Magnon Spin-Momentum Locking: Various Spin Vortices and Dirac Magnons in Noncollinear Antiferromagnets

TL;DR

This paper extends the concept of spin-momentum locking to magnonic systems, analyzing various antiferromagnets to reveal unique spin textures, vortices, and Dirac magnons, with implications for topological magnonics.

Contribution

It introduces a generalized formula for magnon spin expectation values and explores novel spin vortices and Dirac magnons in noncollinear antiferromagnets.

Findings

Magnon spin depends on momentum even with $z$-axis spin symmetry.

Identified a spin vortex with winding number $Q=-2$ in kagome antiferromagnets.

Discovered Dirac magnons with $Q=+1$ analogous to surface states.

Abstract

We generalize the concept of the spin-momentum locking to magnonic systems and derive the formula to calculate the spin expectation value for one-magnon states of general two-body spin Hamiltonians. We give no-go conditions for magnon spin to be independent of momentum. As examples of the magnon spin-momentum locking, we analyze a one-dimensional antiferromagnet with the N\'eel order and two-dimensional kagome lattice antiferromagnets with the 120$^\circ$ structure. We find that the magnon spin depends on its momentum even when the Hamiltonian has the $z$-axis spin rotational symmetry, which can be explained in the context of a singular band point or a $U(1)$ symmetry breaking. A spin vortex in momentum space generated in a kagome lattice antiferromagnet has the winding number $Q=-2$, while the typical one observed in topological insulator surface states is characterized by $Q=+1$. A magnonic analogue of the surface states, the Dirac magnon with $Q=+1$, is found in another kagome lattice antiferromagnet. We also derive the sum rule for $Q$ by using the Poincar\'e-Hopf index theorem.

Figures

Equations

$$H=\frac{1}{2}\sum _{\mathbf{R},{\mathbf{R}}^{\prime }}\sum _{i,j=1}^N\sum _{a,b}{J}_{ij}^{ab}(\mathbf{R},{\mathbf{R}}^{\prime })S_{\mathbf{R},i}^a{S}_{{\mathbf{R}}^{\prime },j}^b,$$

$${\mathbf{S}}_{\mathbf{R},i}\simeq$$

$$+{\mathbf{M}}_i^y\sqrt{2S_0}\frac{a_{\mathbf{R},i}-a_{\mathbf{R},i}^{†}}{2i},$$

$$H=\frac{1}{2}\sum _{\mathbf{k}}({\mathbf{a}}_{\mathbf{k}}^{†},{\mathbf{a}}_{-\mathbf{k}})\cdot {\hat{\mathcal{H}}}_{\mathbf{k}}\cdot \left(\genfrac{}{}{0ex}{}{{\mathbf{a}}_{\mathbf{k}}}{{\mathbf{a}}_{-\mathbf{k}}^{†}}\right),$$

$${\hat{Q}}_{\mathbf{k}}^{†}{\hat{\mathcal{H}}}_{\mathbf{k}}{\hat{Q}}_{\mathbf{k}}=\left(\begin{array}{cc}{\hat{E}}_{\mathbf{k}}& 0\\ 0& {\hat{E}}_{-\mathbf{k}}\end{array}\right),$$

$$H=\sum _{\mathbf{k},\alpha }{E}_{\mathbf{k},\alpha }{b}_{\mathbf{k},\alpha }^{†}{b}_{\mathbf{k},\alpha },$$

$$a_{\mathbf{k},i}=[{\hat{Q}}_{\mathbf{k}}{]}_{i,\alpha }{b}_{\mathbf{k},\alpha }+[{\hat{Q}}_{\mathbf{k}}{]}_{i,\alpha +N}{b}_{-\mathbf{k},\alpha }^{†}.$$

$${\mathbf{S}}_{\mathrm{tot}}=$$

$$=$$

$$+(\mathrm{off}\text{mathchar}45\mathrm{diagonal}\mathrm{terms})$$

$$+(\mathrm{zeroth}\text{mathchar}45\mathrm{and}\mathrm{first}\text{mathchar}45order\mathrm{terms}\mathrm{of}b,b^{†}).$$

$${\mathbf{S}}_{\mathbf{k},\alpha }\equiv ⟨\mathbf{k},\alpha |{\mathbf{S}}_{\mathrm{tot}}|\mathbf{k},\alpha ⟩-⟨0|{\mathbf{S}}_{\mathrm{tot}}|0⟩=\sum _i(-{\mathbf{M}}_i^z)\left\{|[{\hat{Q}}_{\mathbf{k}}{]}_{i,\alpha }{|}^2+|[{\hat{Q}}_{-\mathbf{k}}{]}_{i,\alpha +N}{|}^2\right\}.$$

$$S_{\mathrm{tot}}^a=$$

$$+\sum _{m=1}^{n_{\mathrm{NG}}}\left(C_m^a{b}_{\mathrm{NG},m}+(C_m^a{)}^{\ast }{b}_{\mathrm{NG},m}^{†}\right),$$

$${\tilde{S}}_{\mathrm{tot}}^z{b}_{\mathbf{k},\alpha }^{†}|0⟩$$

$$\propto b_{\mathbf{k},\alpha }^{†}|0⟩.$$

$$⟨\mathbf{k},M|{\tilde{S}}_{\mathrm{tot}}^z|\mathbf{k},M⟩=M,$$

$$⟨\mathbf{k},M|{\tilde{S}}_{\mathrm{tot}}^{x(,y)}|\mathbf{k},M⟩=\pm 1/i⟨\mathbf{k},M|[{\tilde{S}}_{\mathrm{tot}}^{y(,x)},{\tilde{S}}_{\mathrm{tot}}^z]|\mathbf{k},M⟩=0,$$

$$H^{1D}=\sum _{⟨i,j⟩}\left[J{\mathbf{S}}_i\cdot {\mathbf{S}}_j+D\hat{z}\cdot \left({\mathbf{S}}_i\times {\mathbf{S}}_j\right)\right]+K\sum _i{S}_{i,z}^2,$$

$$S_{R,A}^{\pm }=\sqrt{2S_0}{a}_{R,A}^{(†)},S_{R,A}^z=S_0-a_{R,A}^{†}{a}_{R,A},$$

$$S_{R,B}^{\mp }=\sqrt{2S_0}{a}_{R,B}^{(†)},S_{R,B}^z=a_{R,B}^{†}{a}_{R,B}-S_0,$$

$$H^{1D}=\frac{1}{2}\sum _k{\Psi }_k^{†}\left(\begin{array}{cccc}X(k)& 0& 0& Y_{-}(k)\\ 0& X(k)& Y_{+}(k)& 0\\ 0& Y_{+}(k)& X(k)& 0\\ Y_{-}(k)& 0& 0& X(k)\end{array}\right){\Psi }_k,$$

$$H^{2D}=\sum _{⟨i,j⟩}\left[\sum _a{J}^a{S}_i^a{S}_j^a+{\mathbf{D}}_{ij}\cdot ({\mathbf{S}}_i\times {\mathbf{S}}_j)\right],$$

$$\chi (M)=\sum _i{Q}_i,$$

$$\sum _i{Q}_i=0,$$

Full text

PACS numbers

75.30.Ds,75.50.Ee,72.20.-i, 75.76.+j,85.70.-w

††preprint: APS/123-QED

Introduction.—The physics of magnons chumak ; magnonics , the quanta of spin wave excitations, is enriched by their multiband nature. Although magnons have no internal degrees of freedom other than spin, such as atomic orbitals, the presence of chemical and magnetic sublattices allows magnonic systems to exhibit nontrivial band structures. In particular, recent studies have generalized many concepts in topological band theory hasan ; xlq ; niu , established in multiband electron systems, to magnonic systems, e.g. the magnon Hall effect fujimoto ; katsura ; onose ; owerre3 , magnon topological insulators shindou ; lzhang ; chisnell ; owerre , and Weyl fyli (Dirac fransson ) magnons.

Another interesting feature is spin angular momentum carried by one magnon. Magnon spin in multiband systems, which has been studied for a long time kittel , has attracted renewed attention rcheng2 ; zyuzin ; ohnuma ; rcheng motivated by recent developments in spintronic techniques chumak . Except for in simple collinear ferromagnets, magnon spin generally depends on its band properties. For instance, the magnon spin Nernst effect in antiferromagnets, which is a magnonic analogue of the spin Hall effect, is interpreted as the two copies of the magnon Hall effect for magnons with opposite spins rcheng2 ; zyuzin . This example shows the importance of considering the spin direction of each magnon mode in magnon spintronics chumak .

In this Letter, we generalize the concept of spin-momentum locking to magnonic systems. The conventional spin-momentum locking, in which electron spin depends on its momentum, is described by a noninteracting Hamiltonian without rotational symmetry in spin space, such as the Dirac Hamiltonian of topological insulator surface states hasan ; xlq . We define the magnon spin for each band and prove a no-go theorem which states that magnon spin is momentum independent for several conditions. By performing a numerical calculation for kagome lattice antiferromagnets with a 120∘ structure, we find a spin-momentum-locked magnon band characterized by the winding number winding $Q=-2$ [Fig. 1]. Our results for spin Hamiltonians with $z$-axis spin rotational symmetry demonstrate for the first time that spin-momentum locking can be generated through spontaneous symmetry breaking.

Definition of $\bm{k}$-dependent magnon spin.—A general two-body spin interaction Hamiltonian is given by

[TABLE]

where $\bm{S}_{\bm{R},i}=(S^{x}_{\bm{R},i},S^{y}_{\bm{R},i},S^{z}_{\bm{R},i})$ is the spin operator at each site, $\bm{R},\bm{R^{\prime}}$ denote the magnetic lattice vectors, $i,j$ denote the magnetic-sublattice indices, and $N$ is the number of sites in a magnetic unit cell. To rewrite the spin Hamiltonian (1) in terms of spin excitations (magnons) around a classical ground state, we introduce the Holstein-Primakoff boson operators ($a,a^{\dagger}$),

[TABLE]

where $S_{0}$ is the size of the spin and $\{\bm{M}_{i}^{a}\}$ is the set of the basis vectors of the rotating frame in which $\bm{M}_{i}^{z}$ is in the direction of classical spin at $i$. The corresponding quadratic form of boson Hamiltonian is given by

[TABLE]

where $\bm{k}$ is the crystal momentum, $\bm{a}^{\dagger}_{\bm{k}}=(a^{\dagger}_{\bm{k},1},\cdots,a^{\dagger}_{\bm{k},N})$, and $\hat{\mathcal{H}}_{\bm{k}}$ is a $2N\times 2N$ bosonic Bogoliubov-de Gennes Hamiltonian. We ignore magnon-magnon interactions in Eqs. (2) and (3) by assuming $S_{0}\gg 1$ magmageffect . The eigenenergy problem of Eq. (3) can be solved by the Bogoliubov transformation colpa ,

[TABLE]

where $\hat{Q}_{\bm{k}},\hat{Q}^{\dagger}_{\bm{k}}$ are $2N\times 2N$ paraunitary matrices and $\hat{E}_{\bm{k}}=\mathrm{diag}(E_{\bm{k},1},\cdots,E_{\bm{k},\alpha},\cdots,E_{\bm{k},N})$. ($b,b^{\dagger}$) are the magnon field operators, which satisfy

[TABLE]

Using Eqs. (2) and (5), the total spin operator is given by

[TABLE]

Thus, the $\bm{k}$-dependent magnon spin carried by a one-magnon state $|\bm{k},\alpha\rangle\equiv b^{\dagger}_{\bm{k},\alpha}|0\rangle$, where $|0\rangle$ is the Fock vacuum of ($b,b^{\dagger}$), is given by

[TABLE]

No-go conditions for ordered magnets.—In noninteracting electron systems, spin-momentum locking is forbidden for Hamiltonians with rotational symmetries rotsym . In magnonic systems, however, it does not hold, because of spontaneous symmetry breaking. In the following, we write no-go conditions for magnon spin to be independent of momentum. We first consider two-body spin Hamiltonians with $SO(3)$-rotational symmetry in spin space such as isotropic Heisenberg models. Suppose that $|0\rangle$ is a ground state with a spontaneous symmetry breaking. Using $[H,S_{\mathrm{tot}}^{a}]=0$, we can rewrite $S^{a}_{\mathrm{tot}}$ in terms of $(b,b^{\dagger})$ up to the second order supplemental ,

[TABLE]

where ($b_{\mathrm{NG},m},b^{\dagger}_{\mathrm{NG},m}$) are field operators of massless Nambu-Goldstone (NG) modes associated with the spontaneous symmetry breaking, $m$ denotes the index of the independent massless NG modes, $n_{\mathrm{NG}}$ is the total number of the massless NG modes, and $C^{a}_{m}$ are complex numbers. For collinear magnets with the symmetry breaking: SO(3)$$\rightarrow$$U(1), $|0\rangle$ is an eigenstate of the unbroken generator $\tilde{S}^{z}_{\mathrm{tot}}$, which does not include ($b_{\mathrm{NG},m},b^{\dagger}_{\mathrm{NG},m}$). A one-magnon state $|\bm{k},\alpha\rangle$ is also an eigenstate of $\tilde{S}^{z}_{\mathrm{tot}}$:

[TABLE]

Using the notation $|\bm{k},M\rangle$ that is an eigenstate of $\tilde{S}^{z}_{tot}$ with an eigenvalue M instead of $|\bm{k},\alpha\rangle$, we obtain

[TABLE]

where $\tilde{S}^{x(,y)}_{\mathrm{tot}}$ satisfy $[\tilde{S}^{a}_{\mathrm{tot}},\tilde{S}^{b}_{\mathrm{tot}}]=i\epsilon_{abc}\tilde{S}^{c}_{\mathrm{tot}}$. Equation (10) shows that $S^{a}_{\bm{k},\alpha}$ take quantized values. Because quantized spin components are not changed under a small momentum change $\bm{k}\rightarrow\bm{k}+\delta\bm{k}$, we cannot expect the $\bm{k}$-dependent magnon spin in isotropic Heisenberg models with collinear ground states. Note that there is a $trivial$ exception. The above statement assumes the smoothness of $S^{a}_{\bm{k},\alpha}$ on the magnon band $\alpha$. However, when we cannot avoid a singularity such as a band crossing point in a one-dimensional system in the adiabatic deformation $\bm{k}\rightarrow\bm{k^{\prime}}$, $S^{a}_{\bm{k},\alpha}$ can be changed across the singular region. For noncollinear and noncoplanar systems with symmetry breaking: SO(3)$$\rightarrow\{e\}, where $e$ is the identity element, we can also expect the $\bm{k}$-dependent magnon spin since there is no unbroken generator $\tilde{S}^{z}_{\mathrm{tot}}$ such that $\tilde{C}^{z}_{m}=0$ triangular .

Next, we apply a similar argument to the Hamiltonians with the $U(1)$-rotational symmetry around the $z$ axis such as $XY$ models. Since $[H,S^{z}_{\mathrm{tot}}]=0$ and $[H,S^{x,y}_{\mathrm{tot}}]\neq 0$, Eq. (8) holds only for the $z$-component spin. When there is no massless mode, $C^{z}_{m}$ and $(C^{z}_{m})^{*}$ in Eq. (8) are zero. Thus, $|0\rangle$ and $|\bm{k},\alpha\rangle$ are eigenstates of $S^{z}_{\mathrm{tot}}$, and we cannot expect the $\bm{k}$-dependent magnon spin. It is important to note that the above argument does not hold in the presence of the singularity discussed above and symmetry breaking: U(1)$$\rightarrow$$\{e\} in which there is one massless NG mode watanabe , or equivalently, the states are no longer the eigenstates of $S^{z}_{\mathrm{tot}}$. We construct examples for both cases in the following parts.

Trivial example in 1D antiferromagnet.—To gain some insight into the $\bm{k}$-dependent magnon spin, we first consider a simple model of a one-dimensional antiferromagnet, which has been studied in the context of the spin wave field effect transistor rcheng ,

[TABLE]

where $J>0$ is the nearest-neighbor exchange coupling, $D$ is the strength of the Dzyaloshinskii-Moriya (DM) interaction, and $K<0$ is the easy-axis anisotropy. Although the DM interaction and the anisotropy breaks the $SO(3)$ symmetry, they preserve the $U(1)$ symmetry around the $z$ axis.

For sufficiently small $D$, the classical ground state is the Néel state with two sublattices, $A$ with up spin and $B$ with down spin [Fig. 2(a)]. Using the Holstein-Primakoff transformation around the Néel state,

[TABLE]

we can rewrite Eq. (11) in terms of magnons as

[TABLE]

where $k$ is the one-dimensional momentum, $\Psi^{\dagger}_{k}=(a^{\dagger}_{k,A},a^{\dagger}_{k,B},a_{-k,A},a_{-k,B})$, $X(k)=2S(J-K)$, and $Y_{\pm}(k)=-2S(J\cos k\pm D\sin k)$. We set the lattice constant $a=1$. By using a standard Bogoliubov transformation technique colpa , we can find $4\times 4$ paraunitary matrices $\hat{Q}^{1D}_{\bm{k}}$ and $\hat{Q}^{1D\dagger}_{\bm{k}}$ that diagonalize Eq. (13). By performing numerical calculations, we plot magnon energies and the $z$-component magnon spin defined by Eq. (7) for $k$ in Figs. 2 (b) and (c). The band structure has two splitted bands with two crossing points and a finite energy gap. As shown in the previous section, the absence of the $U(1)$ symmetry breaking ensures that $|\bm{k},\alpha\rangle$ is an eigenstate of $S^{z}_{\mathrm{tot}}$. Thus, the $z$-component magnon spin $S^{z}_{\bm{k},\alpha}$ is quantized, while the contribution from each sublattice does not have to be. The fact that each contribution can be over $1$ comes from the quantum nature of the antiferromagnetic magnon quantumcorrection . In the upper and lower bands, the sign of $S^{z}_{\bm{k},\alpha}$ is changed across the band crossing points, which does not conflict with our discussion above. This can be interpreted as a $trivial$ example of the spin-momentum locking with a collinear spin structure in momentum space.

To explore noncollinear spin structures in momentum space such as in the topological insulator surface state, we should consider classical ground states with noncollinear spin structures in real space [see Eq. (7)] noncoplanar .

Magnon spin texture in momentum space.—As an example of a noncollinear structure, we consider the 120∘ structure in kagome lattice antiferromagnets, which have not only magnetic but also chemical sublattices. We analyze the following Hamiltonian:

[TABLE]

where $J^{a}$ are the nearest-neighbor exchange couplings, and $\bm{D}_{ij}=\pm D\hat{z}$ is the DM vector defined in Fig. 1. We consider the two interesting limits: (a) $J^{x}=J^{y}=J^{z}=J>0$, $D>0$ and (b) $J^{x}=J^{y}=J>0$, $J^{z}=D=0$, both of which have classical ground states with the 120∘ structure groundstate and preserve the $U(1)$ symmetry around the $z$ axis. We here choose the ground state shown in Fig. 1(b). By mapping Eq. (14) to the magnon Hamiltonian and performing the numerical Bogoliubov transformation maestro , we plot the magnon band dispersions and $\bm{k}$-dependent magnon spin for each case in Fig. 3 skkim .

The band structure for the case (a) has the finite-energy flat band, which is reminiscence of the zero energy flat band in the classical spin liquid phase of the isotropic Heisenberg model. There is one massless NG mode associated with the symmetry breaking U(1)$$\rightarrow\{e\}, and we can observe noncollinear spin structure in momentum space, as shown in Fig. 3(a). The norm of magnon spin is no longer quantized due to the absence of any spin rotational symmetries. The most striking feature is that spin structures in the highest and flat bands have the winding number $Q=-2$ defined in a closed curve around a $\Gamma$ point, while the original spin-momentum locking in electron systems is characterized by $Q=+1$. Although the vector plot can depend on the choice of the ground state, all plots for $U(1)$-degenerated ground states can be identified up to overall rotation in spin space, which preserves the winding number of the vortexlike spin structures. This model is thought to be realized in KFe3(OH)6(SO4)2 grohol ; matan except for some terms that slightly modify $\bm{S}_{\bm{k},\alpha}$ canted . A similar model has also been investigated in terms of a topological thermal Hall effect owerre2 .

The band structure for the case (b) also has the finite-energy flat band and one massless NG mode for the same reasons. In addition, there are two Dirac points with a finite energy in the $K$ and $K^{\prime}$ points, as shown in Fig. 3(b). For each Dirac cone, a noncollinear spin structure characterized by $Q=+1$ is realized, which is a magnonic analogue of the topological insulator surface state. Note that the magnon spin-momentum locking does not require the relativistic effect, the DM interaction, while the conventional one has been found only in systems with strong relativistic effect, the spin-orbit interaction. In magnonic systems, interesting physics can occur even in the absence of the DM interaction. For instance, Owerre showed that the topological thermal Hall effect occurs in such a situation owerre2 .

Before ending this section, we remark on the relation between a noncollinear spin texture in the two-dimensional Brillouin zone and a mathematical theorem. Although spin is not quantized, there is a mathematical way to characterize such spin structures, i.e., the Poincaré-Hopf index theorem poincare , which states that

[TABLE]

where $M$ is a compact differentiable manifold, $\chi(M)$ is the Euler characteristic of $M$, which is a typical topological invariant of the manifold poincare ; euler , $Q_{i}$ are the winding numbers around isolated zero points of a vector field, and the sum of indices is over all isolated zero points. Using this theorem for the vector field $\bm{S}_{\bm{k},\alpha}$ on the two-dimensional Brillouin zone (the two-dimensional torus $\mathbb{T}^{2}$), we obtain the sum rule for spin-momentum locking,

[TABLE]

where we use $\chi(\mathbb{T}^{2})=0$. In the spin-momentum-locked band with $Q=-2$ discussed above, the isolated zero points are $i=\Gamma$, $K$, $K^{\prime}$ with $Q_{i}=-2,1,1$, respectively.

$Discussion$.—We here briefly discuss the detection of magnon’s spin angular momentum. A current of magnon with finite spin, known as magnon spin current, can be detected by the spin pumping sandweg ; kovalev1 and the spin Seebeck caloritronics ; caloritronics2 ; kovalev2 measurements. In conventional antiferromagnets without an external magnetic field, net spin current in the bulk vanishes due to the degeneracy between the up and down bands ohnuma . The antiferromagnetic examples in Fig. 3, on the other hand, have no band degeneracy except for the crossing regions, and we can expect finite spin Seebeck signals. However, there is no established experimental method to detect directly the momentum-dependent magnon spin, while the spin- and angle-resolved photoemission spectroscopy hasan ; xlq ; hoesch enables us to detect the momentum-dependent electron spin. To this end, we theoretically propose a setup to detect the spin-momentum locking with $Q=-2$ [Fig. 1(b)]. In $Q=-2$ spin structure, the spin direction of the magnon with $(k_{x},0)$ is opposite to that with $(0,k_{y})$. This property would be observed as the spin flip under the magnon propagation in the specular reflection setup. The polarized inelastic neutron scattering chatterji is another possibility. By investigating a change in neutron spin before and after the scattering, it is possible, in principle, to detect magnon spin-momentum locking.

In summary, we presented a theory of the magnon spin-momentum locking. We gave conditions for magnon spin to be independent of momentum and constructed examples of spin-momentum locking by avoiding such conditions. We find the first example of spin-momentum locking induced by spontaneous symmetry breaking.

$Acknowledgments$.—We acknowledge many fruitful discussions with Masao Ogata, Masatoshi Imada, Hiroyasu Matsuura, Yohei Yamaji, Yusuke Kousaka, Tomonari Mizoguchi, Yohei Ema, and Yuta Kikuchi. N. O. is supported by the Japan Society for the Promotion of Science (JSPS) through Program for Leading Graduate Schools (MERIT). N. O. is also supported by JSPS KAKENHI (Grant No. 16J07110).