Spin dynamics simulations of topological magnon insulators: From transverse current correlation functions to the family of magnon Hall effects

We demonstrate theoretically that atomistic spin dynamics simulations of topological magnon insulators (TMIs) provide access to the magnon-mediated transport of both spin and heat. The TMIs, modeled by kagome ferromagnets with Dzyaloshinskii-Moriya interaction, exhibit nonzero transverse-current correlation functions from which conductivities are derived for the entire family of magnon Hall effects. Both longitudinal and transverse conductivities are studied in dependence on temperature and on an external magnetic field. A comparison between theoretical and experimental results for Cu(1,3-benzenedicarboxylate), a recently discovered TMI, is drawn.

Figure 1

Kagome planes of Cu(1,3-benzenedicarboxylate). (a) Chemical structure of Cu(1,3-benzenedicarboxylate) with parallel kagome planes built from ${\mathrm{Cu}}^{2+}$ ions (brown triangles); hydrogen atoms are shown as white spheres. Crystal data are taken from Ref. [18] and Jmol [26] was used for visualization. (b) Artistic sketch of the kagome lattice, which consists of equilateral triangles (green) and hexagons (yellow). The basis vectors of the hexagonal lattice (black arrows) and the three basis sites (red dots) are indicated. (c) Hexagonal Brillouin zone with high-symmetry points $\mathrm{\Gamma }, M$, and $K$. The points $2K$ and $2M$ are shown in addition.

Figure 2

Dynamical structure factor of a kagome ferromagnet with Dzyaloshinskii-Moriya interaction with periodic (a and b) and with open (c) boundary conditions [the extended zone scheme is shown in Fig. 1]. A magnetic field is applied in-plane (a) or perpendicular (b and c) to the two-dimensional cluster. $J=0.3\mathrm{meV}, D=0.045\mathrm{meV}, B=0.1\mathrm{meV}, T=0.01\mathrm{K}, \alpha =0.001$, and $N=14700$.

Figure 3

Effect of various parameters on (normalized) thermal current correlation functions. (a) Longitudinal correlation function for selected cluster sizes $N$; $T=0.2\mathrm{K}, \alpha =0.01, D=0, J=0.6\mathrm{meV}$, and $B=0.1\mathrm{meV}$. (b) Longitudinal correlation function for selected Gilbert damping parameters $\alpha$; $T=0.2\mathrm{K}, N=1200, D=0, J=0.6\mathrm{meV}$, and $B=0.1\mathrm{meV}$. (c) Longitudinal correlation function for selected temperatures $T$; $D=0, \alpha =0.015, N=2700, J=0.3\mathrm{meV}$, and $B=0.1\mathrm{meV}$. (d) Transverse correlation function for selected Dzyaloshinskii-Moriya interactions $D$. For comparison, the longitudinal correlation function is plotted (gray dashed line); $T=0.2\mathrm{K}, \alpha =0.1, N=1200, J=0.6\mathrm{meV}$, and $B=0.1\mathrm{meV}$.

Figure 4

Time-averaged bond thermal currents for $D>0$ (a, left) and $D<0$ (b, right). The darker and longer the arrows, the larger the associated bond current. Only a part of the sample is depicted. $T=0.8\mathrm{K}, B=0.5\mathrm{meV}, D/J=\pm 0.5$, and $\alpha =0.015$.

Figure 5

Longitudinal conductivities vs temperature (left column) and applied magnetic field (right column). The spin conductivity [(a) and (b), top row], the thermal conductivity [(c) and (d), center row], and the magnetothermal conductivity [(e) and (f), bottom row] are computed for $N=4800, J=0.3\mathrm{meV}, D=0.045\mathrm{meV}$, and $\alpha =0.015$.

Figure 6

Same as Fig. 5 but for the transverse conductivities. $\epsilon =0.1$ was used for the numerical time integration of the transverse current correlation functions (see Appendix pp3).

Figure 7

Thermal conductivities of a chain of ferromagnetically coupled spins for $B=0$ vs temperature for selected Gilbert damping constants $\alpha$. Data are normalized to ${\kappa }_0\equiv \kappa \left(0.01\mathrm{K}\right)$ for $\alpha ={10}^{-3}$.

Figure 8

Determination of the cut-off time $t_{\mathrm{c}}$. Quality measure $R$ (black curve) vs time for the transverse thermal-current correlation functions (red and blue curves; not true to scale) with a given error $\epsilon =0.1$. The gray area ($t>t_{\mathrm{c}}$) is excluded from the numerical integration. $J=0.3\mathrm{meV}, D=0.045\mathrm{meV}, T=0.1\mathrm{K}, B=0.1\mathrm{meV}$, and $N=300$.