Nonsymmorphic-Symmetry-Protected Topological Magnons in Three-Dimensional Kitaev Materials

Topological phases in magnetic materials offer novel tunability of topological properties via varying the underlying magnetism. We show that three-dimensional Kitaev materials with nonsymmorphic symmetries can provide a great opportunity for controlling symmetry-protected topological nodal magnons. These materials are originally considered as strong candidates for the Kitaev quantum spin liquid due to the bond-dependent frustrating spin-exchange interactions. As a concrete example, we consider the symmetry and topology of the magnons in the canted zigzag ordered state in the hyperhoneycomb $\beta \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$, which can be obtained by applying a magnetic field in the counter-rotating spiral state at zero field. It is shown that the magnetic glide symmetries and the non-Hermitian nature of the bosonic magnons lead to unique topological protection that is different from the case of their fermionic counterparts. We investigate how such topological magnons can be controlled by changing the symmetry of the underlying spin-exchange interactions.

Figure 1

(a) Field-induced canted zigzag order in the hyperhoneycomb lattice. The magnetic field $\overrightarrow{h}$ is applied along the $\widehat{z}$ direction. The sign ${\xi }_{jk}$ for the off-diagonal exchange is shown. (b) Topological magnons in the first Brillouin zone with $(J,K,\mathrm{\Gamma },h)=(0.063,-1,-0.33,0.065)$. The magnetic glides ($\mathcal{T}{d}_2$, $\mathcal{T}{d}_3$) protect the green nodal lines, and the glide mirror $d_1$ protects the red nodal lines. Red (orange) and blue (green) points represent the Weyl magnons between the first (second) and second (third) lowest energy bands with chirality $-1$ and $+1$, respectively. (c) Generic magnon band crossing without magnon pairing. The magnon pairing gaps out the blue nodal lines.

Figure 2

Magnon phase diagram with $(J,h)=(0.063,0.065)$. $\mathcal{T}{d}_2$- and $\mathcal{T}{d}_3$-protected green nodal lines always exist. Strong $\mathrm{\Gamma }$ gaps out $d_1$-protected red nodal lines between the first and second lowest energy bands, but another $d_1$-protected nodal lines emerge between the third and fourth bands at $\mathrm{\Gamma }>-0.34|K|$. There are four Weyl points between the first and second bands and six Weyl points between the second and third bands at the $q_v=0$ and $q_u=0$ planes, respectively.