Magnon nodal-line semimetals and drumhead surface states in anisotropic pyrochlore ferromagnets

We introduce a type of topological magnon matter: the magnonic pendant to electronic nodal-line semimetals. Magnon spectra of anisotropic pyrochlore ferromagnets feature twofold degeneracies of magnon bands along a closed loop in reciprocal space. These magnon nodal lines are topologically protected by the coexistence of inversion and time-reversal symmetry; they require the absence of spin-orbit interaction (no Dzyaloshinskii-Moriya interaction). We calculate the topological invariants of the nodal lines and show that details of the associated magnon drumhead surface states depend strongly on the termination of the surface. Magnon nodal-line semimetals complete the family of topological magnons in three-dimensional ferromagnetic materials.

Figure 1

(a) Corner-sharing tetrahedra of the pyrochlore lattice with basis sites (red dots, labeled $1,...,4$). Sites 1, 2, and 4 form a kagome lattice with nearest-neighbor interaction $J_{\mathrm{N}}$ (solid bonds); site 3 connects adjacent kagome layers by a weaker interaction $J_{\mathrm{N}}^{\prime }$ (dashed bonds). (b) Brillouin zone of the fcc lattice with indicated high-symmetry points. (c) Magnon spectrum $[{\varepsilon }_n\left(\mathit{k}\right), n=1,...,4$] of the anisotropic pyrochlore ferromagnet along the red path depicted in (b) for $J_{\mathrm{N}}=1\mathrm{meV}, \lambda =0.8$, and $s=1/2$. Band crossings are marked $\mathrm{N}_{12}$ and $\mathrm{N}_{23}$, with insets showing magnifications. The central inset displays the spectrum for the isotropic case ($\lambda =1$).

Figure 2

Perspective views of the nodal lines (red lines) (a) $\mathrm{N}_{12}$ and (b) $\mathrm{N}_{23}$ in the fcc BZ. The high-symmetry points $\mathrm{\Gamma }$ and $\mathrm{L}$ are indicated for all perspectives. Parameters as in Fig. 1.

Figure 3

Integrated Berry phase of band 3 in dependence on the radius $r$ of the loop $C$. The center of $C$ is chosen well outside the nodal line (left inset). The nodal line (red) becomes “woven” with $C$ if its radius exceeds $r_{\mathrm{i}}$ (right inset). For $r>r_{\mathrm{o}}$ the nodal line and $C$ are again separated.

Figure 4

Magnons at the $\left(111\right)$ surface of an anisotropic pyrochlore ferromagnet for two surface terminations; Panels (a–d) show results for a triangular termination (top and central row), while (e) and (f) show results for a kagome termination (bottom row). The surface spectral density $N_0(\varepsilon ,\mathit{k})$ is represented as a color scale (black: zero; white: maximum). Bulk magnons appear as broad features, surface states as sharp light lines. Panels (a), (c), and (e) show constant-energy cuts through the entire surface Brillouin zone for energies indicated by magenta lines in (b), (d), and (f), respectively. Panels (b), (d), and (f) display spectral densities along high-symmetry directions of the surface Brillouin zone. The projected nodal lines $\mathrm{N}_{12}$ and $\mathrm{N}_{23}$ and the respective drumhead surface states $\mathrm{DSS}_{12}$ and $\mathrm{DSS}_{23}$ are marked. Parameters as in Fig. 1.

Figure 5

Magnon nodal lines and invariant planes. $\mathrm{N}_{12}$ (blue), $\mathrm{N}_{23}$ (red), and selected invariant planes $S_{abcd}$ (green) containing four TRIMs ${\mathbf{\Gamma }}_i$ ($i=a,b,c,d$) are depicted and the product of parity eigenvalues denoted below.

Figure 6

Lifshitz transition of nodal line $\mathrm{N}_{23}$ upon variation of the anisotropy parameter $\lambda$. View along (a) $\left[111\right]$ and (b) $\left[100\right]$ directions.