid and dotted lines correspond to the energy current in the bulk band gaps for uniform and edge-defect lattices, respectively. (b) The dispersion relation of the kagome lattice with the parameters of Lu${}_2$V${}_2$O${}_7$. $J=3.4$ meV, $D/J=-0.26$, $H_0=1$ T, $T_L=21$ K, and $T_R=19$ K.