Magnonic Quadrupole Topological Insulator in Antiskyrmion Crystals

We uncover that antiskyrmion crystals provide an experimentally accessible platform to realize a magnonic quadrupole topological insulator, whose hallmark signatures are robust magnonic corner states. Furthermore, we show that tuning an applied magnetic field can trigger the self-assembly of antiskyrmions carrying a fractional topological charge along the sample edges. Crucially, these fractional antiskyrmions restore the symmetries needed to enforce the emergence of the magnonic corner states. Using the machinery of nested Wilson loops, adapted to magnonic systems supported by noncollinear magnetic textures, we demonstrate the quantization of the bulk quadrupole moment, edge dipole moments, and corner charges.

Figure 1

Antiskyrmion crystals support topological magnonic corner states. Magnetic texture of an antiskyrmion crystal in the vicinity of a sample corner. Fractional antiskyrmions that self-assemble along the sample edge allow the emergence of a topological magnonic state whose probability amplitude (depicted in yellow) is corner localized.

Figure 2

Formation of magnonic corner states in confined antiskyrmion crystals. (a),(b) Characterization of confined antiskyrmion crystals at $g{\mu }_B{B}_z/(JS)=0.3$ and $g{\mu }_B{B}_z/(JS)=0.42$, respectively. Left: Classical ground-state magnetic texture. Middle: Magnon spectrum showing corner states (red) and trivial bound states (blue) localized at the fractional antiskyrmions away from the corners. Right: Probability density of the corner states, ${\mathrm{\Gamma }}_{\lambda }$ (defined in Ref. [38]). (c) Magnon spectrum against the applied magnetic field with corner and edge-localized states highlighted as in the above panels. The bottom color bar indicates different configurations with the configuration in (a) and (b) corresponding to the red and green region, respectively.

Figure 3

Bulk symmetries and Wannier spectra in antiskyrmion crystals. (a) Magnetic unit cell of the antiskyrmion crystal with symmetry lines for $C_{2x/2y}$. The $C_{2z}$ rotation axis is at the center of the magnetic unit cell. The first Brillouin zone is also shown. (b) Bulk magnon spectrum of the antiskyrmion crystal. The fourth bulk magnon gap is highlighted in yellow. (c),(d) Wannier spectrum of the four lowest-energy magnon bands showing the Wannier sector ${\nu }_{x/y}^{+}$ in red and ${\nu }_{x/y}^{-}$ in blue. (e),(f) Wannier centers of the Wannier sector ${\nu }_{x/y}^{-}$. For all panels, the magnetic field is $g{\mu }_B{B}_z/(JS)=0.35$ and $G_{x/y}=2\pi /(L_{x/y}a)$ with $L_{x/y}$ denoting the number of lattice sites in the magnetic unit cell along the $x/y$ axis.

Figure 4

Fractional corner charges induced by the quantized bulk quadrupole moment. (a) The value of the magnon charge density $\varphi (\mathit{r})$ of a $60\times 60$ spin lattice at $g{\mu }_B{B}_z/(JS)=0.3$ is indicated for each magnetic unit cell, which are labeled by $R_x$ and $R_y$. Green dashed lines divide the sample into four symmetry sectors whose net magnon charge is denoted by large bold numbers. The corner charge $Q_c$ obtained from each symmetry sector is equal to $\frac{1}{2}$. (b) The magnetic field dependence of $Q_c$, obtained by relaxing the classical ground-state configuration at $g{\mu }_B{B}_z/JS=0.3$ as the field is gradually increased. $Q_c$ remains quantized at $\frac{1}{2}$ below the edge instability critical field and exhibits a sudden increase above it, signaling a topological phase transition.