Topological magnonic dislocations modes

TL;DR

This work analyzes how lattice dislocations influence magnon band topology in a 2D honeycomb ferromagnet. By combining Holstein-Primakoff magnons, paraunitary Bogoliubov diagonalization, the real-space Bott index, and weak $\mathbb{Z}_{2}$ invariants from Wilson loops, it shows that dislocations bind a pair of gapped magnonic modes whose existence is fixed by $N_{\mathrm{dis}}=\frac{1}{2\pi}{\bf B}\cdot{\bf G} \ (\bmod\,2)$. Bulk topology remains robust in the presence of dislocations (and even when the bulk gap closes), with dislocation modes localized at the defect cores and protected against magnetic disorder. The results establish a direct link between real-space lattice defects and the spectrum of topological spin excitations, and point to experimental routes in van der Waals magnets and potential magnonic devices leveraging localized, protected modes.

Abstract

Spin fluctuations in two-dimensional (2D) ferromagnets in the presence of crystalline lattice dislocations are investigated. We show the existence of topologically protected non-propagative modes that localize at dislocations. These in-gap states, coined as {\it magnonic dislocation modes}, are characterized by the $Z_2$ topological invariant that derives from parity symmetry broken induced by sublattice magnetic anisotropy. We uncover that bulk topology existing in the perfect crystal is robust under the influence of lattice defects, which is monitored by the real-space Bott index. It is also revealed that the topology of {magnonic dislocation modes} remains unaffected when bulk topology becomes trivial and is remarkably resilient against magnetic disorder. Our findings point to the intriguing relationship between topological lattice defects and the spectrum of topological spin excitations.

Figures (4)

• Figure 1: Schematic representation of a pair of dislocations in the hexagonal lattice with the in-plane burgers vector ${\bf B}=\pm \sqrt{3}\boldsymbol{ \hat{y}}$, at the left and right side of the dislocations, respectively. The topological magnonic dislocation modes, $\Gamma_L$ (left) and $\Gamma_R$ (right) states, are shown localized at the ends of the dislocation. These modes are gapped due to parity symmetry breaking, and their wave function amplitudes are, accordingly, colored differently.
• Figure 2: Top panel: Magnon spectrum for $S=1$, $K=2 J$, $F=0$, $\Delta_{K}=J$ and $B=0$. Bottom panel: Magnon spectrum for $S=1$, $K=2 J$, $F=J$, $\Delta_{K}=0$ and $B=0$. (a) and (e) Bulk energy spectrum without dislocation along high-symmetry points in the Brillouin zone. The Bott index for the lower and upper bulk bands satisfy $\mathcal{B}_u=-\mathcal{B}_l=0$ and $\mathcal{B}_u=-\mathcal{B}_l=-1$ at panel (a) and (e), respectively. (b) and (f) magnonic energy spectrum with PBCs in presence of a dislocation pair. The energy of dislocation modes are within the band-gap. (c) and (g) real spatial amplitude of the dislocation modes wave function $\Gamma_{L,R}(\boldsymbol{r})$. (d) and (h) Real space localization, $\sigma_{\mathrm{loc}}$, of $\Gamma_{L}(\boldsymbol{r})$ as a function of $\Delta_K/J$ and $F/J$, respectively.
• Figure 3: Left panel (a): polarization $p_{x}(k_{y})$ and right panel (b): polarization $p_{y}(k_{x})$, for the cases with bulk topology trivial $K=2J,\Delta_{K}=J,F=0,B=0$ (blue line), and bulk topology is nontrivial trivial $K=2J,\Delta_{K}=0,F=J,B=0$ (red line). In all plots we assumed $J=1$.
• Figure 4: The energy spectrum of magnons as a function of disorder strength $\eta$ averaged over $20$ realizations with parameters $S=1,K=2J, F=0,\Delta_{K}=J$. Energy gaps associated with both bulk and magnonic dislocation modes are depicted in blue and red lines.