Abstract
Lowenergy eigenmode excitations of ferromagnets are spin waves or magnons that can be triggered and guided in magnonic circuits without Ohmic losses and hence are attractive for communicating and processing information. Here we present new types of spin waves that carry a definite and electrically controllable orbital angular momentum (OAM) constituting twisted magnon beams. We show how twisted beams emerge in magnonic waveguides and how to topologically quantify and steer them. A key finding is that the topological charge associated with OAM of a particular beam is tunable externally and protected against magnetic damping. Coupling to an applied electric field via the AharanovCasher effect allows for varying the topological charge. This renders possible OAMbased robust, lowenergy consuming multiplex magnonic computing, analogously to using photonic OAM in optical communications, and high OAMbased entanglement studies, but here at shorter wavelengths, lower energy consumption, and ready integration in magnonic circuits.
Introduction
Recently, the spatial structuring of photonic, electronic, or neutron beams has been demonstrated^{1,2,3,4,5} enabling to encode additional information in the beam spatial distribution. For instance, a photonic beam can be spatially modulated to have a helical or twisted wavefront embodying a welldefined, internal, meaning originindependent, orbital angular momentum. The external orbital angular momentum (OAM) is origin dependent and is obtained from the photonic angular momentum density L_{γ} = r_{γ} × P_{γ}, where r_{γ} is measured with respect to the beam center and P_{γ} is the linear momentum density^{6}. Similar arguments apply to matter fields^{3,4,5}. The intense research on this topic is fueled not only by fundamental interest but also by the new possibilities and functionalities accomplished by structuring the spatial distributions of the waves. For instance, OAM can take very large values that can be exploited for multiplex communications and quantum information based on entangled photons with large OAM^{1,2,3,4,5,7,8,9}. Here we uncover the existence of twisted magnon beams, which are lowenergy spin waves that carry a welldefined OAM. We demonstrate how they are triggered in a magnonic waveguide (such as in Fig. 1), and how to quantify their topological nature by an associated topological charge, which is found to be related to the OAM of a particular twisted beam. It is shown that this topological charge is electrically tunable and protected against damping. Hence, twisted magnonic beams are ideal candidates for robust, lowenergy cost multiplex magnonic computing using the versatile toolbox for material and spin waves engineering^{10,11,12,13,14,15,16,17}. In addition, we find an OAMdependent magnonic Hall effect that depends on the value of OAM and, hence its value can be steered electrically. We also inspect the nature of thermal magnetic fluctuations around the equilibriumordered phase in a magnetic sample supporting twisted modes and find an anisotropic dependence of the correlation length.
Results
Analytical model for twisted magnon beams
What are the requirements for the emergence of OAMcarrying waves in magnetic systems and what is their relevance? In magnetically ordered systems key lowenergy carriers of information are spin waves or magnons, which are the quanta of lowenergy magnetic excitations. Identifying twisted magnonic beams that carry OAM would open a subfield in magnonics since OAM in principle can have an unbounded value. Thus, let us consider a generic ferromagnetic (FM) cylindrical waveguide modeled by localized magnetic moments at site i described by the magnetic moment operator M_{i} or the corresponding (dimensionless) spin operator S_{i} = −M_{i}/(γħ) (γ is the gyromagnetic ratio). The (Heisenberg) Hamiltonian reads \({\cal{H}} =  \frac{1}{2}\mathop {\sum}\limits_{{ij}} {J_{{ij}}} {\mathbf{M}}_{i}\cdot {\mathbf{M}}_{j}  \frac{{K_{\mathrm{s}}}}{2}\mathop {\sum}\limits_{i} {(\widehat {\mathbf{e}}_{z}\cdot {\mathbf{M}}_{i})^2}  \mathop {\sum}\limits_{i} {\mathbf{B}} \cdot {\mathbf{M}}_{i}\), where J_{ij} > 0 is the exchange coupling between nearestneighbor sites (ij), and K_{s} > 0 is a magnetic anisotropy contribution including the shape anisotropy (the demagnetizing factor tensor D_{ij} at any given direction of the cylindrical nanowire are approximately zero, except for D_{xx} = D_{yy} ≃ 1/2^{18,19}). B = (0, 0, B_{z}) is an external magnetic field along the waveguide (this direction is taken as the zaxis). For magnetically ordered systems longitudinal excitations, meaning changes in the expectation value of M_{i}, caused for example by changes in J_{ij}, are much higher in energy than transversal ones. The latter correspond to a precession of the unit vector field m_{i} = M_{i}/M_{i}, while M_{i} = M = constant and their energy is typically set by K_{s}, which is much smaller than J_{ij}^{20}. Hence, we inspect the transversal spin dynamics governed by the Heisenberg equation of motion \({\mathrm{d}}{\mathbf{m}}_{i}/{\mathrm{d}}t =  i/\hbar [{\mathbf{m}}_{i},{\cal{H}}]\). The dynamics of the magnetization unit vector field proceeds as (we introduce the effective field \({\mathbf{B}}_{\mathrm{s}}: = (K_{\mathrm{s}}M_{{iz}} + B_{z})\widehat {\mathbf{e}}_{z}\))
We are interested in weak transversal fluctuations around the ordered magnetic phase (weak means slow variation in m_{i} within a lattice constant a). Furthermore, we will consider sufficiently large spins where quantum/thermal fluctuations are subsidiary and magnetic reversals are rare. Thus, it is useful to go over to a continuous magnetic texture vector field m_{i}(t) → m(r_{i}, t) and interpret m(r, t) as the local, timedependent expectation value of the field operator m(r, t). The exchange contribution in Eq. (1) appears as the divergence of the spin current^{21}, \(\sum_{j} \gamma MJ_{{ij}}({\mathbf{m}}_{i} \times {\mathbf{m}}_{j}) \to \sum_{\mu \nu } {\partial _{\mu} } {\cal{J}}_{{m}_{\nu}}^{\mu}\). Here \({\cal{J}}_{{m}_{\nu} }^{\mu} = \gamma MJa^{2} \left[ {{\mathbf{m}} \times \partial _{\mu} {\mathbf{m}}} \right]_{\nu}\) stands for the ν polarized spin current along the μ spatial direction. With this Eq. (1) reads
The uniaxial symmetry of B_{s} dictates a conservation of the zcomponent of magnetization, meaning that \(\partial _{t}{m}_{z} + {\nabla}\cdot {\cal{J}}_{{m}_{z}} = 0\). As discussed below, \({\cal{J}}_{{m}_{z}}\) is the only spin current with a nonvanishing time average. Within this general setting, we derive the equations governing the dynamics of small transverse deviations around the zaxis unit vector \(\widehat{\mathbf{e}}_{z}\) by writing \({\mathbf{m}}({\mathbf{r}},{t}) \approx \widehat{\mathbf{e}}_{z} + \widetilde{\mathbf{m}}({\mathbf{r}},{t})\) with \(\widetilde{\mathbf{m}} \bot\widehat{\mathbf{e}}_{z}\) and \(\widetilde {\mathbf{m}} \ll 1\). With \(\widetilde{\mathbf{m}} = (m_{x},m_{y},0)\) and inserting in Eq. (2) we infer the Schrödinger equation
where ψ(r, t) = m_{x}(r, t) − im_{y}(r, t) and
where p = −iħ∇ is the momentum operator and \(m_{J}^ \star = \hbar /(2\gamma JMa^2)\) is an effective mass. The wave function ψ(r, t) vanishes outside the waveguide and, otherwise, is subject to the potential V(r) = −ħγ(K_{s}M + B_{z}). In an extended cylindrical waveguide and in cylindrical coordinates r → (r, ϕ, z), Eq. (3) admits the nondiffractive Bessel solutions (see for instance ref. ^{22})
with \(J_\ell (x)\) is the Bessel function of the first kind with order \(\ell\). The total energy is
Note that similar to optics, conventional (meaning \(\ell = 0\) in Eq. (5)) Bessel modes^{22,23} also exist, but those do not carry any OAM. A striking consequence is that Bessel beams do not interact with electric fields, in contrast to twisted magnon beams and also do not allow for \(\ell\)based operations. This fact is decisive when it comes to constructing OAMsensitive magnonic circuits (similar to ref. ^{11}) that are controlled by electric fields offering so new functionality, as discussed in ref. ^{24}. A further important aspect is the difference between our propagating OAM modes and static/driven magnetization vortices^{25,26,27}. The latter are localized topological excitations that are much higher in energy than a magnon wave (cf. Eq. (5)), which extends over the whole waveguide. The much lower energy set by E(k) is tunable by \(J,m_{J}^{\star}\) and V. Both magnetization vortices and propagating, OAMcarrying magnons have welldefined topological features that can be quantified. To do that in our case, we consider the zcomponent of the magnon current \({{\cal{J}}}_{{m}_{z}}\), which is nothing but the probability current density coiling around the zaxis
meaning that the spin waves spiral around the axis of the waveguide, similarly to the case of photon, electron, or neutron twisted beams^{1,2,3,4,5}. In fact, the analogy runs deeper: let us introduce pseudoelectric and magnetic fields as \(\widetilde {\mathbf{E}} = {\mathbf{m}}_{x} + i{\mathbf{m}}_{y}\) and \(\widetilde {\mathbf{B}} = i({\mathbf{m}}_{x}  i{\mathbf{m}}_{y}).\) Physically, \(\widetilde {\mathbf{B}}\) and \(\widetilde {\mathbf{E}}\) describe the (paraxial) spin wave modes rotating clockwise and anticlockwise, respectively. The magnon density is \({\cal{W}}_{m} = \frac{1}{2}\left( {\widetilde {\mathbf{E}}^2 + \widetilde {\mathbf{B}}^2} \right)\), and the m_{z} polarized magnon current density \({{\cal{J}}}_{{{m}}_{{z}}}\) can be related to the pseudoPoyntinglike vector^{28}, \({\cal{P}}_\mu : = {\cal{J}}_{{m}_{z}}^\mu \propto \frac{1}{2}\left[ {\widetilde {\mathbf{E}}^ \star \times (\partial _\mu \widetilde {\mathbf{B}}) + \widetilde {\mathbf{B}}^ \star \times (\partial _\mu \widetilde {\mathbf{E}})} \right]\). It is straightforward to infer that \({\cal{P}}\) has the proper time and space symmetry of a canonical momentum. Noting that our twisted beam is localized within the waveguide, we can introduce the canonical OAM of the magnon twisted beam as \({\mathbf{L}} = {\mathbf{r}} \times {\cal{P}}\), which is extrinsic and dependent on the choice of the coordinate origin. However, the integral orbital angular momentum over the crosssection is indeed intrinsic, amounting to
with \({\mathbf{k}} \simeq k_{z}\widehat {\mathbf{e}}_{z}\) being the mean wave vector of the spin wave. We have then \(\langle {\mathbf{L}}\rangle \cdot \langle {\cal{P}}\rangle /{\cal{P}} = \ell\), in full analogy with paraxial optical twisted beams^{28}. This \(\ell\) can be identified with the topological charge associated with a particular twisted beam. Generally, our magnons are characterized by the integral momentum \(\langle {\kern 1pt} {\cal{P}}{\kern 1pt} \rangle\) and the orbital angular moment 〈L〉 as two independent properties, meaning that in addition to the intrinsic spin angular moment ħ along the \(\widehat {\mathbf{e}}_{z}\), the magnon in the cylindrical nanowire may carry a longitudinal and intrinsic orbital angular moment \(L_{z}\sim \hbar \ell\) as well, which formally can take arbitrarily large values and can be utilized in magnonic and spintronics applications.
Undamped topological charge of twisted beams
An important feature of twisted magnons is the robustness of the associated topological charge to magnetic damping. This fact is crucial when it comes to utilization of the OAM of twisted magnons as information carriers, analogously to OAMbased optical communications^{7,8}. Generally, the (Gilbert) damping of magnetization procession^{11,17} is governed by a term of the form αm × ∂m/∂_{t}. In Eq. (3), this amounts to substituting ∂_{t} → (1 + iα)∂_{t}, which leads to the timedecaying magnon density \(\rho _\ell (t) = \rho _\ell (t = t_0)\exp (  2\alpha \omega (t  t_0))\) when propagating from time t_{0} to t. The magnon OAM \(\ell\) derives, however, by an averaging of 〈L〉 over the crosssection of the waveguide S, meaning \(\langle {\mathbf{L}}\rangle = \langle {\mathbf{r}} \times {\mathbf{P}}\rangle _{S} = \hat e_{z}({\int} \ell \rho _\ell (t){\mathrm{d}}S)/({\int} {\rho _\ell } (t){\mathrm{d}}S) = \ell \hat e_{z}\). Thus, damping leads to a decaying magnon density (its photon analog is the light intensity), which is however independent of \(\ell\) and hence of OAM. In other words, the OAM (\(\ell\)) is not affected by damping and is conserved during the magnetization evolution. Another point of view is that the damping term would dominate the interfacial spinpumping current by \({\mathbf{m}} \times \partial {\mathbf{m}}/\partial t \propto \Im \left[ {\psi ({\mathbf{r}},t)\partial _t\psi ^ \star ({\mathbf{r}},t)} \right] = \omega \rho _\ell\), which clearly does not depend on the internal phase structure of the twist excitation (and hence on \(\ell\)). These conclusions are supported by the fullfledge numerical simulations presented below.
Simulating an experimental realization
Having laid down the principle existence of twisted OAM and the robustness of the associated topological charge to perturbations that subsume into damping of the precessional dynamics, the experimental feasibility as well as the role of further magnetic interactions that emerge in a realistic setting, have to be clarified. Therefore, we resort to fullfledge numerical micromagnetic simulations (see Methods) for an experimentally realistic waveguide made of insulating yttrium iron garnet (YIG).
The cylindrical YIG waveguide has a diameter of 0.4 μm and a length of 2.0 μm. The saturation magnetization is M_{s} = 1.4 × 10^{5} A/m, the exchange stiffness constant is A_{ex} = 3 × 10^{−12} J/m, the uniaxial magnetocrystalline anisotropy along z is K_{u} = 5 × 10^{3} J/m^{3}, and the Gilbert damping α = 0.01^{24}. The ground state equilibrium magnetization is oriented along the +zaxis, parallel to the main axis of the waveguide. To launch the spin waves, we excite locally at one end z = 0 with a twisted magnetic B field (see Methods) that is linearly polarized in the xdirection and having the amplitude B_{max} = 10 mT and the frequency ω_{B} = 5 GHz. Figure 2a shows a snapshot of the xcomponent of magnetization excitations propagating after 2 ns. More details of excitation mode are gained from slices along the tube in Fig. 2b, which demonstrates the generation of propagating twisted magnon beams. An animated version of m_{x} at the transversal slice z = 1 μm during the magnon propagation is provided as Supplementary Movie 1. The finite damping leads to a decaying magnon density along the YIG tube with the amplitude being normalized with respect to the maximum excitation at z = 0. Remarkably, when evaluating the OAM \(\ell\) from the numerically calculated pseudoPoynting vector, we indeed find that \(\ell\) is independent of the magnon density and is conserved during the spin wave propagation along the tube (cf. Fig. 2c). These full numerical calculations endorse the above formal analysis as well as the experimental feasibility and relevance of twisted magnon beams.
Aharonov–Casher effect and Landau levels
The simulations shown in Fig. 2 as well as our formal analysis evidence that the twisted magnon beams embody a nonvanishing zpolarized spin current density allowing for a coupling to an external electric field \({\cal{E}}\) through the Aharonov–Casher (AC) effect^{29}. For magnonbased information transfer, this fact is crucial as it allows to control the flow of OAMcarrying magnons with electric means. In the presence of an electric field, the exchange interaction in the Heisenberg Hamiltonian becomes anisotropic according to \(J_{ij}{\mathbf{M}}_{i}\cdot {\mathbf{M}}_{j} \to \frac{1}{2}J_{ij}\left( {M_{i}^{\mathrm{ + }}M_{j}^  {\mathrm{e}}^{i\theta _{{ij}}} + M_{i}^  M_{j}^ + {\mathrm{e}}^{  i\theta _{{ij}}}} \right) + J_{{ij}}M_{i}^zM_{j}^z,\) where \(\theta _{ij} = \frac{{g\mu _{\mathrm{B}}}}{{\hbar {c}^{2}}}{\int}_{{\mathbf{r}}_{i}}^{{\mathbf{r}}_{j}} {\mathrm{d}} {\mathbf{r}}\cdot ({{\cal{E}}} \times {\widehat{{\mathbf{e}}}_{z}})\) is the AC phase. The canonical momentum p in Eq. (4) has then the kinetic momentum form \({{\frak{p}}}_{\mathrm{e}} = {\mathbf{p}}  q_{\mathrm{m}}{\mathbf{A}}_{\mathrm{e}}\), where q_{m} = gμ_{B}/c and the electric vector potential A_{e} is given by \({\mathbf{A}}_{\mathrm{e}}({\mathbf{r}}) =  {{\cal{E}}}({\mathbf{r}}) \times {{\widehat {{\mathbf{e}}}_{{z}}}}/c\). For the electric field \({\cal{E}}({\mathbf{r}}) = {\cal{E}}(x/2,y/2,0)\), one finds the symmetric twist vector potential \({\mathbf{A}}_{\mathrm{e}}({\mathbf{r}}) = \frac{{\cal{E}}}{c}(  y/2,x/2,0) = \frac{{{\cal{B}}r}}{2}\widehat {\mathbf{e}}_\phi .\) Our case is analogous to an electron system in uniform perpendicular magnetic field \(\sigma {\cal{B}}\widehat {\mathbf{e}}_z\)^{30}, and \(\sigma = {\mathrm{sign}}({\cal{E}})\) is set by the direction of the applied electric field \({\cal{E}}\). From the gauged Hamiltonian \({\cal{H}}_{\mathrm{m}} = {{\frak{p}}}_{\mathrm{e}}^2/2m_{J}^ \star\) follow the wellknown quantized Landau states having the form of nondiffracting Laguerre–Gaussian (LG) modes^{31,32,33}
where \(L_{n}^{\ell }\) are the associated Laguerre polynomials, and n = 0, 1, 2, … is the quantized radial number. The Landau energy levels read
Here \(w_{\mathrm{e}} = 2\sqrt {\hbar /q_{\mathrm{m}}{\cal{B}}}\) is a characteristic width that depends on the amplitude of the applied electric field, \(\Omega _{\mathrm{e}} = q_{m}\sigma {\cal{B}}/2m_{J}^ \star\) is the Larmor frequency (σ = ±1 correspond to the clockwise or anticlockwise rotation, respectively). Thus, the electric field \({\cal{E}}\) confines the magnons to form paraxial LG beams with
The azimuthal component is directly related to the kinetic orbital angular momentum \({\cal{L}} = {\mathbf{r}} \times {\cal{P}}_{\mathrm{e}}\) as \(\langle {\cal{L}}_{z}\rangle =  \hbar \langle i\partial _\phi + q_{\mathrm{m}}rA_{\mathrm{e}}\rangle\). Unlike the Bessel beams with uniform rotational direction described by the topological charge \(\ell\), the direction of \(\langle {\cal{L}}_{z}\rangle\) of the LG beams depends on \(\ell\) and the twist vector potential A_{e} as well. Note, the \(\langle {\cal{L}}_{z}\rangle\) radial structure depends strongly on σ, that is, the direction of the applied electric field in terms of the first radial maximum of the LG mode \(r_{\ell } = w_{\mathrm{e}}\sqrt {\ell /2}\): For \(\ell \sigma > 0\), we have a uniform, rotationally azimuthal current as usual. However, for \(\ell \sigma < 0\) the currents in the region of \(r < r_\ell\) and \(r > r_\ell\) are countercirculating and are dominated by the topological charge \(\ell\) and the twist vector potential A_{e}, respectively. For an estimate, we take: JM^{2} = 0.1 μeV, a = 10 Å, and \({\cal{E}} = 1\) V/nm, which give w_{e} ≈ 1.4 μm and ħΩ_{e} ≈ 0.33 μeV. Increasing the strength of the applied electric field, higher electric Larmor frequencies Ω_{e} are achievable but the electric Landau radius w_{e} is smaller.
OAMtunable magnon Hall effect
Having clarified the role of an external electric field, one may wonder how the twisted magnon beam is affected by magnetic perturbations. A hallmark of a charge wavepacket propagation in a magnetic field is the transverse deflection, that is, the Hall effect, which in its simplest variant is assigned to the Lorentz force. A similar effect for uncharged particles, such as photons, phonons, or magnons, may thus appear surprising. In our FM case, on the other hand, the Hall effect is also sensitive to the residual magnetization, a phenomenon termed the anomalous Hall effect. Indeed, the Hall effect for magnons has already been observed^{34}. Is this effect OAM dependent and can we control it by tuning the OAM? A positive answer would be an important advance, since OAM can be controlled externally, as demonstrated in Fig. 2. To approach this issue, we note that LG modes have welldefined azimuthal and radial wavefront distributions (quantified by \(\ell\) and n, respectively) and they form an orthogonal and complete basis in terms of which an arbitrary function can be represented. So let us consider a magnonic wave packet centered at (p_{c}, r_{c}) (similar to electronic wave packets^{35,36}). From the above, we conclude that the trajectory spirals around the zaxis. Projecting the local coordinate frame along the curved trajectory, we end up with a noncommutative geometry and a covariant derivative in momentum space^{37,38,39,40}: \(D_{i} = \partial /\partial _{{p}_{i}} + {\cal{A}}_{i}^g({\mathbf{p}})\), where \({{\mathbf{\cal{A}}}}^g({\mathbf{p}}) = i\langle \psi _{\ell ,{\mathrm{n}}}^L\partial /\partial _{\mathbf{p}}\psi _{\ell ,{\mathrm{n}}}^L\rangle\) is the Berry connection. The corresponding Berry gauge field (curvature), \({\cal{B}}_{\mathrm{g}} = \partial/\partial _{\mathbf{p}} \times {\cal{A}}^{g}({\mathbf{p}})\) gives rise to a deflection of the wave packet center. Given that the zaxis is now locally directed along p, the vector \(({\widehat {\mathbf{e}}}_{x} + i{\widehat {\mathbf{e}}}_{y})\) is orthogonal to p and moves on the unit sphere with p/p under variation of p, which results in a magneticmonopoletype field^{41,42}, \({{\cal{B}}}_{{\mathrm{g}}} =  \ell {\mathbf{p}}/p^{3}\) for the orbital motion of magnons (since \(\exp (i\ell \phi) = ({\widehat {\mathbf{e}}}_{x} + i{\widehat {\mathbf{e}}}_{y})^{\ell}\)). As a result, we infer the Hamiltonian equations of motion^{35,39}
The anomalous velocity term \(\hbar \ell {\dot{\mathbf{p}}}_{c} \times {\mathbf{p}}_{c}/p^3\) is perpendicular to p_{c} causing a transversal motion of the wave packet. The corresponding transverse shift δr_{c} is cast as \(\delta {\mathbf{r}}_{c} = \hbar \ell {\int}_{L} {({\mathbf{p}}_{c} \times {\mathrm{d}}{\mathbf{p}}_{c})} /p_{c}^{\mathrm{3}}\) and is determined by the geometry of the contour L in momentum space and \(\ell\). This is in so far remarkable, as arbitrarily large values of \(\ell\) are possible giving rise to a large deflection at small driving force.
For concreteness, let us consider a weak spatially varying, perpendicular magnetic field B_{z}, say along the xdirection^{43,44}. The driving force is \({\mathbf{F}}_{x} =  \partial _{x}B_{z}({\mathbf{r}})\widehat {\mathbf{e}}_{x}\) and thus \({\dot{\mathbf{p}}}_{c} = {\mathbf{F}}_{x}\) and \(\delta {\dot{\mathbf{r}}}_{c} = \hbar \ell {\mathbf{F}}_{x} \times {\mathbf{p}}_{c}/p_{c}^{\mathrm{3}}\). Assuming paraxial magnonic propagation, meaning \({\mathbf{p}}_{c} \simeq p_{z}\widehat {\mathbf{e}}_{z}\), the transverse deflection reads (an exact but cumbersome analytical expression is available) \(\delta {\mathbf{r}}_{c} \simeq \delta {\mathbf{y}} \propto \frac{{\hbar \ell F_{x}z}}{{p_{c}^3}}\widehat {\mathbf{e}}_y.\) Different LG modes with different topological charges \(\ell\) split at the waveguide end (cf. Fig. 3a). Similar to photon^{41} and electron^{42} twisted beams, we predict a magnonic OAMdependent Hall effect, which is qualitatively different from the usual magnon Hall effect^{10,34}.
For a cylindrically inhomogeneous waveguide, for instance due to inhomogeneity in the demagnetization field^{18,19}, we may formally write the potential in Eq. (4) as \(V(r) = \bar V + V_2r^2/2\) with \(V_2 \ll \bar V\) causing further \(\delta {\mathbf{r}}_{c} = \delta {\boldsymbol{\phi }} \propto \frac{{\hbar \ell V_2z}}{{p_{c}^3}}\widehat {\mathbf{e}}_\phi .\) The displacement direction is along \(\widehat {\mathbf{e}}_\phi\), meaning that LG modes are rotated uniformly clockwise or anticlockwise depending on \(\ell\) (cf. Fig. 3b), a result in line with the optical Magnus effect^{6,45}.
Another aspect arises when the waveguide is an FM metallic wire, in which case the electric field vanishes inside the wire. However, we may still have nontrivial magnetoelectric effects. Starting from a general symmetrical but inhomogeneous electric potential V_{e}(r), the energy shift due to the AC effect is given to the first order in the perturbation V_{e}(r) by \(\delta E \simeq \frac{{q_{m}}}{{m_{J}^ \star c}}\langle \frac{1}{r}\frac{{\partial V_{\mathrm{e}}}}{{\partial r}}\left[ {\widehat {\mathbf{e}}_{z}\cdot ({\mathbf{r}} \times {\mathbf{p}})} \right]\rangle = \frac{{q_{m}}}{{m_{J}^ \star c}}\langle \frac{{L_{z}}}{r}\frac{{\partial V_{\mathrm{e}}}}{{\partial r}}\rangle .\) For a stepfunction profile of the wire electric potential V_{e}(r) = UΘ(r − R) (here Θ(r) is the Heaviside function and R is the radius of the metallic wire), we find ∂V_{e}/∂r = Uδ(r − R). No electric field is present in the nanowire except for the surface. Nonetheless, the twisted magnon beam acquires an additional phase upon traveling a distance z along the nanowire \(\varphi (\ell ,z) = U\rho _\ell (R)\frac{{q_{m}}}{{\hbar c}}\frac{\ell }{{k_{z}}}\frac{z}{R}.\) This phase accumulation may result in an interference between the different LG modes, for example, \(\left( {\psi _{  \ell } + \psi _\ell } \right) \propto \cos [\ell \phi + \varphi (\ell ,z)]\).
Anisotropic correlation length of magnetic fluctuations
While the control of twisted magnons via external electromagnetic fields is crucial for applications, one may wonder about the stability of the FM state to twisted excitations. To explore this aspect we have to study the influence of vorticity on the magnetic fluctuations and in particular the correlation length. Both are indicators on how magnetic ordering reacts to excitations^{46}. Let us start with the Ginzburg–Landau free energy density of the FM waveguide at temperature T (in unit of 1/M^{2}), \({\cal{F}} = \frac{{Ja^2}}{2}({\bf{\nabla }}{\mathbf{m}})^2 + (k_{\mathrm{B}}T  \frac{{K_{\mathrm{s}}}}{2})m_{z}^{\mathrm{2}} + K_4m^4  B_{z}m_{z}\) (with k_{B} is Boltzmann constant and m_{z} ≤ 1). The quartic term with K_{4} > 0 stabilizes the longrange FM ordering below T_{c} ≈ K_{s}/2k_{B}. Because of the uniaxial anisotropy and the cylindrical demagnetizing field, the SU(2) spin symmetry is reduced to SO(2)\(\times {\Bbb Z}_2\). Lowenergy Goldstone modes are due to the continuous SO(2) symmetry.
For T < T_{c} the saddle point approximation delivers the meanfield value of m_{z} in the absence of an external magnetic field: \(\bar m_{z}(T) = \pm \sqrt {k_{\mathrm{B}}(T_{\mathrm{c}}  T)/2K_4}\) in the FM sector. The stiffness of deformations around the saddle point solution is inferred from the longitudinal (along \(\widehat {\mathbf{e}}_z\)) and transverse (along \(\widehat {\mathbf{e}}_{r}\) and \(\widehat {\mathbf{e}}_\phi\)) fluctuations^{46}, \({\mathbf{m}} \simeq [{\cal{R}}({\mathbf{r}}),\Phi ({\mathbf{r}}),\bar m_{z} + {\cal{Z}}({\mathbf{r}})]\). The fluctuations energy cost, up to second order, is \(\delta {\cal{F}} = \;\frac{{Ja^2}}{2}\left[ {({\bf{\nabla }}{\cal{R}})^2 + ({\bf{\nabla }}\Phi )^2 + ({\bf{\nabla }}{\cal{Z}})^2} \right]  2k_{\mathrm{B}}(T  T_{\mathrm{c}}){\cal{Z}}^2.\) For nondiffracting Bessel modes at \(T \ll T_{\mathrm{c}}\), we find the anisotropic characteristic length scales, \(\frac{1}{{\xi _{z}}} \simeq \sqrt {\frac{{2K_{\mathrm{s}}}}{{Ja^2}}} ,\frac{1}{{\xi _{\mathrm{r}}^{\ell n}}} \approx \frac{{\chi _{\ell {n}}}}{R}\,\,{\mathrm{and}}\,\,\frac{1}{{\xi _\phi ^\ell }} \approx \frac{\ell }{R}\), where \(\chi _{\ell {n}}\) is the nth root of the Bessel prime function \({\mathrm{d}}J_\ell (r)/{\mathrm{d}}r_{{r} = {R}} = 0\). The definition of χ_{00} = 0 gives \(\frac{1}{{\xi _{\mathrm{r}}^{00}}} = 0\) and \(\frac{1}{{\xi _\phi ^0}} = 0\), which implies that the transverse excitations correspond to the Goldstone modes of the SO(2) symmetry.
For the paramagnetic region T > T_{c}, the quartic K_{4} term is not relevant and the meanfield equilibrium solution is \(\bar m_{z} = B_{z}/k_{\mathrm{B}}(T  T_{\mathrm{c}})\). For transverse fluctuations around the saddle point solution \(\bar m_{z}\), we write \({\mathbf{m}} \simeq \sqrt {1  \rho ^2} \widehat {\mathbf{e}}_{z} + \rho \widehat {\mathbf{e}}_{r},\) with 0 ≤ ρ ≤ 1. Substituting in \(\delta {\cal{F}}\) and neglecting constants and higher order terms in ρ leads to \(\delta {\cal{F}}_\rho = \frac{{Ja^2}}{2}({\bf{\nabla }}\rho )^2  \alpha (T)\rho ^2 + \beta \rho ^4\) with α(T) = [k_{B}T − (K_{s} + B_{z})/2] and β = B_{z}/8. Interestingly, transverse fluctuations are most probable not at the boundary ρ = 0 but at \(\rho _0 = \sqrt {\alpha /2\beta }\), the global minimum of the free energy \(\delta {\cal{F}}_\rho\). For cylindrical waveguide the transverse fluctuations are subject to a nonzero restoring force and the correlation length is \(\frac{1}{{\xi _{r}}} \simeq \sqrt {\frac{{4\alpha (T)}}{{Ja^2}}} .\) Thus, the transverse scattering density is Lorentzian.
Generation of twisted magnon beams
In Fig. 2 we demonstrated how to trigger twisted magnons via magnetic fields. In fact, there are various other ways to accomplish this goal: one may use, for instance, an engineered magnetic spiral phase plate or fabricate a hologram such as a pitch fork for scattering of incoming magnons before entering the waveguide. Another possibility is the following (cf. also Fig. 1): let us consider a uniformly charged thin wire with a linear charge density λ_{e} located in the region z ∈ [d_{1}, d_{2}] on the axis of a FM cylindrical waveguide. The radial electric field \({\cal{E}}({\mathbf{r}}) = \frac{{\lambda _{\mathrm{e}}}}{{2\pi \epsilon r}}\widehat {{e}}_{r}\) in the FM nanowire within z ∈ [d_{1}, d_{2}] (\(\epsilon\) is the electric permittivity of the FM). For zpolarized magnons, the interaction of magnons with the radial electric field \({\cal{E}}\) involves the vector potential \({\mathbf{A}}_{\mathrm{e}}({\mathbf{r}}) =  {{\cal{E}}} \times \widehat {\mathbf{e}}_{{z}}/c = \frac{{\lambda _{\mathrm{e}}}}{{2\pi \epsilon cr}}\widehat {\mathbf{e}}_\phi ,\) which is intimately related to the socalled Dirac phase for magnons moving along a contour C, \(\Phi _{\mathrm{D}} =  \frac{{q_{\mathrm{m}}}}{\hbar }{\int}_{C} {{\mathbf{A}}_{\mathrm{e}}} ({\mathbf{r}})\cdot {\mathrm{d}}{\mathbf{r}},\) producing a phase structure \(\exp ({\mathrm{i}}\ell \phi )\) with the quantized number \(\ell =  \frac{{q_{\mathrm{m}}\lambda _{\mathrm{e}}}}{{2\pi \hbar \epsilon c}}\). Similar to twisted electron beams^{47}, the twisted magnons can be produced by considering a transition of a conventional magnon with (\(\ell = 0\)) in the region z ≤ d_{1} without any flux (λ_{e} = 0) to the region z ∈ [d_{1}, d_{2}] with the nonzero flux (λ_{e} ≠ 0). For an estimate, we have \(\frac{{q_{\mathrm{m}}\lambda _{\mathrm{e}}}}{{2\pi \hbar \epsilon c}}\sim 5.6\lambda _{\mathrm{e}} \times 10^{  15}\) with the linear charge density λ_{e} in unit of e/m. For ~10^{22} cm^{−3} carrier density in copper and magnon wavelength (governed by the exchange interaction) ~μm, the linear density λ_{e} ~ 10^{14}–10^{16} e/m is experimentally feasible.
Demonstration of OAM magnon beams evolution
As evidenced, twisted magnon beams are controlled in numerous ways including external electric and magnetic fields, geometry design and holographic means. For instance, with suitably designed holograms, different superpositions of twisted beams can be created such as the OAMbalanced superposition. This is determined by two LG eigenmodes with the same radial number n but opposite topological charges \(\pm \ell\), meaning \(\psi _1 = \psi _{  \ell ,{n}}^L + \psi _{\ell ,{n}}^L.\) This superposition has zero net canonical OAM 〈L_{z}〉 = 0 exhibiting a flowerlike symmetry pattern: \(\psi _1^2 \propto \cos ^2(\ell \phi )\) at z = 0, as shown in Fig. 4a. The OAMunbalanced superposition consists of two LG eigenmodes with the same radial number n but different topological charge 0 and \(\ell\) means that \(\psi _2 = \psi _{0,{n}}^L + \psi _{\ell ,{n}}^L.\) This superposition carries a finite net canonical OAM, 〈L_{z}〉 ∝ l, and is characterized by a pattern with \(\ell\) offaxis vortices (cf. Fig. 4b).
In the paraxial approximation with a relatively small transverse kinetic energy and considering that \(\Omega _{\mathrm{e}} \propto \sigma {\cal{B}}\), we infer the allowed wave number as k_{z} ≃ k + Δk_{z}, with \(\Delta k_{\mathrm{z}} =  [\sigma \ell + (2n + \ell  + 1)]/z_{\mathrm{e}}\). Here \(z_{\mathrm{e}} = w_{\mathrm{e}}\sqrt {E/\hbar \Omega _{\mathrm{e}}}\) is a characteristic longitudinal scale. On the propagation distance z, the electric field modifies the longitudinal wave vector resulting in an additional phase φ = Δk_{z}z, which strongly depends on the direction (σ) of the applied electric field. As for \(\ell \sigma > 0\), the induced phase φ shows a linear dependence on the topological charge \(\ell\). In contrast, for \(\ell \sigma < 0\), φ = −(2n + 1)z/z_{e} is independent of \(\ell\). These phenomena are caused by partial cancellation of countercirculating azimuthal currents produced by the beam with \({\mathrm{exp}}(i\ell \phi )\) and by the twist vector potential A_{e}.
The propagation of twisted magnon beams in the waveguide can give rise to a Faraday effect similar to the case of electron vortex states^{32,33}. As for the OAMbalanced superpositions, the induced phases for the modes \(\psi _{ \pm \ell ,{n}}^L\) reads \(\varphi _ \pm = \mp \sigma \ell z/z_{\mathrm{e}}\), which results in rotation of the interference pattern by the angle Δφ = σz/z_{e}, as displayed in Fig. 4a. For the OAMunbalanced superpositions, the mode \(\psi _{\ell ,{n}}^L\) acquires an additional phase \(\Delta \varphi =  (\sigma \ell + \ell )z/z_{\mathrm{e}}\), as compared to the \(\psi _{0,{n}}^{L}\) mode. The superposition with \(\sigma \ell > 0\) shows a rotation of the interference pattern by the angle Δφ = 2σz/z_{e}, as demonstrated by the lower panels in Fig. 4b with σ = −1, whereas the superposition with \(\sigma \ell < 0\), has no rotation at all (Δφ = 0, as evidenced by the σ = −1 panels in Fig. 4b.
All these electric field tunable phases, including the mixed cases with \(\psi _3 = \psi _{  \ell ,{n}}^L + \psi _{0,{n}}^L + \psi _{\ell ,{n}}^L\), as shown in Fig. 4c, indicate rich magnetoelectric patterns of the interference intensity, which also may serve as a marker for identifying the twisted magnon beam.
Discussion
Imprinting an orbital angular momentum on a magnon beam puts a new twist on magnonics, as OAM can be functionalized as an independent robust, meaning damping resistant parameter. Together with the fact that OAM can be tuned to very large values, the twisted magnon beams offer new opportunities for reliable multiplex communications. The susceptibilities of the OAMcarrying magnons to electric and magnetic fields are thereby a key advantage, as they offer versatile tools to extract and steer the flow of information. Further immediate consequences of the topological nature of the discovered twisted beams are that the internal global phases, related to OAM, result in a controllable interference pattern when such beams interfere. This we may employ to imprint spatially modulated magnon density on a length scale well below the beam’s extensions. The parameters of the twisted beams are directly reflected in the magnon density spatiotemporal pattern. Thus, in addition to the wellestablished, energysaving applications of magnonics and magnonbased spintronics, this new offspring is useful for OAMbased, electrically controlled functionalities in magnonicbased information transfer and processing. Future studies are focused on the scattering of twisted magnon beams from a magnetization vortex core^{48} and its possible application in a long distance (at short wavelength^{49}) electrically controlled multiplex communication channel between different vortex cores.
Methods
Micromagnetic simulations
We used the opensource, graphical processing unit (GPU)accelerated software package MuMax^{3}^{50} for the micromagnetic simulations. The (0.4 × 0.4 × 2.0) μm^{3} large system is discretized into (100 × 100 × 500) cubic cells with a size of (4 nm)^{3}. The spatial profile of the excitation field applied at the z = 0 layer reads \(B_\ell (\rho ,\phi ,t) = \Re \left\{ {\hat eB_{{\mathrm{max}}}(\frac{\rho }{{w_0}})^{\ell }\exp \left[ {\frac{{\ell }}{2}\left( {1  \frac{{\rho ^2}}{{w_0^2}}} \right)} \right]\exp \left[ {  i(t\omega _{B} + \ell \phi )} \right]} \right\}\). The waist parameter w_{0} denotes the distance of maximum field strength B_{max}, \(\hat e\) corresponds to the polarization vector, which is equal to \(\hat e_{\mathrm{x}}\) in our case, whereas w_{0} = 75 nm. The time evolution is computed using a Dormand–Prince algorithm with adaptive timesteps. To avoid artifacts due to finite size simulations, we apply open boundary conditions (disregard demagnetizing fields).
Code availability
For obtaining the results plotted in Fig. 2 as well as the movie file, we used the opensource, GPUaccelerated software package MuMax^{3}^{50} for the micromagnetic simulations. For reproducing the data, one has to use the paramter for YIG, as given in the text and discretize the (0.4 × 0.4 × 2.0) μm^{3} large cylinder into (100 × 100 × 500) cubic cells with a size of (4 nm)^{3}. Full details on the code are in ref ^{50}.
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (Nos. 11474138 and 11834005), the German Research Foundation (No. SFB 762 and SFB TRR 227), and the Program for Changjiang Scholars and Innovative Research Team in University (No. IRT16R35).
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C.J. initiated and performed the analytic modeling, D.M. and A.F.S. performed and analyzed the numerical calculations. C.J. and J.B. supervised the project and wrote the paper. All authors discussed and analyzed the results and contributed to the final version of the paper.
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Jia, C., Ma, D., Schäffer, A.F. et al. Twisted magnon beams carrying orbital angular momentum. Nat Commun 10, 2077 (2019). https://doi.org/10.1038/s41467019100083
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DOI: https://doi.org/10.1038/s41467019100083
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