Bosonic Bott Index and Disorder-Induced Topological Transitions of Magnons

We investigate the role of disorder on the various topological magnonic phases present in deformed honeycomb ferromagnets. To this end, we introduce a bosonic Bott index to characterize the topology of magnon spectra in finite, disordered systems. The consistency between the Bott index and Chern number is numerically established in the clean limit. We demonstrate that topologically protected magnon edge states are robust to moderate disorder and, as anticipated, localized in the strong regime. We predict a disorder-driven topological phase transition, a magnonic analog of the “topological Anderson insulator” in electronic systems, where the disorder is responsible for the emergence of the nontrivial topology. Combining the results for the Bott index and transport properties, we show that bulk-boundary correspondence holds for disordered topological magnons. Our results open the door for research on topological magnonics as well as other bosonic excitations in finite and disordered systems.

Figure 1

(a) Schematic of a normal honeycomb magnet. The ground state is out of plane. (b) From left to right: the spin-wave spectra for infinite samples (along $k_x$ for $k_y=0$), zigzag strips of width $N_y=100$ along $x$, and finite samples of $N_x=20$ and $N_y=20$. (c) Schematic of a squeezed honeycomb magnet of $\theta =5\pi /12$. (d) Spin-wave spectra for (c).

Figure 2

Comparison between the Chern number (${\mathcal{C}}_u$) and Bott index (${\mathcal{B}}_u$) for finite and clean systems. The equivalence is established as a function of the staggered anisotropy $\mathrm{\Delta }K$ and for a system size of $40\times 40$.

Figure 3

Comparison between the Bott index and the total transmission as a function of the disorder strength $W/J$, when the clean-limit system is (a) topologically nontrivial ($\mathrm{\Delta }K=0$) and (b) trivial ($\mathrm{\Delta }K=1.35J$). The inset in (b) is the band structure near the gap of a 100-wide zigzag strip. The real-space wave function is depicted for $F=7J$ and $W=6J$ in (c) $\mathrm{\Delta }K=0$ and (d) $\mathrm{\Delta }K=1.35J$, corresponding to the circled data points in (a) and (b), respectively. This is represented by the expectation of the in plane spin components $⟨S_x⟩$ and $⟨S_y⟩$, the spatial distribution of the eigenstate whose energy is closest to ${ϵ}_0$. The size of the circles indicates the amplitude, and the color encodes the azimuthal angle.