Magnon Hall effect and topology in kagome lattices: A theoretical investigation

Ferromagnetic insulators with Dzyaloshinskii-Moriya interaction show the magnon Hall effect, i.e., a transverse heat current upon application of a temperature gradient. In this theoretical investigation we establish a close connection of the magnon Hall effect in two-dimensional kagome lattices with the topology of their magnon dispersion relation. From the topological phase diagram we predict systems which show a change of sign in the heat current in dependence on temperature. Furthermore, we derive the high-temperature limit of the thermal Hall conductivity; this quantity provides a figure of merit for the maximum strength of the magnon Hall effect. Eventually, we compare the temperature and field dependence of the magnon Hall conductivity of the three-dimensional pyrochlore ${\mathrm{Lu}}_2{\mathrm{V}}_2{\mathrm{O}}_7$ with experimental results.

Figure 1

Pyrochlore and kagome lattices. (a) Three-dimensional pyrochlore lattice, with $\left(111\right)$ planes representing stacked two-dimensional kagome lattices (marked by blue bold lines). (b) Atomic positions, labeled by numbers, in the pyrochlore lattice. (c) Two-dimensional kagome lattice with lattice vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$. 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 $+z$ ($-z$) for a counterclockwise (clockwise) chirality: A-B-C (C-B-A).

Figure 2

Function $c_2\left(\varepsilon \right)$, as defined in Eq. (13), versus energy $\varepsilon$ for selected temperatures (as indicated). The broken line marks the high-temperature limit of ${\pi }^2/3\approx 3.28987$.

Figure 3

Relation between magnon band structure, Chern numbers, and thermal conductivity. The magnon dispersion relation for $J_{\mathrm{N}}=4$ meV, $D=1$ meV, and $J_{\mathrm{NN}}=0$ is shown in (a). The band- and energy-resolved Chern numbers $C_i\left(\varepsilon \right)$ and the energy-resolved thermal conductivity ${\kappa }^{xy}\left(\varepsilon \right)$ at $T=30$ K are displayed in (b) and (c), respectively. The step width of the energy mesh is 1/20 meV. The energy scale is compressed to account for the finite temperature (cf. Sec. 3d).

Figure 4

Dzyaloshinskii-Moriya interaction in a kagome lattice. Lattice vectors $\mathit{R}$ from atom $i=\mathrm{A},\mathrm{B},\mathrm{C}$ to the basis of both nearest (red) and next-nearest (blue) neighbors of type $j=\mathrm{A},\mathrm{B},\mathrm{C}$ are given by arrows. $\pm$ represent the signs of the Dzyaloshinskii-Moriya interaction, in accordance with the chirality. Lattice vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ are defined in Fig. 1. The dot ($•$) denotes the origin of the coordinate system.

Figure 5

Magnon band structures of a kagome lattice for selected nearest- (N) and next-nearest- (NN) neighbor Heisenberg exchange parameters (as indicated). The Dzyaloshinskii-Moriya parameter $D$ equals $J_{\mathrm{N}}=1$ meV. (a) Closing and reopening of a band gap at $K$, from left to right, according to the phase boundary given by Eq. (27). (b) Degeneracy along the $\Gamma$-K line. The bands are distinguished by colors.

Figure 6

Analysis of band degeneracies. (a) Parameter combinations $D/J_{\mathrm{N}}$ and $J_{\mathrm{NN}}/J_{\mathrm{N}}$ for which two neighboring bands are degenerate; red (blue) lines indicate degeneracy of the lower (upper) two bands 1 and 2 (2 and 3). (b) Point of degeneracy, $\zeta$, along $\Gamma$-K line as a function of the parameter ratio $J_{\mathrm{NN}}/J_{\mathrm{N}}$. (c) Boundary between antiferromagnetic (AF) and ferromagnetic (F) phases. (d) Complete topological phase diagram with regions characterized by sets $(C_1,C_2,C_3)$ of Chern numbers. The antiferromagnetic phase is colored red.

Figure 7

Transverse thermal conductivity ${\kappa }^{xy}$ versus temperature $T$ (solid lines) for $D/J_{\mathrm{N}}=1/4$ and selected ratios $J_{\mathrm{NN}}/J_{\mathrm{N}}$ (as indicated), with $J_{\mathrm{N}}=4$ meV. The values of ${\kappa }_{\lim }^{xy}$ are given within each panel and represented by broken lines.

Figure 8

Band structure (left) and integrand of Eq. (34) (right). Arrows represent the reciprocal lattice vectors and the dashed green line indicates the Brillouin zone. $J_{\mathrm{N}}=10J_{\mathrm{NN}}=8D=4$ meV.

Figure 9

Topological phase diagram of the high-temperature transverse thermal conductivity ${\kappa }_{\lim }^{xy}$, shown as color scale (right). Regions with positive (negative) values are marked $+$ ($-$). The broken lines represent the analytically derived phase boundaries given by Eq. (27).

Figure 10

Sign of the transverse thermal conductivity ${\kappa }^{xy}$ within the topological phase space for selected temperatures $T$ (as indicated). $J_{\mathrm{N}}=4$ meV.

Figure 11

Transverse thermal conductivity ${\kappa }^{xy}$ versus temperature $T$ for $J_{\mathrm{NN}}/J_{\mathrm{N}}=1/2$, $D/J_{\mathrm{N}}=1/4$, and $J_{\mathrm{N}}=4$ meV (solid line). The dashed line represents the limit ${\kappa }_{\lim }^{xy}=-2.12\times {10}^{-12}$ W/K; the dotted line marks the temperature $T_{\mathrm{s}}=113$ K of vanishing conductivity.

Figure 12

Transverse thermal conductivity ${\kappa }^{xy}$ of Lu${}_2$V${}_2$O${}_7$ versus temperature $T$. Theoretical data for Dzyaloshinskii-Moriya interactions $D=11\mu \text{eV}$ (dashed line) and $15\mu \text{eV}$ (wide-dashed line) are compared with experimental data from Ref. [5] (dots). The external magnetic field of 0.5 T is chosen slightly larger than the saturation field in the experiment. The range of the high-temperature limits of ${\kappa }^{xy}$ is indicated by the gray area.

Figure 13

Transverse thermal conductivity ${\kappa }^{xy}$ of Lu${}_2$V${}_2$O${}_7$ versus applied magnetic field $H$ along $\left[111\right]$ at $T=20$ K. The theoretical result has been obtained for Dzyaloshinskii-Moriya interaction $D=15\mu \text{eV}$ (broken line). Experimental data (red dots) for the magnetic field direction along $\left[100\right]$ are reproduced from Ref. [5].