Bosonic Bott Index and Disorder-Induced Topological Transitions of Magnons

Abstract

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” inelectronic 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 [1].We start from a ferromagnetic honeycomb layer [Fig. 1(a), (c)]. The honeycombs can be distorted, controlled by the bond angle θ, as shown in Fig. 1. We consider nearest-neighbor (NN) ferromagnetic exchange energy, NN pseudodipolar interaction [2], and on-site anisotropy energy. For strong enough perpendicular easy-axis anisotropy, the out-of-plane ferromagnetic state is an equilibrium state. We then calculate the spin wave spectra upon this state using the standard methods. Because of the bosonic commutation relation, we result in a non-Hermitian generalized eigenvalue problem η H T=Tη E, where H is the Hermitian Hamiltonian, E is the diagonal matrix of eigenvalues, η is a metric due to the bosonic commutation relation, and T is the Bogoljiubov transformation matrix diagonalizing the Hamiltonian. We define a bosonic Bott index, which takes the same form as the electronic Bott index [3] but with a modified projector P=TηΓT† η, where Γ is a matrix electing certain states.We first consider the clean limit. Figure 1(b) and (d) show spin wave spectra for infinite system, 100-wide zigzag strip, and 40*40 finite flake (from left to right) for normal and squeezed honeycomb lattices, respectively. Gapped bulk spectra in infinite and periodic systems and gapless (crossing) edge states for an open strip can be observed, indicating a nontrivial topology. A topological transition happens at θ=π/2. When θ>π/2 such as Fig. 1(a) (θ=2π/3), the upper (lower) band has Chern number +1 (-1), and the Bott indices for the finite flake give the same values as the Chern numbers. For θ<π/2 such as Fig. 1(c) (θ=5π/12), the signs of Chern numbers as well as Bott indices flip. The system can also be tuned to be topologically trivial by staggered anisotropy. The Bott indices are still consistent with Chern numbers during this process.Then we add a non-correlated random anisotropy ranged [-W,W] onto each spin in the normal honeycomb lattice to study the effect of disorder. In this non-periodic system, the k-space Chern number cannot be directly used, and the Bott index becomes powerful. When the system is nontrivial in clean limit (Bott index is 1 for the upper band), for moderate disorder, the Bott index is still one, meaning that the system keeps nontrivial. When the disorder is strong enough, a topological transition occurs and the system becomes topologically trivial [Fig. 2(a)]. This phenomenon is consistent with the common wisdom that the topology is quite robust since very strong disorder is needed to break the topology. A more remarkable phenomenon occurs when the disorder affects a topologically trivial system [Fig. 2(b)]. For a trivial system in clean limit (the band structure of a strip near the gap in the clean limit is shown in the inset), there are no gapless edge states inside the bulk gap. Surprisingly, as the disorder strength increases, the Bott index increases from zero and reaches a plateau of B=1 and then drops to zero large disorder. This finding indicates that there exists a disorder-induced topological phase, similar to the topological Anderson insulator phase in electronic systems [4].We further demonstrate the bulk-boundary correspondence in our topological magnon model by studying the transport properties. We calculate the total transmission of magnons at the mid-gap energy across a disordered sample sandwiched by two clean leads, and compare them with the Bott index results in Fig. 2(a) and (b). The agreement is quite good, demonstrating the fact that when the Bott index is nontrivial, there exists robust edge channel inside the bulk band gap, which is the common wisdom of bulk-boundary correspondence.The existence of edge states in strongly disordered magnets is further confirmed by the calculation of the real-space wave functions. Eigenstates whose energies are closest to the mid-gap energy for a certain disorder configuration are calculated, and the expectation values 〈Sx〉, 〈Sy〉 are plotted in Fig. 2(c) and (d), corresponding to the circled data points in (a) and (b), respectively. **