To investigate the low-energy dynamics of magnetizations in a ferromagnetic nanotube, we take the continuum limit. The normalized magnetization m is considered as a field with two spatial variables z and φ, which are the axial coordinate and the azimuth, and a temporal variable t. The axis of the nanotube is chosen to be the z-axis. For theoretical description of a ferromagnetic nanotube, we use the continuum Heisenberg model with easy-axis anisotropy, whose potential energy density is given by U = A 2 ∂ z m 2 + 1 ρ 2 ∂ φ m 2 − K 2 m z 2 , (1) where ρ is the radius of the nanotube, A is the exchange constant, and K > 0 is the anisotropy constant [25]. The dynamics of the magnetization stems from the kinetic part of the Lagrangian density L = L kin − U , which is given by L kin = − s ⁡ cos ⁡ θ ∂ t ϕ , and the Rayleigh dissipation R = ( α s / 2 ) | ∂ t m | 2 , where α is the Gilbert damping coefficient. The variational principle with respect to m yields the Landau-Lifshitz-Gilbert equation [51] s ∂ t m − α s m × ∂ t m = h eff × m , (2) where s is the spin density and h _eff is the effective magnetic field conjugate to m given by h eff = − δ U δ m = A ∂ z 2 m + A ρ 2 ∂ φ 2 m + K m z z ̂ , (3) where U = ∫ d z d φ ρ U . Solving Eq. 2 for the case of ∂ _t m = 0, we can obtain static solutions. We will consider a set of solutions of the Skyrmion-textured domain wall as a family of the slow modes with an angle representation m ₀ = (sin θ ₀ cos ϕ ₀, sin θ ₀ sin ϕ ₀, cos θ ₀), cos θ 0 = tanh z − Z λ , ϕ 0 = Q φ − ϒ + Φ , (4) where λ ≡ λ 0 / 1 + Q 2 λ 0 2 / ρ 2 and λ 0 ≡ A / K [25]. Here, Z, ϒ, Φ and Q are the position of the domain wall, the angle of the Skyrmionic texture, the global rotation angle, and the Skyrmion charge, respectively. Since the system is invariant under translations along the z axis, orbital rotations about the axis of the nanotube, and spin rotations about the easy-axis, linear momentum, orbital angular momentum, and spin angular momentum are conserved. The domain-wall solution (4) breaks these three symmetries and the three collective coordinates Z, ϒ, Φ represent corresponding zero-energy modes. We decompose the magnetization into a slow mode m ₀ (z, φ) and a perturbative fast mode δ m (z, φ, t) which are orthogonal. The magnetization can be written as m z , φ , t = 1 − | δ m | 2 m 0 z , φ + δ m z , φ , t . (5) With this decomposition, the equation of motion for the fast mode can be derived from the expansion of the Landau-Lifshitz equation up to linear order of the fast mode δ m, s 1 − α m × ∂ t δ m = A ∇ 2 m 0 + K m 0 ⋅ z ̂ z ̂ × δ m + A ∇ 2 δ m + K δ m ⋅ z ̂ z ̂ × m 0 , (6) where ∇ 2 = ∂ z 2 + ∂ φ 2 / ρ 2 . To obtain magnon modes on top of the domain wall, we neglect the Gilbert damping by following previous literature on the interaction of magnons with magnetic textures, e.g., Refs. [52–54], which allows us to invoke the conservation of angular momentum to understand the interaction with transparent physical picture. Note that the effect of the damping on magnons can be captured simply by introducing a finite lifetime, τ ∝ (αω)⁻¹ with ω the magnon frequency as has been done in, e.g., Refs. [13, 52, 55]. Plugging the domain-wall solution (4) into Eq. 6, the equation of motion for the fast mode on top of the Skyrmion-textured domain wall becomes s ∂ t δ m = m 0 × − A ∇ 2 δ m − K δ m ⋅ z ̂ z ̂ + A / λ 0 2 1 − λ 0 / λ 2 2 ⁡ sin 2 ⁡ θ δ m . (7)

It is convenient to use the following as an orthonormal basis { e ̂ 1 , e ̂ 2 , e ̂ 3 } e ̂ 1 = θ ̂ = cos θ 0 ⁡ cos ϕ 0 , cos θ 0 ⁡ sin ϕ 0 , − sin θ 0 , e ̂ 2 = ϕ ̂ = − sin ϕ 0 , cos ϕ 0 , 0 , e ̂ 3 = m 0 = e ̂ 1 × e ̂ 2 . (8)

The magnetic texture of the spin-wave excitation with the solution (4) is illustrated in Figure 1. For the convenience, we define a complex field Ψ = δm ₁ − iδm ₂, where δ m j = δ m ⋅ e ̂ j and  j = 1,2 . The complex filed Ψ contains full information of the spin wave. Reassembling the equations of motion for each component δm _j yields the equation of motion for the complex filed Ψ. The complex field Ψ obeys the “Schrödinger” equation ¹ i s ∂ t Ψ = H Ψ (9) with the “Hamiltonian” H = A ℏ 2 ℏ i ∇ − a 2 + V , (10) with vector potential a = a φ φ ̂ , a φ = Q ℏ ⁡ cos θ 0 / ρ and scalar potential V = A [ 1 / λ 0 2 − ( 2 / λ 2 − Q 2 / ρ 2 ) sin 2 θ 0 ] . The independency of the Hamiltonian (Eq. 10) on the spin azimuthal angle ϕ ₀ is due to the spin rotational symmetry. The independency on the spatial azimuthal angle φ is due to the orbital rotational symmetry. Conservation of spin and orbital angular momenta of the system stems from these two rotational symmetries.

Before solving the Hamiltonian (Eq. 10) exactly, we can invoke the following simple argument to compute orbital angular momentum change of magnons. Magnons on the Skyrmion-textured domain wall experience an emergent magnetic field [56] b = ∇ × a = − Q ℏ λ ρ sech 2 ( z − Z λ ) ρ ̂ . For the magnetic field which is a curl of a, we only take its surface-normal component, since the nanotube is considered as a two-dimensional surface ² . The orbital angular momentum change of each magnon can be evaluated from the Lorentz force, Δ l z = Δ ( ρ p φ ) = ∫ ρ ( v × b ) ⋅ φ ̂ d t = − 2 Q ℏ , where v is the magnon velocity and p _φ is momentum in the azimuthal direction. Since the total orbital angular momentum of the domain wall and magnons is a conserved quantity, an orbital transfer torque to the domain wall per magnon is 2Qℏ. Figure 2 depicts the bent trajectory of the magnon due to the surface-normal emergent magnetic field. In this classical viewpoint, we can calculate the orbital angular momentum transfer, but it does not offer concise reflection property. In the section below we solve the reflection problem exactly.

Since the field θ ₀ of the domain-wall solution (4) has z-dependence only, the “Hamiltonian” H (Eq. 10) is invariant under global translation of the azimuthal angle φ → φ + δφ. Therefore we can write the wavefunction as an eigenmode Ψ (z, φ, t) = ψ (z)e ^i(lφ−ωt), l = 0, ±1, ±2, ⋯. The equation of motion for ψ (z) is written as a time independent Schrödinger equation sωψ (z) = H _eff ψ (z) with the effective Hamiltonian H eff = − A ∂ z 2 + V eff , where the effective potential is V eff = A l 2 ρ 2 − 2 Q l ρ 2 tanh z − Z λ + 1 λ 2 1 − 2 sech 2 z − Z λ . (11)

Then the original two-dimensional problem is simplified as a one-dimensional quantum mechanics. The effective potential is also known as the Rosen-Morse potential [57]. The tanh term plays role of an energy barrier which reduce longitudinal momentum of incoming magnon. The sech² term represents a potential well, which gives us possibility of existence of bound modes.

The problem can be solved by using a method of SUSY QM [48, 49]. In terms of creation and annihilation operators a = ∂ z + tanh z − Z / λ / λ + β , a † = − ∂ z + tanh z − Z / λ / λ + β , the effective Hamiltonian and its SUSY partner Hamiltonian can be written as H eff = A a † ⁡ a − β 2 + l 2 ρ 2 = − A ∂ z 2 + V eff , (12) H ̃ eff = A a a † − β 2 + l 2 ρ 2 = − A ∂ z 2 + V ̃ eff , (13) where β = −Qlλ/ρ ² and V ̃ eff = A − 2 Q l ρ 2 tanh z − Z λ + 1 λ 2 + l 2 ρ 2 . (14)

Note that, in the case of Q = 0, the partner potential V ̃ eff is a constant potential so that there is no reflection of magnons and the linear momentum is preserved. Also, the original potential V _eff becomes the Pöschl-Teller potential [58, 59] which is a well known reflectionless potential as shown in Figure 3. Here, we can conclude that, for moving magnons, the Skyrmionic texture of the domain wall generates the potential barrier. Since H ̃ eff = H eff + A a , a † , the commutation relation a , a † = 2 λ 2 sech 2 z − Z λ (15) eliminate the sech 2 z − Z / λ term in Eq. 11 and yields Eq. 14. The problem with the potential V _eff is simplified to the problem with V ̃ eff . Eigenfunctions of the two Hamiltonians are easily transformed into each other’s ones by annihilation and creation operators: H ̃ eff a ψ z = a H eff ψ z = s ω a ψ z , (16) H eff a † ψ ̃ z = a † H ̃ eff ψ ̃ z = s ω ̃ a † ψ ̃ z , (17) where s ω ̃ ψ ̃ = H ̃ eff ψ ̃ . Since the a (a ^†) operation on the eigenfunction of H eff ( H ̃ eff ) is the eigenfunction of H ̃ eff ( H eff ) , we can switch the problem to its SUSY partner which turn out to be easier to tackle than the original one.

One can obtain the bound state solution by solving aψ (z) = 0, which is given by ψ z b = sech z − Z λ e Q l λ ρ 2 z , (18) ω b = A l 2 s ρ 2 1 − Q 2 λ 2 ρ 2 , (19) where ω _b is the frequency of the bound solution. The condition of normalizability is |l| < ρ ²/(|Q|λ ²). This mode is local [60, 61] and the magnon position δ obeys tanh δ − Z λ = Q l λ 2 ρ 2 , (20) where the position is defined as the maximum position of wavefunction’s amplitude. Figure 4 shows the positions of bound magnons schematically, the white lines are trajectories of bound magnon. Note that, for the non-zero Skyrmion charge, the bound modes with different l are separated [47, 53, 62–65] unlike the bound modes with Q = 0 [66, 67].

Let us consider a scattering problem of the spin wave. We now can write the wavefunction ψ (z) as ψ z → − ∞ ∼ e i k − z + r e − i k − z , (21) ψ z → + ∞ ∼ t e i k + z , (22) where k + = k − 2 − 4 | Q l | ρ 2 (23) and k ₋ is the wavenumber of the incoming magnon. Because of the asymmetric potential, wavenumbers of incoming and outgoing magnons are different. Using the relations (Eqs 16, 17), we can obtain asymptotic behaviors of the corresponding partner wavefunction: a ψ z → − ∞ ∼ i k − − 1 λ + β e i k − z + r − i k − − 1 λ + β e − i k − z , (24) a ψ z → + ∞ ∼ t i k + + 1 λ + β e i k + z . (25)

Then we can check the fact that the reflection probabilities of original and partner potentials are equivalent: R ̃ k = − i k − − 1 λ + β i k − − 1 λ + β 2 | r | 2 = | r | 2 = R k , (26) where R k ( R ̃ k ) is the reflection probability of the original (partner) potential. Now the problem is simplified as a hyperbolic-tangent-potential-barrier problem which is exactly solvable. The reflection probability of the SUSY partner is given by R k = sinh 2 π λ 2 k − − k + sinh 2 π λ 2 k − + k + , if  k − > 2 ρ | Q l | 1 , if  k − < 2 ρ | Q l | (27)

[68–71]. Here, we use the Euler’s reflection formula [72] to simplify the reflection probability. We can assume that l is negative and Q is positive, without loss of generality, since the reflection probability is symmetric under the exchange of k ₋, k ₊. The transmission probability is determined by T _k = 1−R _k for both Hamiltonians. Figure 5 illustrates trajectories of reflected, transmitted, and incoming spin waves.

The wavefuntion Ψ is defined in the spatially varying local frame { θ ̂ , ϕ ̂ , m 0 } (Eq. 8). Thus, to see scattering properties of the magnon in the laboratory frame, we obtain wavefunction in the laboratory frame. Since there are two domains, local axis of a moving magnon is flipped passing by the domain wall and it makes difference between functional forms of incoming and outgoing wavefunctions: ψ Lab in = δ m ⋅ x ̂ + i y ̂ = − 1 Ψ e i ϕ 0 as z → − ∞ ∼ ψ z e i l + Q φ − ω t , ψ Lab out = δ m ⋅ x ̂ − i y ̂ = Ψ e − i ϕ 0 as z → + ∞ ∼ ψ z e i l − Q φ − ω t , (28) up to constant phase factors ³ . Since Φ₀ = Q (φ−ϒ) + Φ, orbital wavenumbers of incoming and outgoing magnons are different. An orbital angular momentum of a transmitted magnon is changed from ℏ (l + Q) to ℏ (l−Q). Hence the change of magnon’s orbital angular momentum after passing the domain wall is Δl _z = −2ℏQ and orbital-transfer torque applying to the domain wall per magnon is 2ℏQ. The resultant torque derived from the calculation by SUSY QM is consistent with the result from the emergent field dynamics of Section 2.2. This is one of our main results: By using a Skyrmion-textured domain wall, we can generate an orbital angular momentum of a magnon.