1.   Introduction

Spintronics is a multidisciplinary field that studies the active control and manipulation of spin in solid state systems based on spin ^{[ 1 ]} Traditional spintronics takes electron spin as the research object, but the inevitable Joule heating caused by the movement of charge makes it difficult to achieve low-energy information transport in the electron system. Spin wave is an elementary excitation spontaneously generated in a magnet due to the exchange interaction between electron spins. It is called spin wave because it has the form of a wave, and its quantized quasiparticle is also called magnon. Because magnon does not ^{[ 2 - 4 ]} .

The Hall effect has always been the core of the information transmission process and the design of spintronic memory devices. Different types of Hall effects can realize the conversion of electron charge flow and spin flow between different propagation directions. Specifically, the traditional Hall effect realizes the conversion between longitudinal electric field and transverse current through an external magnetic field. ^{[ 5 ]} The anomalous Hall effect is the conversion between spin-polarized currents without an external magnetic field. ^{[ 6 - 9 ]} The spin Hall effect realizes the transition between longitudinal charge flow and transverse spin flow through spin-orbit coupling. ^{[ 10 - 13 ]} The topological Hall effect describes the phenomenon that the topologically induced longitudinal charge flow in real space is converted into a transverse spin flow. ^{[ 14 - 17 ]} On the other hand, the Hall effect is divided into the extrinsic Hall effect induced by extrinsic factors and the intrinsic Hall effect dominated by intrinsic factors. The intrinsic Hall effect is strongly dependent on the electron spin and the Berry curvature of the reciprocal space. The Berry curvature originates from differential geometry and characterizes the wave function with wave vector. k The rate of change of k The relevance of spatial band structure. This Berry curvature has been discussed in the context of the anomalous Hall effect since the 1970s, but it was not called Berry curvature at that time. Until Berry ^{[ 18 ]} Its mathematical properties and role in physical phenomena were further clarified by pioneering work published in 1984. In addition to inverted-space topology, real-space topology also induces the Hall effect. Specifically, when an electron passes through a topological magnetic texture, its spin evolves adiabatically along the spin of the magnetic structure, and the corresponding angular momentum transfer process induces the generation of a virtual magnetic field, which in turn deflects the electron or magneton. This is called the topological Hall effect. ^{[ 14 ]} The concept of Skyrme originated from Skyrme. ^{[ 19 ]} The proposed nonlinear σ Model, which was subsequently found in central inversion symmetry breaking magnetic materials ^{[ 20 - 22 ]} , is considered to be a magnetic structure with topological protection. But the spin and k The band topology of space is not exclusive to electronic systems. As a typical boson, magnons also have spin angular momentum and corresponding band topology, and the interaction between magnons and widespread topological defects is unavoidable. Exploring the magnonic Hall effect caused by the topology of inverse space and real space is an important research direction in magnetonics.

In this review, we will review the development of magnonic Hall effect, introduce the magnonic Hall effect induced by spin-orbit coupling, the topological Hall effect based on topological magnetic solitons, the nonlinear Hall effect associated with Berry curvature dipole, and the newly reported magnonic nonlinear topological spin Hall effect based on three-magneton process. Finally, we will briefly discuss the future of this field, and explore some potential research directions and unsolved problems.

2.   Magnonic Hall effect

For both bosons and fermions, the intrinsic spin Hall effect usually comes from the virtual magnetic field induced by the Berry phase, that is, the Berry curvature. Specifically, similar to the electromagnetic field constructed by the vector potential in electrodynamics, the Berry phase can be approximated as the magnetic flux in momentum space, while the Berry curvature corresponds to the magnetic field in momentum space. The equivalent magnetic field will produce a transverse velocity proportional to the Berry curvature for the transported electrons, which leads to the electron Hall effect. $\psi_{n{\boldsymbol k}} ({\boldsymbol r}) = u_n({\boldsymbol k}, {\boldsymbol r}){\mathrm{e}}^{{\mathrm{i}}{\boldsymbol k}\cdot{\boldsymbol r}}$ . Although this Bloch wave function is spatially extended, the electron wave packet can be constructed as a linear combination of Bloch waves, which in parameter space ( ${\boldsymbol k}, {\boldsymbol r}$ The semiclassical equation of motion for) is ^{[ 23 ]} 。

$$\dot{{\boldsymbol r}}= \frac{1}{\hbar} \dfrac{\partial E_n({\boldsymbol k})}{\partial {\boldsymbol k}}-{\dot {\boldsymbol{k}}}\times {\boldsymbol \varOmega}_n({\boldsymbol k}),\; \hbar { \dot{\boldsymbol{k}}}=-e {\boldsymbol E},$$

(1)

Among, $E_n$ Is the electron th n The eigenvalue of an eigenstate, e Is the electron charge, $\hbar$ Is the reduced Planck constant, ${\boldsymbol \varOmega}_n({\boldsymbol k})$ Is the Berry curvature of the electron. (1) The second term on the right side of the first equal sign is perpendicular to the electric field E Direction, which is the source of the electron intrinsic Hall effect.

2.1   Magnon Hall effect of exchange spin waves

By analogy with the semiclassical equation of motion of electron, the wave packet equation of motion of short wavelength spin wave without considering dipolar interaction is constructed. ^{[ 24 ]} :。

$$\dot{{\boldsymbol r}}= \frac{1}{\hbar} \frac{\partial \varepsilon_n({\boldsymbol k})}{\partial {\boldsymbol k}}-{\dot{\boldsymbol {k}}}\times {\boldsymbol \varOmega}_n({\boldsymbol k}), \;\hbar { \dot{\boldsymbol{k}}}=-\nabla U({\boldsymbol r}),$$

(2)

Among $\varepsilon_n$ Is the magneton th n The eigenvalue of an eigenstate, $U({\boldsymbol r})$ Is the potential energy of the system. Similarly, the Berry curvature of the reciprocal space。

$${\boldsymbol \varOmega}_n= -\varepsilon_{\alpha\beta\gamma}{\mathrm{Im}}\bigg\langle \dfrac{\partial u_{n, {\boldsymbol k}}}{\partial k_\alpha} \bigg|\dfrac{\partial u_{n, {\boldsymbol k}}}{\partial k_\beta}\bigg\rangle,$$

Is also the key factor determining the magnonic Hall effect, where the $u_{n, {\boldsymbol k}}$ Is the periodic part of the Bloch function, $\varepsilon_{\alpha\beta\gamma}$ Because the magneton has no charge, it is not regulated by the electric field, but the energy change in the magnetic system can form an effective electric field force: $\nabla U({\boldsymbol r})$ For collinear magnets, magnons at the boundary of the system will experience a non-zero effective electric field force, which directly leads to the anomalous velocity of magnons at the boundary. ${ \dot{\boldsymbol{k}}} \times {\boldsymbol\varOmega}_n$ , and the corresponding boundary magneton current。

$$I_y = -\displaystyle\int_a^b\mathrm{d}x\partial_xU\left\{ \frac{1}{\hbar V}\sum\limits_{n,\boldsymbol{k}}^{ }\rho\left[\varepsilon_{n,\boldsymbol{k}}+U(\boldsymbol{r})\right]\varOmega_{n,z}(\boldsymbol{k}) \right\},$$

Among x = a And x = b They represent inside and outside the system respectively. $\rho(\varepsilon)$ Is the distribution function of magnetons. The magneton current is independent of the shape of the boundary, so it is very robust and can move around the whole boundary, such as Fig. 1 (B) And Fig. 1 (C) Indicated ^{[ 24 ]} .

Similar to the electron system, in addition to the motion around the boundary, the magneton wave packet also undergoes the so-called rotation in the process of motion. By analogy with the expression of the electron's rotation angular momentum, the rotation angular momentum of the magneton wave packet can be obtained:。

$$l_z^{{\mathrm{self}}}=-\frac{2}{\hbar V} {\rm Im} \sum\limits_{n, {\boldsymbol k}} \rho_n\bigg\langle \frac{\partial u_n}{\partial k_x}\bigg|(H-\varepsilon_{n{\boldsymbol k}})\bigg|\frac{\partial u_n}{\partial k_y} \bigg\rangle.$$

(3)

Consider the normalization condition for the wave function: $\displaystyle\sum\nolimits_{n} |u_n({\boldsymbol k})\rangle \langle u_n({\boldsymbol k}) | = 1$ , Eq. (3) Can be rewritten as。

$$\begin{split} &\\[-5pt] l_z^{{\mathrm{self}}}=\;&-\dfrac{2}{\hbar V} {\rm Im} \displaystyle\sum\limits_{n,n' {\boldsymbol k}} \rho_n \\ &\times\dfrac{\left\langle u_{n} \bigg|\dfrac{\partial H}{\partial k_x}\bigg| u_{n'}\right\rangle \left\langle u_{n'} \bigg|\dfrac{\partial H}{\partial k_y} \bigg| u_n \right\rangle}{\varepsilon_{n'}-\varepsilon_n}. \end{split}$$

(4)

The Berry curvature can be expressed as。

$$\begin{split} {\boldsymbol \varOmega}_n=\;&-\dfrac{2}{\hbar V} {\rm Im} \displaystyle\sum\limits_{n, n' {\boldsymbol k}} \rho_n \\ &\times \dfrac{\left\langle u_{n} \bigg|\dfrac{\partial H}{\partial k_x}\bigg| u_{n'}\right\rangle \left\langle u_{n'} \bigg|\dfrac{\partial H}{\partial k_y}\bigg| u_n \right\rangle}{(\varepsilon_{n'}-\varepsilon_n)^2} \end{split},$$

The equation ( 4 ) is highly similar to Berry curvature.

Therefore, the rotation behavior of the magneton wave packet and the rotation process around the boundary are determined by the finite Berry curvature in momentum space. Fig. 1 (a) The spin motion of the magnons and the motion around the boundary are given ^{[ 24 ]} When the system is in equilibrium, the magneton current circulates along the system boundaries and is equal in magnitude at the two opposite boundaries, resulting in zero total heat flow through the magnet. Fig. 1 (d) ), the magnetons will flow from the high temperature region to the low temperature region, which breaks the heat flow balance at the two opposite boundaries, resulting in the generation of a transverse non-zero thermal Hall magneton flow. ^{[ 24 ]} In addition, Zhang et al. ^{[ 25 ]} It is shown that these boundary magneton currents are actually chiral boundary States caused by nontrivial topological magneton bands. These boundary States have the characteristics of unidirectional propagation and topological protection, so they are immune to the interference of defects and disorder. In a word, the magneton Hall effect with robust characteristics originates from the nontrivial band structure of magnons.

In electronic systems, the spin Hall effect usually comes from the Berry curvature induced by spin-orbit coupling, which usually introduces the Dzyaloshinskii-Moriya (DM) interaction into the spin Hamiltonian in systems with broken central inversion symmetry. ^{[ 26 , 27 ]} 2010, Katsura et al. ^{[ 28 ]} The thermal Hall effect of the magneton flow in the Kagome lattice system with DM interaction is predicted, which was predicted by Onose et al. ^{[ 29 ]} In the collinear ferromagnetic insulator Lu ₂ V ₂ O ₇ Confirmed in. Lu ₂ V ₂ O ₇ The system can be described by the following Hamiltonian:。

$$\begin{array}{l} H=\displaystyle\sum\limits_{\langle i,j \rangle}[-J {\boldsymbol S}_{i}\cdot{\boldsymbol S}_{j}+{\boldsymbol D}_{ij}\cdot({\boldsymbol S}_{i}\times{\boldsymbol S}_{j})]-{\boldsymbol h}\cdot \displaystyle\sum\limits_{i}{\boldsymbol S}_{i}, \end{array}$$

(5)

Where the first term represents the Heisenberg exchange energy, the second term represents the DM interaction, and the third term is the Zeeman energy due to the magnetic field. The strength and form of the DM interaction are given by $D_{ij}={\boldsymbol D}_{ij}\cdot \boldsymbol n$ It was decided that ${\boldsymbol D}_{ij}$ Is the direction of the system DM vector, n Stands for the unit vector along the direction of the magnetic field.In order to calculate the transmission matrix elements between different magnetic substates, the Hamiltonian is transformed into an expression for the spin operator:。

$$\begin{split} H=\;&-J {\boldsymbol S}_{i}\cdot{\boldsymbol S}_{j}+{\boldsymbol D}_{ij}\cdot({\boldsymbol S}_{i}\times{\boldsymbol S}_{j}) \\ =\;& -\dfrac{J}{2} (S_i^+ S_j^-+S_i^-S_j^+) \\ & +{\mathrm{i}}\dfrac{D_{i,j}}{2}(S_i^+S_j^--S_i^-S_j^+), \end{split}$$

(6)

Among, $S_i^+=S_{i, x}+{\mathrm{i}}S_{i, y}$ , $S_i^-=S_{i, x}-{\mathrm{i}}S_{i, y}$ The raising and lowering operators corresponding to the spin respectively; J Since the system has translational symmetry, the magnetic substate can be assumed to be a Bloch state. $|k\rangle = \dfrac{1}{\sqrt N}\times \displaystyle\sum\nolimits_i {\mathrm{e}}^{{\mathrm{i}}{\boldsymbol k}\cdot {\boldsymbol r}_i} |i\rangle$ , the transmission between different magnetic substates is determined by the following matrix elements:。

$$\begin{split} \langle i| &- J {\boldsymbol S}_{i}\cdot{\boldsymbol S}_{j}+{\boldsymbol D}_{ij}\cdot({\boldsymbol S}_{i}\times{\boldsymbol S}_{j}) |j\rangle\\= \;&\langle i| -\frac{J}{2} (S_i^+S_j^-+S_i^-S_j^+)\\ &+{\mathrm{i}} \dfrac{D_{i,j}}{2}(S_i^+S_j^--S_i^-S_j^+) |j\rangle=A{\mathrm{e}}^{{\mathrm{i}}\phi_i}, \end{split}$$

(7)

In formula $A{\mathrm{e}}^{{\mathrm{i}}\phi_i}=J+{\mathrm{i}}D_{ij}$ , its argument $\phi_i$ Determined by the strength of the DM interaction.

( 7 The imaginary part of the matrix element in the equation indicates that the DM interaction is similar to a vector potential in the process of magnon transport, and acts as an "orbital magnetic field", which produces an effective Lorentz force acting on the magnon, and then induces the Hall effect of the magnon. It is worth noting that the vector potential depends on the direction of the DM vector on the one hand, that is, when the DM vector is perpendicular to the spin equilibrium state, it does not contribute to the vector potential, and on the other hand, it also depends strongly on the lattice ^{[ 30 ]} Taking pyrochlore and perovskite structures as examples, the argument induced by DM interaction $\phi_i$ It does not cancel in the unit cell of pyrochlore, which in turn induces a finite total magnetic flux, such as Fig. 2 (a) In the distorted perovskite structure, the total argument induced by the DM interaction is zero, and the corresponding total magnetic flux is also zero, as shown in Fig. 2 (B) Shown.

The above results show that the spin-orbit coupling of electrons can induce the Hall effect of both electrons and magnons. However, there is a certain difference in the spin angular momentum between electrons and magnons. Does the corresponding spin-orbit coupling also exist in the magnonic system? That is, the magnonic polarization varies with the momentum space, and can the corresponding coupling induce the Hall effect of magnons? In order to solve these problems, Shen ^{[ 31 ]} The effect of dipolar interaction on the polarization and transport of magnon in antiferromagnetic system is studied. The eigenfunction of magnon in this system can be obtained by considering the dipolar interaction in the Hamiltonian and considering the long wavelength approximation:。

$$\begin{array}{l} \psi_{\boldsymbol k}^{\pm}=\dfrac{1}{\sqrt{2}}(\alpha_{\boldsymbol k}\pm {\mathrm{e}}^{2{\mathrm{i}}\phi_{\boldsymbol k}}\beta_{\boldsymbol k}), \end{array}$$

(8)

Where, $\alpha_{\boldsymbol k}$ And $\beta_{\boldsymbol k}$ Corresponding to the left- and right-handed magneton wave functions in the antiferromagnetic $\phi_{\boldsymbol k}$ For the wave vector in x - y The eigenwave function shows the coupling of left-handed and right-handed magnons, resulting in a wave vector k Dependent spin precession trajectory (i.e., wavevector-dependent magnon polarization), as Fig. 3 (a) This is similar to the spin-orbit coupling of an electron system. Consider magnons propagating along the in-plane direction and refer to the Berry curvature expression for the induced spin Hall effect in an electron system. ^{[ 11 ]} , one can obtain the Berry curvature for both magnonic modes:。

$$\begin{array}{l} \varOmega_{z}^{\pm}=\pm \dfrac{c_{\mathrm{s}}^2}{\varDelta_{\boldsymbol k}\omega_{\boldsymbol k}}\sin^2(\phi_{\boldsymbol k}), \end{array}$$

(9)

Among, $c_{\mathrm{s}}$ Determined by the Heisenberg exchange energy and the lattice parameters of the system, $\varDelta_{\boldsymbol k}$ Then by the magneton dispersion and $c_{\mathrm{s}}$ Determined by the equation ( 9 It can be seen that the two magnons will acquire opposite Berry curvatures, which in turn leads to the magnonic spin Hall effect, such as Fig. 3 (B) Shown.

2.2   Magnon Hall effect of magnetostatic spin waves

The magnonic Hall effect described above is based on the Hall effect of short wavelength spin waves, that is, exchange spin waves. However, the dipolar interaction that exists widely in magnetic systems plays an extremely important role in the transport of long wavelength spin waves. According to the wavelength, spin waves can be divided into magnetostatic spin waves, exchange magnetostatic spin waves and exchange spin waves ^{[ 32 , 33 ]} , such as Fig. 4 (a) Magnetostatic spin waves can also be divided into three types according to their equilibrium States and the relative directions of their wave vectors, as shown in Fig. 4 (B) Similar to the exchange spin wave, the magnetostatic spin wave can also experience the magnon Hall effect induced by the Berry curvature. However, the expression for the Berry curvature needs to be rewritten as $-\varepsilon_{\alpha\beta\gamma}$ Im $\times \left\langle \dfrac{\partial {\boldsymbol m}_{n, {\boldsymbol k}}}{\partial k_\alpha} \bigg|\sigma_y \bigg|\dfrac{\partial {\boldsymbol m}_{n, {\boldsymbol k}}}{\partial k_\beta}\right\rangle$ ^{[ 34 ]} , where $\varepsilon_{\alpha\beta\gamma}$ Is the antisymmetric tensor, which is determined by the special normalization relation of the magnetostatic spin wave, namely $\langle {\boldsymbol m}_{n, {\boldsymbol k}}|\sigma_y| {\boldsymbol m}_{n, {\boldsymbol k}} \rangle = 1$ When the spin equilibrium state ${\boldsymbol M}_0$ When lying in the face ( $\theta=\pi/2$ ) The system remains unchanged under the combined operation of time reversal and in-plane rotation of 180 °, so that the Berry curvature is zero. This corresponds to the magnetostatic backward volume mode (BVMSW) and the magnetostatic surface spin wave (MSSW) mode. On the other hand, for the magnetostatic forward volume mode (FVMSW), because its saturation magnetization is in the out-of-plane direction ( θ = 0) ^{[ 35 ]} So the Berry curvature is not zero.

When only dipolar interactions are present, the system th n Eigenvalues ( n Is an integer) the corresponding magnetostatic spin wave function is。

$${\boldsymbol m}_{n,{\boldsymbol k}} = \left( \begin{array}{ccc} {\mathrm{i}}\kappa k_x + vk_y \\ -vk_x + {\mathrm{i}}\kappa ky \end{array} \right)\cos\left(\sqrt{p}kz + \dfrac{n\pi}{2}\right),$$

(10)

Among, $\kappa = \dfrac{\omega_{\mathrm{M}}\omega_{\mathrm{H}}}{\omega_H^2-\omega_n^2},\; v = \dfrac{\omega_{\mathrm{M}}\omega_n}{\omega_{\mathrm{H}}^2-\omega_n^2},\; p = -1-\kappa > 0$ . $\omega_{\mathrm{H}}=\gamma H_0$ Determined by the static magnetic field, $\omega_{\mathrm{M}}= \gamma M_{\mathrm{s}}$ From the saturation magnetization of the system $M_{\mathrm{s}}$ Decision, $\omega_n$ Then is the reciprocal space of n Substituting the corresponding wave function into the Berry curvature expression of the magnetostatic wave, we can obtain the first eigenvalue. n Berry curvature of a magnetostatic wave mode:。

$$\begin{array}{l} \varOmega_{n,z}({\boldsymbol k})=\dfrac{1}{2\omega_{\mathrm{H}} k}\dfrac{\partial\omega_n}{\partial k}\left(1-\dfrac{\omega_{\mathrm{H}}^2}{\omega_n^2}\right). \end{array}$$

(11)

Fig. 5 (a) And Fig. 5 (B) Respectively given in $H_0=M_{\mathrm{s}}$ The eigenvalue of the magnetostatic forward bulk mode and the Berry curvature pair for the case of k And n Dependency of.

From the Berry curvature of the magnetostatic spin wave, it can also be estimated in the $k_{\mathrm{B}} T \gg\hbar \omega_{\mathrm{H}}$ The thermal Hall conductance in the case ^{[ 34 ]} :。

$$\kappa_{xy}\approx \dfrac{k_{\mathrm{B}} \omega_{\mathrm{M}} N}{8\pi}\Big[1-r\log\Big(1+\dfrac{1}{r}\Big)\Big],$$

(12)

Among, $r=H_0/M_{\mathrm{s}}$ , $N=L/l_{{\mathrm{ex}}}$ , $l_{{\mathrm{ex}}}$ Is the exchange length, L Then is the layer thickness of the magnet. 12 ) shows that the thermal Hall conductance of magnetostatic spin waves is independent of temperature.

3.   Topological magnon Hall effect

除倒空间拓扑之外, 实空间拓扑也同样可以诱导出携带自旋的粒子的霍尔效应. 具体来讲, 当携带自旋角动量的粒子经过空间非共线磁织构时, 由于自旋角动量的转移, 粒子会经历一个感生磁场, 在有效洛伦兹力的作用下, 粒子运动轨迹发生偏转, 这一现象被称为拓扑霍尔效应. 该效应最早在电子体系中被报道^[14]. 同样, 携带自旋角动量的磁子在经过实空间的拓扑磁织构时也会经历由虚拟磁场诱导的有效洛伦兹力, 这一现象被称为拓扑磁子霍尔效应^[36].

3.1   虚拟电磁场理论

磁子在经过拓扑磁织构时感受到的有效磁场的来源可以这样来理解: 当把非共线磁织构中的自旋通过局域旋转矩阵变换到z轴时, 规范变换会将原本的空间导数转变为协变导数的形式$\partial_{\mu} + {\boldsymbol{A}}_{\mu}, \mu=x, y$. 这里, ${\boldsymbol{A}}_{\mu}={{{\boldsymbol{R}}}}^{-1}\partial_\mu {{{\boldsymbol{R}}}}$是一个$3\times3$的反对称矩阵, ${{{\boldsymbol{R}}}}$是旋转矩阵:

$$\begin{array}{l} {{{\boldsymbol{R}}}}=\left( \begin{array}{ccc} \cos\theta\cos\phi & -\sin\phi & \cos\phi\sin\theta \\ \cos\theta\sin\phi & \cos\phi & \sin\phi\sin\theta \\ -\sin\theta & 0 & \cos\theta \end{array} \right). \end{array}$$

(13)

式中, θ和$\phi$分别代表自旋的极化角和方位角. 对自旋应用Holstein-Primakoff (HP)变换^[37]:

$$\begin{split}& S_x= \dfrac{1}{\sqrt{2S}}\left(a+a^{\dagger}-\dfrac{a^{\dagger}aa+a^{\dagger}a^{\dagger}a}{4S}\right),\\ & S_y= -\dfrac{{\mathrm{i}}}{\sqrt{2S}}\left(a-a^{\dagger}-\dfrac{a^{\dagger}aa-a^{\dagger}a^{\dagger}a}{4S}\right),\\ &S_z= 1- {a^{\dagger}a}/{S}, \end{split}$$

(14)

代入磁体系的拉氏量:

$${\cal{L}} = \displaystyle\int {\boldsymbol{\varLambda}}\cdot \dot {{\boldsymbol{S}}} -\big [A(\nabla{\boldsymbol S})^2 + D{\boldsymbol S}\cdot(\nabla\times{\boldsymbol S})-KS_z^2\big ]{\mathrm{d}}{\boldsymbol r},$$

(15)

其中${\boldsymbol{\varLambda}}=({\boldsymbol{\varOmega}}\times{\boldsymbol{S}})/(1+{\boldsymbol{S}}\cdot{\boldsymbol{\varOmega}})$是沿着方向${\boldsymbol{\varOmega}}$的磁单极的磁矢势. 保留到玻色算子a, $a^{\dagger}$的二阶项, 可以得到不考虑磁子-磁子相互作用的二阶哈密顿量:

$$\begin{split} {\cal{L}}_2=& \displaystyle\int [-{\mathrm{i}}A(a^{\dagger}\partial_{\boldsymbol r} a-a\partial_{\boldsymbol r}a^{\dagger})-a^{\dagger}a(\cos\theta \partial_t\phi)\\ &+{\mathrm{i}} a^{\dagger}\dot{a}-A\partial_{\boldsymbol r}a^{\dagger}\cdot\partial_{\boldsymbol r}a-Ka^{\dagger}a]{\mathrm{d}}{\boldsymbol r}. \end{split}$$

(16)

进一步地, 假设磁子呈现波包的形式, 并且假设其波型在经过斯格明子时近似不变, 进而可以对磁子波包应用集体坐标理论^[38,39]. 在该近似下, 自旋波波函数对时间的偏导转变为$\partial_t \psi= \partial_{\boldsymbol r} \psi\cdot {\boldsymbol v}$, 其中${\boldsymbol v}=\partial{\omega}/\partial{\boldsymbol k}$为磁子群速度, 这一近似在描述斯格明子和畴壁的动力学中被广泛使用^[40,41]. 通过这一假设可以进一步简单地将体系的拉氏量写作波包的位置和动量的函数^[38]:

$${\cal{L}}_2 = 2a^{\dagger}a\displaystyle \int \bigg[{\boldsymbol v}\cdot\Big({\boldsymbol k}-{\boldsymbol A}_{12} + \dfrac{D{\boldsymbol S}_0}{2J}\Big)-A k^2 \bigg]{\mathrm{d}}{\boldsymbol r},$$

(17)

其中, $\omega$为磁子频率, ${\boldsymbol S}_0$为自旋平衡态, ${\boldsymbol A}_{12} = -{ \triangledown}\phi \cos \theta$代表来自交换能的有效矢势. 值得注意的是磁子的产生湮灭算符通常和磁子在某个态下的占据数有关, 而拉氏量中的$a^{\dagger}a$就对应着磁子的粒子数. 本文认为磁子的粒子数和自旋波振幅的平方相关. 随后对体系拉氏量应用欧拉-拉格朗日方程, 可以得到如下的动力学方程:

$$\begin{array}{l} a^{\dagger}am_{\rm sw}\dot{{\boldsymbol v}}-a^{\dagger}a\omega{\boldsymbol v}\times {\boldsymbol B}=0, \end{array}$$

(18)

式中, $m_{\rm sw}=1/2 A$代表归一化的磁子波包有效质量. 方程(18) 表明磁子在经过斯格明子时会经历来自交换能和DM相互作用诱导的虚拟磁场${\boldsymbol B}= B_z{\boldsymbol e}_z$的作用, 其中

$$\begin{split} \;& B_z=\dfrac{\hbar}{e}\left[{\bf \nabla} \times \left({\boldsymbol A}_{12}+\dfrac{D{\boldsymbol S}_0}{2 J}\right)\right]_z\\ =\;& \dfrac{\hbar}{e}\left[{\boldsymbol S}_0\cdot(\partial_x{\boldsymbol S}_0\times\partial_y{\boldsymbol S}_0)+\left({\bf \nabla} \times\dfrac{D{\boldsymbol S}_0}{2 J}\right)_z\right] \end{split}.$$

通常来讲, 由交换能诱导的虚拟磁场的总磁通等于斯格明子的拓扑荷, 而DM相互作用诱导的虚拟磁场的总磁通近似为零^[36]. 因此, 磁子的运动轨迹主要取决于磁织构的拓扑荷. 当拓扑荷不为零时, 磁子会经历一个斜散射(skew scattering)过程. 而当拓扑荷为零时, 磁子则会经历一个边跳跃(side jump) 过程^[38], 这类似于电子被杂质散射后导致的反常霍尔效应, 如图6(a)所示.

在反铁磁或者亚铁磁中, 由于体系具有相反自旋的两套子晶格, 使得磁子的极化会拥有全自旋自由度. 因此, 磁子会经历所谓的拓扑磁子自旋霍尔效应, 即具有相反极化的磁子流经历自旋依赖的相反的有效洛伦兹力, 继而被斯格明子分离^[42-45], 如图6(b)和图6(c)所示.

3.2   散射理论

3.1节的虚拟电磁场理论适用于斯格明子尺寸大于磁子波长的情况, 并且忽略了一些粒子数不守恒的项, 如$a^{\dagger}a^{\dagger}$和aa等. 当磁子的波长大于斯格明子直径时, 可以采用散射理论去处理该情况下的磁子-斯格明子散射行为. 该方法最早于2014年由Iwasaki等^[46]用于研究磁子和斯格明子的耦合体系.同年由Schütte等^[47]完成了更系统的研究. 考虑包含交换能、体DM相互作用和塞曼能的系统的哈密顿量为

$$\begin{array}{l} {{\cal{H}}}=\dfrac{A}{2} (\nabla {\boldsymbol S})^2+D{\boldsymbol S}\cdot(\nabla\times{\boldsymbol S})-hS_z. \end{array}$$

(19)

为了描述磁子, 可以使用由3个相互正交的单位向量$({\boldsymbol e}_1, {\boldsymbol e}_2, {\boldsymbol e}_3)$定义的局域坐标系, 其中${\boldsymbol e}_3= {\boldsymbol S}_0/|{\boldsymbol S}_0 |={\boldsymbol e}_1\times{\boldsymbol e}_2$, ${\boldsymbol S}_0$代表自旋的平衡态. 相应的磁子波函数为$\psi=(\delta_1\pm {\mathrm{i}}\delta_2)/\sqrt{2}$, 其中$\delta_{1(2)}$描述了自旋波在两个正交方向的振幅分量. 将自旋哈密顿量表达为磁子波函数的函数, 并保留其二阶微扰部分, 可以得到相应的本征方程:

$$\begin{array}{l} H(m)\psi=\varepsilon \tau^z \psi, \end{array}$$

(20)

式中, m代表散射磁子携带的轨道角动量, 哈密顿量$H(m)=H_0(m)+V(m)$, $H_0(m)$和V(m)分别为不存在斯格明子时系统的哈密顿量和斯格明子带来的有效散射势, 分别表示为

$$H_0(m) = A\left[{\boldsymbol I} \Big( -\partial_\rho^2 - \dfrac{\partial_\rho}{\rho} + \dfrac{m^2 + 1}{\rho^2} + \dfrac{h}{A} \Big) - {\boldsymbol{\tau}}^z\dfrac{2m}{\rho^2}\right],$$

(21)

$$\begin{split} &V(m) =A[v_z{\boldsymbol{\tau}}^z+v_0{\boldsymbol I}+v_x{\boldsymbol{\tau}}^x],\\ &v_z =-2m\left(\dfrac{\cos\theta-1}{\rho^2}-\frac{D\sin\theta}{A\rho}\right),\\ &v_0 =\dfrac{3[\cos(2\theta)-1]}{4\rho^2}-\dfrac{3D\sin(2\theta)}{2A\rho}\\&\qquad+\dfrac{h(\cos\theta-1)}{A}-\dfrac{D\theta'}{A}-\dfrac{\theta'^2}{2},\\ &v_x =\dfrac{\sin^2(\theta)}{2\rho^2}+\dfrac{D}{A\sin(2\theta)}{2\rho}-\dfrac{D\theta'}{A}-\dfrac{\theta'^2}{2}, \end{split}$$

(22)

式中, ${\boldsymbol{\tau}}^{x(z)}$代表泡利矩阵, I代表单位矩阵, $\rho$为自旋和斯格明子中心的距离. 在没有斯格明子的情况下, 方程(22)的本征值和波函数分别为$\varepsilon= A \Big(\dfrac{h}{A}+k^2 \Big)$, $\psi_{m, 0}=\left( \begin{array}{ccc} 1 \\ 0 \\ \end{array} \right)\dfrac{1}{\sqrt Aa}{\mathrm{J}}_{m-1}(k\rho)$, 本征函数中的${\mathrm{J}}_m$为贝塞尔函数, a为体系晶格参数. 对应的本征值代表了基态情况下铁磁磁子的色散关系. 在高能散射的情况下(磁子频率较高的情况), 非对角元的散射矩阵可以忽略(v_x), 可以只考虑$v_z$和$v_0$带来的影响. 而在非高能散射的情况下(磁子频率较低的情况), 由于斯格明子势是空间依赖的, 则需要对实空间上每个位置的本征方程进行对角化, 求得对应的本征值和波函数. 在远离斯格明子的区域, 磁子的波函数可以写为

$$\begin{array}{l} \left( \begin{array}{ccc} 1 \\ 0 \\ \end{array} \right)\left[{\mathrm{e}}^{{\mathrm{i}}{ k}\cdot{ r}}+f(\chi)\dfrac{{\mathrm{e}}^{{\mathrm{i}}k\rho}}{\sqrt{\rho}}\right], \end{array}$$

(23)

其中, $\chi$为斯格明子所在平面的方位角, $f(\chi)$代表方向依赖的磁子的散射强度, 通常由无穷远处散射波的相移决定. 采用半经典的散射理论可以计算$f(\chi)$以及对应的微分散射截面和散射角的依赖性^[48], 如图6(d)所示. 显而易见, 相对于磁子入射方向($\chi=0$), 微分散射截面具有很强的非对称性, 代表着明显的偏向散射过程. 通过计算该体系波函数在实空间的分布, 可以明显看到磁子经过斯格明子之后的多峰散射过程, 即彩虹散射过程 (rainbow scattering), 如图6(e)所示.

3.3   磁子的朗道能级

当在二维电子气中施加外磁场时, 会出现分立的量子化能级, 称为朗道能级 (Landau level). 与之对应的, 磁子在经过由斯格明子诱导的虚拟磁场时, 也可能会出现对应的磁子朗道能级. Kim等^[43]考虑磁子在斯格明子晶体中的情况, 并将斯格明子诱导的虚拟磁场做一个空间平均化处理. 此时, 在斯格明子体系中的磁子可由如下薛定谔方程描述:

$${\mathrm{i}}\hbar \partial_t \psi=\bigg[\dfrac{1}{2m}\left(\dfrac{\hbar}{{\mathrm{i}}}\nabla -q{\boldsymbol A}_{12}\right)^2\bigg]\psi,$$

(24)

其中q对应磁子的手性, 类比为磁子的有效电荷. 有效磁场${\boldsymbol B}=B_z{\boldsymbol e}_z=\displaystyle\int \dfrac{\hbar}{e}\big [{\bf \nabla} \times ({\boldsymbol A}_{12})\big ]_z {\mathrm{}}{\mathrm{d}}{\boldsymbol r}/V {\boldsymbol e}_z$可以近似为斯格明子拓扑荷对空间的平均, 其中V是系统的体积. 相应的磁子回旋频率为$\omega_{\mathrm{c}}= B_z/m$, 其本征值给出磁子体系中的朗道能级$\varepsilon_n= \hbar \omega_{\mathrm{c}}\left(n + {1}/{2}\right)$, n为整数.

4.   磁子的非线性霍尔效应

前两节的磁子霍尔效应和拓扑霍尔效应都对应着霍尔磁子流对外界激励的线性响应. 2015年, Sodemann和Fu^[49]在理论上首次预测了存在于时间反演对称性体系中的霍尔电流, 该电流与外界电场有二阶响应关系. 由于这种非线性响应, 他们将其命名为非线性霍尔效应. 为了求得非线性霍尔电流的表达式, 可从霍尔电流的密度$j_{\mathrm{a}}=-e\displaystyle\int_k f(k)v_{\mathrm{a}} {\mathrm{d}}k$ 出发, 其中f(k)为电子分布函数, $v_{\mathrm{a}}$为电子速度. 分布函数$f(k)$可以通过级数展开至电场的二阶项:

$$\begin{array}{l} f={\rm Re}\{f_0+f_1+f_2\}, \end{array}$$

(25)

其中, $f_0$是没有外场下的电子分布函数, $f_1$和$f_2$为

$$\begin{split} &f_1 =f_1^\omega {\mathrm{e}}^{{\mathrm{i}}\omega t},\; f_1^\omega=\dfrac{e\tau \varepsilon_a \partial_a f_0}{1+{\mathrm{i}}\omega\tau},\\ &f_2 =f_2^0+f_2^{2\omega}{\mathrm{e}}^{2{\mathrm{i}}\omega t},\;f_2^0=\dfrac{(e\tau)^2 \varepsilon_a^*\varepsilon_b \partial_{ab}f_0}{2(1+{\mathrm{i}}\omega \tau)},\\ &f_2^{2\omega} =\dfrac{(e\tau)^2\varepsilon_a\varepsilon_b \partial_{ab}f_0}{2(1+{\mathrm{i}}\omega \tau)(1+2{\mathrm{i}}\omega \tau)}, \end{split}$$

(26)

式中, τ为电子的弛豫时间. 相应地, 可以得到保留到电场二阶项的霍尔电流$j_{\mathrm{a}}={\mathrm{Re}}(j_{\mathrm{a}}^0+j_{\mathrm{a}}^{2\omega}{\mathrm{e}}^{2 {\mathrm{i}}\omega t})$, 其中,

$$\begin{split} &j_{\mathrm{a}}^0 = \dfrac{{e}^2}{2}\displaystyle\int_k \varepsilon_{abc} \varOmega_b \varepsilon_{{c}}^*f_1^\omega {\mathrm{d}}k-{e}\displaystyle\int_k f_2^0\partial_a\varepsilon(k) {\mathrm{d}}k,\\ & j_{\mathrm{a}}^{2\omega} = \dfrac{{{e}}^2}{2} \displaystyle\int_k \varepsilon_{abc} \varOmega_b \varepsilon_{{c}}f_1^\omega {\mathrm{d}}k - {e} \displaystyle\int_k f_2^{2\omega}\partial_a\varepsilon(k) {\mathrm{d}}k.\end{split}$$

(27)

(27)式中的第二项是完全和贝里曲率无关的项, 当考虑时间反演对称的体系时, 该项也会消失. 因此可以着重于与拓扑相关的第一项, 将霍尔电流重写为$j_{\mathrm{a}}^0=\chi_{abc}\varepsilon_b \varepsilon_c^*$以及$j_{\mathrm{a}}^{2\omega}=\chi_{abc}\varepsilon_b \varepsilon_c$, 其中

$$\chi=-\varepsilon_{abc}\dfrac{e^3\tau}{2(1+{\mathrm{i}}\omega\tau)}\displaystyle\int_k f_0(\partial_b\varOmega_d) {\mathrm{d}}k.$$

显而易见, 在存在时间反演对称性的体系中, 电子的霍尔电流由贝里曲率偶极子$\displaystyle\int_k f_0(\partial_b\varOmega_d) {\mathrm{d}}k$决定. 图7(a)给出了由贝里曲率诱导的反常霍尔效应和贝里曲率偶极子诱导的非线性霍尔效应^[50].

相应地, 类似的处理方法可以映射到磁子体系. 以由温度梯度诱导的反常能斯特效应为例, 其对应的霍尔磁子流的表达式为

$$\begin{array}{l} J_y=-\dfrac{1}{\hbar V}\displaystyle\sum\limits_{n,\boldsymbol k} \varOmega_n({\boldsymbol k}) \displaystyle\int {\mathrm{d}}\varepsilon \dfrac{\partial}{\partial x} \rho[E_n({\boldsymbol k})+\varepsilon, T(x)], \end{array}$$

(28)

其中$E_n(\boldsymbol k)$, $\varOmega_n(\boldsymbol k)$, $\rho (E, T(x))$分别代表能量本征值、第n条能带的贝里曲率和能量为E的磁子在温度$T$下的分布函数, $T(x)=T_0-x\nabla T$是随空间变化的温度分布. 为了获得和温度梯度呈现非线性关系的贝里曲率, 可从体系的玻尔兹曼输运方程出发:

$$\begin{split} & \left(\dfrac{\partial}{\partial t} + \dot{\boldsymbol x}\cdot \frac{\partial}{\partial {\boldsymbol x}} + \dot{\boldsymbol k}\cdot \frac{\partial}{\partial {\boldsymbol k}}\right)\rho [E_n({\boldsymbol k}) + \varepsilon, T(x)]= \\ & - \frac{\rho [E_n({\boldsymbol k}) + \varepsilon, T(x)]-\rho_0 [E_n({\boldsymbol k}) + \varepsilon, T(x)]}{\tau}, \\[-1pt] \end{split}$$

(29)

其中, τ和$\rho_0$分别代表磁子的弛豫时间和平衡分布函数. 考虑这是一个没有外场的稳态系统, 上述方程可以进一步约化为

$$\begin{split} \rho [E_n({\boldsymbol k})+& \varepsilon, T(x)]=\rho_0 [E_n({\boldsymbol k})+\varepsilon, T(x)]\\ &-\tau\dot{\boldsymbol x}\cdot \frac{\partial}{\partial {\boldsymbol x}}\rho [E_n({\boldsymbol k})+\varepsilon, T(x)]. \end{split}$$

(30)

将$\dot{{\boldsymbol{x}}}$重写为$(1/\hbar)\partial_{k_x}E_n(\boldsymbol k)$, $\partial/\partial{x}$重写为$-\nabla T\partial/\partial T_0$, 并将(29)式代入磁子流的表达式, 可以得到与温度梯度呈现二阶响应的磁子流表达式:

$$\begin{split}J_y=\; & \dfrac{\nabla T}{\hbar V}\displaystyle\sum\limits_n^{ }\int_{\mathrm{BZ}}^{ }\mathrm{d}^2kc_1\left\{\rho_0\left[E_n(k),T_0\right]\right\}\varOmega_n(\boldsymbol{k}) \\ & +\dfrac{\tau(\nabla T)^2}{\hbar^2VT_0}\displaystyle\sum\limits_n^{ }\int_{\mathrm{BZ}}^{ }\mathrm{d}^2kc_1\left\{\rho_0\left[E_n(k),T_0\right]\right\} \\ & \times\dfrac{\partial}{\partial k_x}\left[E_n(\boldsymbol{k})\varOmega(\boldsymbol{k})\right],\end{split}$$

(31)

其中, $c_1(\rho_0) = (1 + \rho_0)\ln(1 + \rho_0) - \rho_0\ln(\rho_0)$, 方程(30) 的第一项对应线性磁子的能斯特效应; 第二项则代表磁子流对温度梯度的非线性响应(二阶), 其和磁子的贝里曲率偶极子相关联. 如果考虑方程更高的阶数, 可以得到磁子流对温度梯度的三阶、四阶响应. 图7(b)给出了反铁磁蜂巢晶格中的非线性磁子流和交换系数的关系^[51]. 非线性磁子流由其在k空间的能带分布(图7(c))、贝里曲率(图7(d))和贝里曲率偶极子(图7(e))决定^[51].

5.   磁子非线性拓扑自旋霍尔效应

当自旋波的激发振幅较大时, 线性化方程已不足以描述磁子行为, 需要讨论自旋波高阶相互作用项的影响. 这些非线性高阶项通常表述为磁子-磁子散射过程(magnon-magnon scattering). 其中最常见的非线性过程是三磁子和四磁子散射过程. 三磁子散射包括三磁子融合和三磁子分裂两种类型, 一般情况下是由磁偶极相互作用或者非共线磁织构诱导^[52]. 三磁子融合是指两个磁子融合为一个磁子, 其逆过程为三磁子分裂, 对应一个磁子分裂为两个磁子(图8(a)), 三磁子过程可以用于产生磁子频率梳^[53,54], 如图8(b)所示. 四磁子散射主要是指两个磁子转变为另外两个磁子^[55], 主要由交换相互作用诱导产生, 如图8(c)所示. 需要指出的是, 磁子作为玻色子, 在磁子-磁子散射过程中粒子数可以不守恒, 但需要遵循能量和动量守恒.

在光学体系中, 非线性光在传输过程中和线性光一样会获得贝里相位^[56-58]. 而在磁子体系中, 非线性磁子也同样可能会经历由非共线磁织构导致的虚拟磁场. 最近, Jin等^[59]系统研究了由于三磁子过程诱导的磁子非线性拓扑霍尔效应, 揭示了非线性磁子在经过斯格明子时会感受到额外的规范场, 继而具有更大的霍尔角, 如图9(a)所示. 具体来说, 该工作考虑了在反铁磁系统中入射磁子和斯格明子呼吸模之间的相互作用, 并以此来诱导三磁子过程, 产生磁子频率梳. 为了考虑哈密顿量中不同磁子模式的贡献, 将玻色算子展开为$a= a_{\mathrm{s}}{\mathrm{e}}^{{\mathrm{i}}{\boldsymbol k}_{\mathrm{s}}\cdot{\boldsymbol r}}+ a_{\mathrm{p}}{\mathrm{e}}^{{\mathrm{i}}{\boldsymbol k}_{\mathrm{p}}\cdot{\boldsymbol r}}+a_{\mathrm{q}}{\mathrm{e}}^{{\mathrm{i}}{\boldsymbol k}_{\mathrm{q}}\cdot{\boldsymbol r}}+a_{\mathrm{r}}\psi_{\mathrm{r }}$, 其中${\boldsymbol k}_{\mathrm{s}}$, ${\boldsymbol k}_{\mathrm{p}}$和${\boldsymbol k}_{\mathrm{q}}$分别为入射模$a_{\mathrm{s}}$、合频模$a_{\mathrm{p}}$和差频模$a_{\mathrm{q}}$在无穷远处的波矢, $\psi_{\mathrm{r}}$为呼吸模$a_{\mathrm{r}}$的波函数. 相应的哈密顿量为(考虑归一化的自旋矢量S = 1) ${{\cal{H}}}={{\cal{H}}}_2+{{\cal{H}}}_3$. 其中代表磁子线性过程的二阶哈密顿量为

$$\begin{split} {{\cal{H}}}_2=\;&\displaystyle\sum\limits_{i={\mathrm{s,p,q}}}2a_i^{\dagger}a_i\displaystyle\int \bigg[{\dfrac{1}{J}\omega_i^2{\boldsymbol v}_i^2} -\left(2{\boldsymbol A}_{12}+\dfrac{D{\boldsymbol l}_0}{J}\right)\cdot\omega_i{\boldsymbol v}_i \bigg]{\mathrm{d}}{\boldsymbol r}. \end{split}$$

(32)

哈密顿量中的 ${\boldsymbol l}_0$为平衡态下反铁磁体系的奈尔矢量. 而代表非线性过程的三阶哈密顿量为${{\cal{H}}}_3= {{\cal{H}}}_{3 {\mathrm{s}}}+{{\cal{H}}}_{3 {\mathrm{p}}}+{{\cal{H}}}_{3 {\mathrm{q}}}$, 其中${{\cal{H}}}_{3 {\mathrm{s}}}$, ${{\cal{H}}}_{3 {\mathrm{p}}}$和${{\cal{H}}}_{3 {\mathrm{q}}}$分别表示入射、融合和分裂磁子模所贡献的哈密顿量, 具体表达式为

$$\begin{split} &{{\cal{H}}}_{3{\mathrm{s}}}= \int\omega_{\mathrm{s}}{\boldsymbol v}_s\cdot\Big \{-\dfrac{1}{\sqrt{2}} \Big[({\mathrm{i}}{\boldsymbol A}_{13}+{\boldsymbol A}_{23})+\dfrac{D}{2J}({\mathrm{i}}{\boldsymbol e}_{\phi}+{\boldsymbol e}_{\theta})\Big] \Big[3a_{\mathrm{q}}a_{\mathrm{r}}a_{\mathrm{s}}^{\dagger}{\mathrm{e}}^{{\mathrm{i}}(-{\boldsymbol k}_{\mathrm{s}}+{\boldsymbol k}_{\mathrm{q}})\cdot{\boldsymbol r}}+a_{\mathrm{p}}^{\dagger}a_{\mathrm{r}}a_{\mathrm{s}}{\mathrm{e}}^{{\mathrm{i}}({\boldsymbol k}_{\mathrm{s}}-{\boldsymbol k}_{\mathrm{p}})\cdot{\boldsymbol r}}\Big]\psi_{\mathrm{r}}+{\rm H.c.}\Big\}{\mathrm{d}}{\boldsymbol r},\\ &{{\cal{H}}}_{3{\mathrm{p}}}= \int\omega_{\mathrm{p}}{\boldsymbol v}_{\mathrm{p}}\cdot\Big\{-\dfrac{3}{\sqrt{2}}\Big[({\mathrm{i}}{\boldsymbol A}_{13}+{\boldsymbol A}_{23})+\dfrac{D}{2J}({\mathrm{i}}{\boldsymbol e}_{\phi}+{\boldsymbol e}_{\theta})\Big] a_{\mathrm{s}} a_{\mathrm{r}}a_{\mathrm{p}}^{\dagger}{\mathrm{e}}^{{\mathrm{i}}({\boldsymbol k}_{\mathrm{s}}-{\boldsymbol k}_{\mathrm{p}})\cdot{\boldsymbol r}}\psi_{\mathrm{r}}+{\rm H.c.}\Big\}{\mathrm{d}}{\boldsymbol r}, \\ &{{\cal{H}}}_{3{\mathrm{q}}}= \int\omega_{\mathrm{q}}{\boldsymbol v}_{\mathrm{q}}\cdot\Big\{ -\dfrac{1}{\sqrt{2}}\Big[({\mathrm{i}}{\boldsymbol A}_{13}+{\boldsymbol A}_{23})+\dfrac{D}{2J}({\mathrm{i}}{\boldsymbol e}_{\phi}+{\boldsymbol e}_{\theta})\Big] a_{\mathrm{s}}a_{\mathrm{r}}^{\dagger}a_{\mathrm{q}}^{\dagger}{\mathrm{e}}^{{\mathrm{i}}({\boldsymbol k}_{\mathrm{s}}-{\boldsymbol k}_{\mathrm{q}})\cdot{\boldsymbol r}}\psi_{\mathrm{r}}^*+{\rm H.c.}\Big\}{\mathrm{d}}{\boldsymbol r},\\[-1pt] \end{split}$$

(33)

其中, ${\boldsymbol A}_{\nu\nu'}={\cal{A}}_{x, \nu\nu'}{\boldsymbol e}_x+{\cal{A}}_{y, \nu\nu'}{\boldsymbol e}_y$代表由局域坐标变换带来的规范场($\nu, \nu'=1, 2, 3$); ${\boldsymbol e}_{\theta}$和${\boldsymbol e}_{\phi}$为极坐标系中的两个单位矢量; 而$\omega_{\mathrm{s}}$, $\omega_{\mathrm{p}}$, $\omega{\mathrm{_q}}$和$\omega_{\mathrm{r}}$是入射模、融合模、分裂模和斯格明子呼吸模的频率, 并且符合能量守恒$\omega_{{\mathrm{p}}({\mathrm{q}})}=\omega_{{\mathrm{s}}}\pm\omega_{\mathrm{r}}$. 显而易见, 传统的规范场${\boldsymbol A}_{12}$只出现在二阶哈密顿量中, 而规范场${\boldsymbol A}_{13}$和${\boldsymbol A}_{23}$则出现在磁子的非线性过程中. 对体系的总拉氏量运用欧拉-拉格朗日方法, 可以得到不同模式波包的运动方程:

$$a_i^{\dagger}a_i\dfrac {\hbar\omega_i^2}{eJ}\dot{{\boldsymbol v}_i}-a_i^{\dagger}a_i\omega_i{\boldsymbol v}_i \times {\boldsymbol B}-{\boldsymbol F}_i^{{\rm nl}}=0,\;i = {\mathrm{s,p,q}},$$

(34)

其中B与方程(17)中的虚拟磁场等价, 是导致传统磁子霍尔效应的虚拟磁场. 这里$\hbar$和e是约化普朗克常数和单位电荷, 而${\boldsymbol F}_i^{{\rm nl}}$代表来自于三磁子非线性过程的额外的有效洛伦兹力:

$$\begin{array}{l} {\boldsymbol F}_i^{{\rm nl}}=c_i{\boldsymbol v}_i\times {\boldsymbol B}',\;i={\mathrm{s,p,q}},\\ \end{array}$$

(35)

$$\begin{split} \;& {\boldsymbol B}'=B_z'{\boldsymbol e}_z, \\ \;& B_z'=\dfrac{\hbar}{e}\left({\boldsymbol \nabla} \times {\boldsymbol A}_{23}\right)_z+\dfrac{\hbar D}{2 Je}({\boldsymbol \nabla} \times{\boldsymbol e}_{\theta})_z\\ =\;&\dfrac{\hbar}{e}\left[\partial_y{\boldsymbol l}_0\cdot\partial_x \Big({\boldsymbol n}\times\dfrac{{\boldsymbol e}_z}{\sin\theta}\Big) - \partial_x{\boldsymbol l}_0\cdot\partial_y \Big({\boldsymbol n}\times\dfrac{{\boldsymbol e}_z}{\sin\theta}\Big)\right]\\ &+\dfrac{\hbar D}{2 Je}\left(\nabla\times\dfrac{{\boldsymbol e}_z+\cos\theta{\boldsymbol l}_0}{\sin\theta}\right)_z\end{split}$$

代表来自于三磁子过程贡献的额外虚拟磁场. 由于斯格明子的旋转对称性, 矢势${\boldsymbol A}_{13}$的贡献为零. 这里

$$\begin{split} &c_{\mathrm{s}}= ({\omega_{\mathrm{s}}}/{4})(g_{\mathrm{p}}a_{\mathrm{p}}^{\dagger}a_{\mathrm{r}}a_{\mathrm{s}}+3 g_{\mathrm{q}}a_{\mathrm{q}}^{\dagger}a_{\mathrm{r}}^{\dagger}a_{\mathrm{s}}+ {\rm H.c.}), \\ &c_{\mathrm{p}}= ({3\omega_{\mathrm{p}}}/{4})(g_{\mathrm{p}}a_{\mathrm{s}}a_{\mathrm{r}}a_{\mathrm{p}}^{\dagger}+{\rm H.c.}), \\ & c_{\mathrm{q}}= ({\omega_{\mathrm{q}}}/{4})(g_{\mathrm{q}}a_{\mathrm{s}} a_{\mathrm{r}}^{\dagger}a_{\mathrm{q}}^{\dagger}+{\rm H.c.}), \\ &g_{\mathrm{p}}=\dfrac{1}{\sqrt{2 S}V}\displaystyle\int {\mathrm{e}}^{{\mathrm{i}}({\boldsymbol k}_{\mathrm{s}}-{\boldsymbol k}_{\mathrm{p}})\cdot{\boldsymbol {{r}}}}\psi_{\mathrm{r}}{\mathrm{d}}{\boldsymbol r}, \\ &g_{\mathrm{q}}= \dfrac{1}{\sqrt{2 S}V} \displaystyle\int {\mathrm{e}}^{{\mathrm{i}}({\boldsymbol k}_{\mathrm{s}}-{\boldsymbol k}_{\mathrm{q}})\cdot{\boldsymbol r}}\psi_{\mathrm{r}}^*{\mathrm{d}}{\boldsymbol r}, \end{split}$$

代表了磁子的重叠积分, 其中V是系统的体积. 在整个磁子和斯格明子的散射过程中, 自旋波波包可以看作是在虚拟磁场B和B'下运动的点粒子. 进一步考虑4种磁子模式之间的关系:

$$\begin{split} &a_{\mathrm{p}}=\dfrac{g}{\varepsilon+{\mathrm{i}}\alpha(\omega_{\mathrm{s}}+\omega_{\mathrm{r}})}a_{\mathrm{s}}a_{\mathrm{r}},\\ &a_{\mathrm{q}}=\dfrac{g}{\varepsilon-{\mathrm{i}}\alpha(\omega_{\mathrm{s}}-\omega_{\mathrm{r}})}a_{\mathrm{s}}a_{\mathrm{r}}^{\dagger},\end{split}$$

(36)

可以将方程(33) 转变为

$$m_{{\rm sw},\,i}\dot {\boldsymbol v}_i-e{\boldsymbol v}_i\times \sigma({\boldsymbol B}+\lambda_i{\boldsymbol B}')=0, \;i={\mathrm{s,p,q}}.$$

(37)

这里, $m_{{\rm sw}, i}=\hbar\omega_i/J$为反铁磁中自旋波波包的有效质量, $\sigma=\mp1$表示左右旋磁子, e为元电荷, $\lambda_{\mathrm{s}} = n_{\mathrm{r}}\left(\dfrac{gg_{\mathrm{p}}}{4\varepsilon} +\dfrac{3 gg_{\mathrm{q}}}{4\varepsilon} +{\rm H.c.}\right)$, $\lambda_{\mathrm{p}} = \dfrac{3}{4}\left(\dfrac{\varepsilon g_{\mathrm{p}}}{g} + {\rm H.c.}\right)$, $\lambda_{\mathrm{q}}=\dfrac{1}{4}\left(\dfrac{\varepsilon g_{\mathrm{q}}}{g}+{\rm H.c.}\right)$, 其中$n_{\mathrm{r}}=\langle a_{\mathrm{r}}^{\dagger}a_{\mathrm{r}}\rangle$为斯格明子呼吸模的粒子数, g为三磁子耦合强度, $\varepsilon = \omega - \omega_{\mathrm{r}}$. 图9(b)给出了虚拟磁场$B_z/B_0$和$B'_z/B_0$的空间分布, 其中$B_0=\hbar/{a^2 e}$, a是体系的晶格参数. 可以看出它们的大小在同样的数量级. 通过数值求解方程(35)在${\boldsymbol B}'=0$和${\boldsymbol B}'\neq 0$的情况, 可以得到对应的磁子运动轨迹(图9(b)). 显而易见, 三磁子过程诱导的虚拟磁场会诱导更大的磁子霍尔角. 进一步, 微磁学模拟也论证了上述的理论分析, 即相较于入射波模式, 差频和合频模式具有更为显著的磁子霍尔角, 如图9(c)所示. 值得注意的是, 随着非线性阶数m的增加, 磁子的霍尔角也几乎呈现线性增加的趋势, 如图9(d)所示. 这一过程可以类比为光经过大气层之后经历的多次折射现象.

在该体系中, 散射磁子流和入射磁子流之间存在非线性响应. 具体来说, 当磁子经过斯格明子时, 通过三磁子过程激发出斯格明子的呼吸模式. 随后入射磁子和斯格明子呼吸模耦合产生融合与分裂模. 假设呼吸模的磁子数正比于入射波的磁子数即 $a_{\mathrm{r}}=ca_{\mathrm{s}}$, 根据方程(35), 相应的融合和分裂模式的磁子数可以表示为

$$\begin{split} &n_{\mathrm{p}}=\dfrac{(cg)^2}{\varepsilon^2+\alpha^2(\omega_{\mathrm{s}}+\omega_{\mathrm{r}})^2}n_{\mathrm{s}}^2,\\ &n_{\mathrm{q}}=\dfrac{(cg)^2}{\varepsilon^2+\alpha^2(\omega_{\mathrm{s}}-\omega_{\mathrm{r}})^2}n_{\mathrm{s}}^2, \end{split}$$

(38)

其中$n_{\mathrm{s}}$, $n_{\mathrm{p}}$和$n_{\mathrm{q}}$分别代表入射磁子、融合和分裂模式磁子的磁子数. 显而易见, 其散射磁子数和入射磁子数确实呈现非线性的关系. 这种全新的非线性霍尔效应起源于四点. 1)玻色子粒子数不守恒的内禀性质: 单个磁子可以分裂为多个磁子, 也可以与其他磁子合并成一个磁子, 这在低能费米子体系中没有对应. 2) 非线性三磁子散射形成的磁子频率梳: 频域上一串离散的具有相同频率间隔的自旋波谱线. 3)隐藏的规范场: 分析表明剩下的两个规范场矩阵元出现在磁织构的非线性磁子输运中, 它们作用在磁子频率梳上, 产生巨大的磁子霍尔角. 4)反铁磁磁子具有的两种自旋态: 分别对应右旋和左旋的磁矩进动模式.

6.   总结与展望

本文回顾了磁子霍尔效应最新的研究进展. 以电子霍尔效应为起点, 介绍了由能带拓扑和实空间拓扑引起的磁子霍尔效应和磁子拓扑霍尔效应. 进一步地, 考虑磁子流对外界刺激的二阶响应, 可以得到由贝里曲率偶极子诱导的磁子非线性霍尔效应. 而这一非线性效应也可以推广到由实空间磁织构引起的拓扑霍尔效应当中, 导致磁子非线性拓扑自旋霍尔效应. 值得指出的是, 近年来, 磁子霍尔效应家族正在不断发展壮大. 除了本文主要介绍的霍尔效应外, 还包括磁子能斯特效应和磁子塞贝克效应等^[60,61]. 隐藏在磁子传输过程中更多新奇的物理效应值得进一步挖掘. 具体来说, 斯格明子晶体中磁子的朗道能级通常具有非零的陈数, 继而对磁子的霍尔效应产生贡献. 但是目前大部分基于磁子朗道能级的研究都局限于线性区域, 不涉及非线性效应. 单个斯格明子的存在破坏了体系的空间平移对称性, 而大量斯格明子构成的斯格明子周期结构(斯格明子晶体)能恢复这种对称性. 在磁子体系中, 三磁子相互作用所主导的非线性哈密顿量会对磁子能带引入厄密和非厄密的自能修正. 其中厄密的自能项会对磁子能带进行重整化而非厄密自能则会影响磁子能带的展宽^[62]. 基于上述磁子非线性作用对倒空间拓扑可能带来的影响, 斯格明子晶体中隐藏在磁子-磁子相互作用中的规范场对磁子朗道能级的影响也值得进一步探索.

在真实材料中, 磁子-声子散射通常是不可避免的, 其可以通过两种不同的方式去影响磁子. 首先, 声子激发造成的晶格畸变可能会影响磁矩之间交换作用的大小和各向异性. 其次, 晶格振动可能会产生显著的磁子-声子耦合形成磁子极化子, 并在耦合体系的能谱中形成反交叉. 在磁性绝缘体中, 由温度梯度形成的热霍尔流可以由磁子或声子单独携带, 也可以由新的准粒子-磁子极化子携带^[63]. 具体来说, 当磁化方向上的镜像对称性被破坏时, 自旋-晶格相互作用将会引起热霍尔效应, 并且热霍尔电导率由磁子和声子之间的耦合强度决定. 但是这种磁体系中的非本征霍尔效应都集中在磁子的线性输运过程中. 非本征磁子非线性霍尔效应的理论也值得进一步的发展和探索.

迄今为止, 磁子霍尔效应的研究对象大多局限于传统的铁磁或反铁磁材料体系. 近年来, 转角范德瓦耳斯层因为其中的拓扑平带具有诸多新奇拓扑物态现象, 包括非平凡超导和Mott绝缘态等^[64,65]而广受关注. 如在转角石墨烯中, 无序散射和贝里曲率偶极子均会诱导电子的非线性霍尔效应^[66,67]. 而转角体系也会形成特殊的磁子拓扑保护态, 继而对磁子的霍尔效应产生影响^[68]. 另外, 由于转角体系中的摩尔超晶格也存在斯格明子等拓扑磁结构, 其中的非线性磁子输运会受到倒空间和实空间拓扑的共同调制, 其背后的物理机制有待进一步探索和阐明. 此外, 也可以考虑偶极-偶极相互作用对非线性霍尔效应的影响, 探索线性霍尔磁子流和非线性磁子流对于翻转磁矩的作用, 利用非线性霍尔磁子流的巨大霍尔角设计相关的磁子学器件, 挖掘在不同对称性磁体中磁子的非线性霍尔效应, 探索静磁自旋波的拓扑霍尔效应等. 总之, 对磁子霍尔效应的研究为理解相关的基本物理概念和现象提供了新的视角. 可以断定, 对凝聚态物理和拓扑物理等领域中不同物理机制诱导的霍尔效应的研究以及相关应用的开拓方兴未艾, 有着诱人的前景.