Higher-order Topological Phases of Magnons in van der Waals Honeycomb Ferromagnets

In recent years, higher-order topological phases of matter have attracted much attention due to the unconventional bulk-boundary correspondence WABenalcazar ; ZSong ; JLangbehn ; WABenalcazar2 ; FSchindler ; MEzawa ; XZhu ; MGeier ; KKudo ; ZYan ; XWLuo ; YJWu ; CChen ; YRen ; WWang . Conventional topological insulator (TI) and superconductor (TSC) have one-dimensionally (1D) lower boundary or surface modes MZHasan ; XLQi . As a contrast, $r$th-order $(r\geq 2)$ TIs and TSCs in $d$ dimension host $(d-r)$-dimensional topologically protected modes, dubbed corner or hinge modes. These higher-order topological phases are first proposed in the electronic systems, predicted and observed in quite a few materials CChen ; CYue ; ZWang ; YXu2 ; XLSheng ; BLiu ; YBChoi ; BJack , and then extended to the bosonic systems, such as photons JNoh ; XDChen ; BYXie , phonons MSGarcia ; XZhang ; LLuo , and the magnons ZLi ; ASil ; THirosawa ; ZXLi1 ; ZXLi2 ; AMook ; MJPark ; ZXLi3 in magnetic systems. For now, the works on higher-order topological phases of magnons are mainly distributed in the magnetic solitonic system. These works in the spin systems utilize either specially designed coupling AMook ; ASil or externally controlled antiferromagnetic noncollinear order  MJPark , which are not realized experimentally due to technique difficulty.

Since the experimental discovery of two-dimensional (2D) ferromagnetism BHuang ; CGong ; ZFei , the van der Waals magnets provide new and versatile platforms to investigate the fundamental physics and also new opportunities for spintronic device design. Recent experimental progresses reveal topological phases in 2D magnetic materials, such as CrI₃ and MnBi₂Te₄, etc LChen1 ; LChen2 ; FZhu ; YDeng ; DZhang ; CLiu ; BWei . The van der Waals magnets show a store of in-plane and out-of-plane magnetic orders. For example, in the collinear phase, there are four representative ground states, the combination of intralayer ferromagnetic (FM)/antiferromagnetic (AFM) and interlayer FM/AFM orders. When involving non-ferromagnetic orders, the magnon behavior usually does not have direct counterpart to electronic and other (quasi)particle systems YHLi ; CWang ; JSklenar ; HYYuan ; GChen ; HKondo ; YMLi . The interlayer magnetic orders and coupling strength can be even tuned by the external field, pressure, stacking and the interlayer distance PJiang ; XKong ; TLi ; TSong . As a result, the van der Waals magnets offer us more degrees of freedom to seek for intriguing properties of magnons, with experimental feasibility by use of the state-of-the-art van der Waals engineering.

In this Letter, motivated by the AFM interlayer coupling in chromium trihalides, we theoretically proposed that AA-stacked honeycomb ferromagnets with AFM interlayer coupling host second-order topological phases for magnons. The monolayer is a Chern insulator in the presence of next-nearest-neighboring (NNN) Dzyaloshinskii-Moriya interactions (DMI). The interlayer AFM coupling endows a pseudo-time-reversal symmetry (PTRS), realizes $Z_{2}$ topological phases and supports helical edge states in bilayer and anisotropic topological surface modes (TSMs) in bulk located on the surfaces parallel to the stacking direction. An easy-plane anisotropic term breaks the PTRS and also the $Z_{2}$ topological phase. But a magnetic two-fold rotational symmetry is preserved. Along the lines in the Brillouin zone leaving the system invariant under the rotational symmetry, the Hamiltonian can be decomposed into two parts with different eigenvalues of the symmetry operator. A half-quantized rotation-graded topological polarizations are obtained, indicating the system hosts a second-order topological phase with corner modes in bilayer and hinge modes along the stacking direction. Besides, a staggered interlayer coupling builds a $Z_{2}\times Z$ topology, giving rise to gapped TSMs located on the surfaces perpendicular to the stacking direction, carrying non-zero Chern numbers. According to the bulk-boundary correspondence, chiral hinge modes propagate along the horizontal hinges with opposite chirality on the top and bottom horizontal hinges in a cuboid geometry. We would like to emphasize that our conclusion can be applied to different stacking, not limited to the AA stacking. This makes our proposal easier to be realized experimentally.

Model–We consider the AA-stacked van der Waals honeycomb ferromagnets for simplicity, as shown in Fig. 1 (a). The interlayer coupling is AFM-type, and the strength is staggered controlled by the interlayer distance. The minimal spin interaction Hamiltonian is given by

where $l$ is the layer index. $J_{\perp}\pm\delta J_{\perp}>0$ describe the AFM coupling and the intralayer spin Hamiltonian

$J_{\parallel}>0$ characterizes FM intralayer coupling and $D$ represents the NNN DMI strength. $\nu_{ij}=\pm 1$ with $+$ ($-$) for counterclockwise (clockwise) circulation. The last two terms denote the single-ion easy-plane (EPA) and easy-axis anisotropy. There are two ground states of the spin configuration, the down layer pointing up while the upper layer pointing down, or reversed. For convenience, we choose one of the two spin configurations without loss of generality. By applying the Holstein-Primakoff transformation $S_{+}=\sqrt{2S}a$, $S_{-}=\sqrt{2S}a^{\dagger}$, $S_{z}=S-a^{\dagger}a$ for first layer and $S_{-}=\sqrt{2S}a$, $S_{+}=\sqrt{2S}a^{\dagger}$, $S_{z}=-S+a^{\dagger}a$ for second layer, and making the Fourier transformation $a_{i}^{l}=\frac{1}{N}\sum_{\mathbf{k}}e^{i\mathbf{k}\cdot\mathbf{r}_{i}}b_{\alpha,\mathbf{k}}^{l}$, $a_{i}^{l\dagger}=\frac{1}{N}\sum_{\mathbf{k}}e^{-i\mathbf{k}\cdot\mathbf{r}_{i}}b_{\alpha,\mathbf{k}}^{l\dagger}$ ($\alpha=A,B$, the sublattice index), the Hamiltonian in $\mathbf{k}$-space is expressed as $H=\sum_{\mathbf{k}}\Psi_{\mathbf{k}}^{\dagger}H_{\mathbf{k}}\Psi_{\mathbf{k}}$ with bosonic Bogoliubov-de Gennes (BdG) Hamiltonian in $\mathbf{k}$-space SM

The basis is $\Psi_{\mathbf{k}}=(\psi_{\mathbf{k}},\psi_{-\mathbf{k}}^{\dagger})^{T}$ with $\psi_{k}=(b_{A,\mathbf{k}}^{1},b_{B,\mathbf{k}}^{1},b_{A,\mathbf{k}}^{2},b_{B,\mathbf{k}}^{2})^{T}$. $h_{1}(\mathbf{k})=h_{0}\sigma_{0}+\sum_{i=x,y,z}h_{i}\sigma_{i}$, $h_{2}(\mathbf{k})=h_{1}^{*}(-\mathbf{k})$, $h_{\perp,k_{z}}=(J_{\perp}+\delta J_{\perp})S+(J_{\perp}-\delta J_{\perp})Se^{-ik_{z}}$. $\sigma_{0}$ is $2\times 2$ identity matrix and $\sigma_{i=x,y,z}$ the Pauli matrices. $h_{0}=(2K_{z}+K_{x}+3J_{\parallel}+2J_{\perp})S$, $h_{x}=Re(\gamma_{\mathbf{k}})$, $h_{y}=-Im(\gamma_{\mathbf{k}})$, $h_{z}=-2DS\sum_{i=1}^{3}\sin(\mathbf{k}\cdot\mathbf{a}_{i})$, $\gamma_{\mathbf{k}}=-J_{\parallel}S(1+e^{-i\mathbf{k}\cdot\mathbf{a}_{3}}+e^{i\mathbf{k}\cdot\mathbf{a}_{2}})$, $\mathbf{a}_{1}=(\sqrt{3},0,0)$, $\mathbf{a}_{2}=(-\frac{\sqrt{3}}{2},\frac{3}{2},0)$, $\mathbf{a}_{3}=(-\frac{\sqrt{3}}{2},-\frac{3}{2},0)$ intralyer NNN vectors. Diagonalizing the quadratic form in Eq. (3) invokes the Bogoliubov transformation, $T_{\mathbf{k}}^{\dagger}H(\mathbf{k})T_{\mathbf{k}}=diag\{E_{\mathbf{k}},E_{-\mathbf{k}}\}$, with paraunitary eigenvectors satisfying $T_{\mathbf{k}}^{\dagger}\tau_{z}T_{\mathbf{k}}=\tau_{z}$ and $\tau_{z}$ is the Pauli matrix acting on the particle-hole space RShindou . Typical magnon bands are shown in Fig. 2 (a).

In the absence of EPA term, the Hamiltonian in Eq. (3) hosts a PTRS, $\Theta\tau_{z}H_{\mathbf{k}}=\tau_{z}H_{-\mathbf{k}}\Theta$ with $\Theta=i\tau_{z}s_{y}\sigma_{0}\mathcal{K}$, where $s_{y}$ is Pauli matrix acting on the layer index and $\mathcal{K}$ denotes the complex conjugation. We can see this PTRS coming from the AFM interlayer coupling, as the intralayer ferromagnetism breaks the TRS. As a result, the magnon bands show Kramers degeneracy (Fig. 2 (a)), and also host nontrivial Z₂ topological phase. The previous work LFu on quantum spin Hall phase and topological crystalline insulators told us four Z₂ invariants were needed to form the combination $[\nu_{0},(\nu_{1},\nu_{2},\nu_{3})]$ to describe the topological phase in 3D. As the bosonic Hamiltonian (3) does not host an inversion symmetry. We calculate the Z₂ invariants by enlarging the unit cell along $y$-direction to get a cuboid BZ RYu and obtain $\nu_{0}=0$, $(\nu_{1},\nu_{2},\nu_{3})=(001)$, indicating the system is in a “weak” TI phase SM . Consequently, topological surface modes (TSMs) appear on the surfaces parallel to the stacking direction, i.e. the $z$-axis, labelled as type-I. In Fig. 2 (b), we plot the magnon bands in the projected $k_{x}$-$k_{z}$ plane with finite width along the $y$ axis. The anisotropic gapless TSMs emerge in the gap and show a nodal line along the $k_{z}$ axis, not even number of surface Dirac cones exhibited in the previous works LFu ; YYang ; SHKooi . The $Z_{2}$ invariant for a bilayer such system should be equal to $\nu(k_{z}=0)=1$, indicating a 2D $Z_{2}$ TI phase and gapless Dirac dispersive edge states SM .

Second-order topology–In the presence of EPA term, the PTRS is no longer kept. The magnon band degeneracy is lifted, as shown in Fig. 2 (a), thus the $Z_{2}$ topological phase is broken. As a consequence, type-I TSMs at zigzag-terminated surfaces are gapped, as shown by the blue curves in Fig. 2 (b). Interestingly, when we consider a parallelogram pillar geometry with two zigzag-terminated surfaces encounter with an armchair connection, hinge modes arise in the surface gap, as shown by the red curves in Fig. 2 (c), where the magnon dispersion along $k_{z}$ direction is plotted. The local density distribution of the hinge modes is shown in the inset. Here these hinge modes come from nontrivial bulk band topology. Although the EPA breaks the PTRS and drives the 3D TI into a trivial insulator, it preserves various crystalline symmetries, i.e., a magnetic two-fold rotation symmetry in our case shown in Fig. 1 (a), which make the system to exhibit topological crystalline phase.

The magnetic two-fold rotational symmetric operation, illustrated in Fig. 1 (a), exchanges the A (B) sublattice in the first layer with the B(A) sublattice in the second layer and flips the spin. Acting on the magnon Hamiltonian ((3)), we have $\mathcal{M}_{y}\tau_{z}H_{\mathbf{k}}(k_{x},k_{y},k_{z})\mathcal{M}_{y}=\tau_{z}H_{\mathbf{k}}(k_{x},-k_{y},k_{z})$, with $\mathcal{M}_{y}=\tau_{0}s_{x}\sigma_{x}$, behaving like a mirror symmetry changing $y$ to $-y$. As $\mathcal{M}_{y}^{2}=1$, $\mathcal{M}_{y}$ has two eigenvalues $\pm 1$. This indicates that we can decompose the Hamiltonian ((3)) into decoupled two parts labelled by the eigenvalues of $\mathcal{M}_{y}$ along the lines $l_{\mathcal{M}_{y}}$ in the BZ that leave the Hamiltonian invariant after applying $\mathcal{M}_{y}$. Here we regard the $k_{z}$ as a parameter. There are two types of lines: along $k_{x}$ at $k_{y}=0$ and $2\pi/3$. We mainly discuss the case $k_{y}=0$, which is directly related to the hinge modes. The Hamiltonian at $k_{y}=0$ can be decomposed into two parts $H_{\mathbf{k}}^{\pm}(k_{y}=0)$ by using the $\mathcal{M}_{y}$ eigenvectors

where $h_{x}^{0}$ and $h_{z}^{0}$ take the value of $h_{x}$ and $h_{z}$ at $k_{y}=0$ and $\pm$ denotes the subspace with $\pm 1$ eigenvalues.

We can diagonalize the bosonic Hamiltonians $H_{\mathbf{k}}^{\pm}$ by employing the Bogoliubov transformation, $T_{\mathbf{k},\pm}^{\dagger}H_{\mathbf{k}}^{\pm}T_{\mathbf{k},\pm}=diag\{E_{\mathbf{k}}^{\pm},E_{-\mathbf{k}}^{\pm}\}$, with paraunitary eigenvectors satisfying $T_{\mathbf{k},\pm}^{\dagger}\tau_{z}T_{\mathbf{k},\pm}=\tau_{z}$. Then we have the rotation-graded Berry connection $A_{\mu}^{n,\pm}=i\mathrm{Tr}[\Gamma^{n}\tau_{z}T_{\mathbf{k},\pm}^{\dagger}\tau_{z}(\partial_{k_{\mu}}T_{\mathbf{k},\pm})]$ for n-th band, where $\Gamma^{n}$ is the diagonal matrix taking $+1$ for n-th digaonal component and zero otherwise. The rotation-graded topological polarization $P_{\mathcal{M}_{y}}^{\pm}(k_{z})=\frac{1}{2\pi}\sum_{n}\int_{l_{\mathcal{M}_{y}}}\mathbf{A}_{n}^{\pm}\cdot d\mathbf{l}_{\mathcal{M}_{y}}$. The summation is over the occupied particle bands. In our case, the integration interval is $(0,\frac{4\pi}{\sqrt{3}})$ along $k_{x}$ at $k_{y}=0$. We calculate numerically the topological polarization $P_{\mathcal{M}_{y}}^{\pm}$ for rotation-graded mangnon bands and find that $P_{\mathcal{M}_{y}}^{\pm}=1/2$ at any $k_{z}$. This half-quantized polarization confirms that the system is in a topological crystalline phase TMLec . Gapless TSMs would exist on the armchair surface even at nonvanishing EPA, different from the TSMs at zigzag surfaces that are gapped by the EPA. Besides, when two zigzag surfaces encounter with an armchair connection, in-gap modes will emerge in the surface bandgap, as verified above in Fig. 2 (c). As the $P_{\mathcal{M}_{y}}^{\pm}$ is independent on $k_{z}$, the hinge modes are not chiral and are nearly flat due to the weak interlayer coupling. Thus they are not robust against disorders. But as long as the surface gap remains, these hinge modes retain. In Fig. 2 (d), we plot the magnon dispersion along $k_{z}$ at $\delta J_{\perp}=0$. The surface gap closes at $k_{z}=\pi$, the hinge modes disappear near $k_{z}=\pi$. The above discussions can also be applied to the bilayer case. The bilayer is also in second-order topological phase with corner modes SM , as shown in Fig. 1 (c). These corner modes are also protected by the magnetic rotational symmetry.

We now discuss the topology along the $k_{z}$ direction. The EPA term breaks the Z₂ topology of the system and gives topological hinge modes. Here we show that at nonzero $K_{x}$, there exists another Z₂ topology along the stacking direction and will generate another type of TSMs and also chiral hinge modes. We start from the Bosonic BdG Hamiltonian in Eq. (3) and introduce the Berry connection ${A}_{\mu}^{n}(\mathbf{k})=i\mathrm{Tr}[\Gamma^{n}\tau_{z}T_{\mathbf{k}}^{\dagger}\tau_{z}(\partial_{k_{\mu}}T_{\mathbf{k}})]$ and the Berry curvature $\bm{\Omega}^{n}=\nabla_{\mathbf{k}}\times\mathbf{A}^{n}(\mathbf{k})$. To characterize the bulk polarization along the stacking direction, we compute the Wannier bands $w_{z}^{n}=(1/{2\pi})\int_{-\pi}^{\pi}\mathbf{A}_{z}^{n}dk_{z}$. We find when $\delta J_{\perp}<0$, $w_{z}=1/2$ for all the bands and for all $(k_{x},k_{y})$ at any finite $K_{x}$. We have checked $w_{z}=0$ for $\delta J_{\perp}>0$. This indicates a quantized bulk polarization along staking direction $P_{z}=[1/V_{\Omega}]\int_{\Omega}w_{z}dk_{x}dk_{y}=1/2$ for all the bands, where $\Omega$ is projected ontop BZ shown in Fig. 1 (b) and $V_{\Omega}$ the area. The above analysis shows an additional $Z_{2}$ invariant in our system, coming from the staggered interlayer coupling, which gives a Su-Schrieffer-Heeger (SSH) configuration of antiferomagnetic order.

The quantized polarization will give rise to a different type of TSM, labelled by type-II. In Fig. 3 (a), we plot the magnon bands in $k_{x}-k_{y}$ plane with finite layers along stacking direction. Gapped TSMs emerge between the split bands, not in the bulk bandgap. These TSMs are doubly degenerate, distributed on the top and bottom surfaces, respectively. We calculate the Chern numbers for the type-II TSMs with lower energy and find the Chern number is $+1$ ($-1$) for type-II TSMs located at bottom (top) surface. Due to the nontrivial Chern numbers, a class of $Z_{2}\times Z$ topology is established even the PTRS is broken. According to the bulk-boundary correspondence, there exists topological modes closing the TSM bandgap. In Fig. 3 (b), we further impose a finite width constraint on the $y$-direction, and verified the existence of gapless states. These states are thus chiral hinges modes, robust against disorders, and propagate along the horizontal hinges in a cuboid geometry, shown in Fig. 1 (d) SM . As type-II TSMs at top and bottom surfaces host opposite Chern numbers, the corresponding chiral hinge modes show opposite chirality. Note that chiral hinges states coexist with the gapped type-I TSMs in the bulk bandgap. Although the $Z_{2}\times Z$ topology is built at finite $K_{x}$, where the PTRS is broken, the topology is still preserved at $K_{x}=0$, as the bulk polarization along stacking direction is independent on $K_{x}$. In this situation, both the type-I SMs and chiral hinge modes are gapless and coexist in the bandgap SM .

Discussions and summary–Finally, we discuss the possibility of experimental realization of our proposal. The CrI₃, CrBr₃ and other honeycomb ferromagnets with AFM interlayer coupling are promising candidates, and even the kagome ferromagnets. There are already experimental observations of the topology of magnons in monolayer honeycomb ferromagnets, due to the DM interaction LChen1 ; LChen2 ; FZhu , which is necessary in our proposal. The stacking type is not restricted in the AA-type. For example, the bilayer CrI₃ have two stable stacking, i.e., rhombohedral stacking with FM interlayer coupling and monoclinic stacking with AFM interlayer coupling. The monoclinic stacking bilayer has an interlayer $\frac{1}{3}$-glide along the zigzag direction compared to AA stacking. With the EPA term, we can also get the second-order topological phase SM when the two layers have an interlayer $\frac{1}{3}$-glide, as the combined symmetry of $C_{2}1^{\prime}$ and the glide plays as the effective mirror symmetry of $\mathcal{M}_{y}$ in AA stacking. The $Z_{2}\times Z$ topology is independent on the stacking type, the chiral hinge modes are maintained in any stacking type. In addition, the magnetic anisotropy terms, especially EPA term, can be controlled by light, mechanical strain and electronic gating DAfanasiev ; KNakamura ; YWang ; IAVerzhbitskiy in 2D monolayer and van der Waals magnets.

In summary, we find the second-order topological phases of magnons in van der Waals ferromagnets with AFM interlayer coupling. There are two different hinge modes coexisting arising from two different mechanisms. Firstly, the AFM interlayer coupling realizes a magnonic Z₂ TI phase, protected by the PTRS. In the presence of the EPA, the PTRS is broken, but the magnetic two-fold rotational symmetry gives rise to rotation-graded topological polarization, which protects a second-order topological phase. Secondly, the introduced staggered AFM interlayer coupling brings a $Z_{2}\times Z$ topology, giving rise to type-II gapped TSMs carrying finite Chern numbers, thus supporting chiral hinge modes along horizontal directions. Our work reveals the higher-order topological phase of magnons in 2D Van der Waals magnets, and paves an new way for dissipationless magnonic devices using van der Waals magnets.

Y.-M. Li thanks Dr. Weinan Lin for his helpful discussions. This work is partly supported by MOST of China (Grants No. 2017YFA0303400). Y.-M. Li is supported by the startup funding from Xiamen University.