Domain wall of a ferromagnet on a three-dimensional topological insulator

Topological insulators (TIs) show rich phenomena and functions which can never be realized in ordinary insulators. Most of them come from the peculiar surface or edge states. Especially, the quantized anomalous Hall effect (QAHE) without an external magnetic field is realized in the two-dimensional ferromagnet on a three-dimensional TI which supports the dissipationless edge current. Here we demonstrate theoretically that the domain wall of this ferromagnet, which carries edge current, is charged and can be controlled by the external electric field. The chirality and relative stability of the Neel wall and Bloch wall depend on the position of the Fermi energy as well as the form of the coupling between the magnetic moments and orbital of the host TI. These findings will pave a path to utilize the magnets on TI for the spintronics applications.

shows the domain wall energy E DW with the width ξ = 2.0 as a function of ϕ for J 3 = 0 ((a)) and J 3 = J 0 ((b)). The qualitative behavior is the same as that of Fig. 3 for ξ = 4.0 in the main text. Figure S2 shows E DW as a function of ξ. It is seen that the smaller ξ is preferred for all the cases. For the illustrative purpose, we set ξ = 4.0 in the main text so that we can capture the difference between various domain wall structures.

S2. Effective action
We derive an effective action of the 2D Weyl Hamiltonian. The effective action is obtained by calculating the susceptibility in the one-loop approximation, where J x,y = J ∥ and J z = J ⊥ . For the region qv F ≪ |m|, one can expand χ αβ (q, 0) with respect to q, while for the region qv F ≫ |m|, the mass m can be put to be zero for the derivation. We derive the DM interaction and the exchange interaction by calculating the non-diagonal element of χ αβ in the massive Weyl Hamiltonian. We estimate the domain wall width by calculating the diagonal element of χ αβ in the massless Weyl Hamiltonian.

The DM and exchange interaction:
Firstly, we derive the DM interaction and the exchange interaction from the massive Weyl Hamiltonian. The continuum theory of the 2D Hamiltonian is described by The Green's function reads With the use of the Green's function, the susceptibility is given by We put iω l = 0, and take the T = 0 limit. We can show The susceptibility is inverted between µ ←→ −µ. The DM interaction reads  with ∥ . This is Eq. (18) in the main text. We calculate the diagonal terms χ αα (q, 0) up to O(q 2 ), They contain the ultra-violet momentum cut-off Λ, which is naturally given by the 3D band or the lattice constant. The qindependent terms yield which describes the easy-axis anisotropy, while the q-dependent terms yield which acts as the exchange interaction.

The domain wall width:
Next, we derive the effective action for µ = 0, m = 0. We make the following decomposition of Eq. (S4), and so on. The results for Eq. (S12) read up to O(q 2 ) as The domain wall width ξ is estimated as follows. First, assume that q ∼ 1/ξ is much smaller than |m|/v F , and use the expansion Eqs. (S15) and (S17). Then, we demand the anisotropy energy, i.e., the difference between χ zz (0, 0) and χ xx (0, 0) = χ yy (0, 0), is equal to the elastic energy. However, the obtained q is much larger than |m|/v F since |m| ≫ v F Λ. Therefore, we need to look for q ∼ 1/ξ in the region q ≫ |m|/v F , i.e., using Eq. (S21). This results in which satisfies v F /ξ ≫ |m| and hence gives the self-consistent estimation.

S3. Edge channel along the domain wall
We investigate the edge channel along the domain wall, The exact solution is obtained when J ∥ = 0, which is known as the Jackiw-Rebbi solution. The continuum theory of the 2D Hamiltonian in Eq. (2) is described by The eigenequation for k y = 0, The solution which is localized around the domain wall is which is Eq. (12) in the main text, with the normalization constant In the presence of n x and n y terms for small J ∥ , we may estimate the energy as which is in Eq. (14) in the main text. Figure S3 shows numerical results on the shift of energy at k y = 0 as a function of ϕ. They are well fitted by the cosine curve obtained by the first-order perturbation Eq. (S22) for J 3 /J 0 = 0.2.  Figure S4 shows the set of figures corresponding to Fig. 3 in the main text with various J 3 .

S5. Symmetry Analysis
Here, we summarize the symmetry properties of the models presented in the main text.

3D Hamiltonian:
The 3D tight-binding Hamiltonian in Eq. (4) is given by where µ is the chemical potential. Its complex conjugate is which transforms the Hamiltonian as On the other hand, the particle-hole operator transforms the Hamiltonian as This operator is the generator of the particle-hole transformation together with k → −k, J 3 → −J 3 and µ → −µ. The energy spectrum is symmetric (asymmetric) between the positive and the negative energy when J 3 = 0 (J 3 ̸ = 0). The time-reversal operator is given by which transforms the Hamiltonian as The chiral symmetry is defined by the product of the particle-hole symmetry and the time-reversal symmetry, which transforms the Hamiltonian as The chiral transformation is preserved when n = 0, J 0 = 0 and µ = 0. We make the mirror symmetry along y direction The domain wall energy is the same for ϕ and 2π−ϕ due to the mirror symmetry along the y direction as σ y → −σ y , n y → −n y . Therefore it is enough to show the results for 0 ≤ ϕ ≤ π.

2D Hamiltonian:
The 2D Hamiltonian for the surface state is given by Its complex conjugate is The time-reversal operator is given by which transforms the Hamiltonian as −n, µ) . (S37) When n = 0, the time-reversal transformation symmetry is preserved, On the other hand, when µ = 0, there is a symmetry with the time-reversal symmetry operator Θ even when n ̸ = 0. The electron density distribution is almost the same between the case of J3 = 0 and the Bloch domain wall with J3 = J0 because the divn term is zero. There is a double peak structure for the J3 = J0 Neel wall due to the divn term. Figure S5 shows the electron density distribution for the cases of Neel and Bloch walls. There are two peak structures for J 3 = J 0 for the Neel domain wall, which come from divn term in Eq. (22) in the main text.

S7. Charging energy
We evaluate the charging energy for three cases: (i) heavily doped case where the Fermi energy is in the 2D conduction band, (ii) half-filling case, and (iii) the case where the Fermi energy is deviated from the half-filling but in the 2D bang gap, and show that the charging energy can be neglected compared with the band energy.
We take the lattice constant a ≈ 10Å as the unit. Then, the unit of the Coulomb energy is where, we have used ε ≈ 20 for the TIs.
(i) Heavily doped case. For the heavily doped case, we can neglect the long-range Coulomb interaction and consider only the onsite repulsion because of the screening. According to numerical calculations, the electron distribution amplitudes is less than 10 −2 , and the width is about 10. Therefore, the charging energy is where we have taken into account the two domain walls due to the periodic boundary condition. On the other hand, the order of the band energy is where we have used t ≈ 3eVÅ for the TIs. E band is about ten times larger than E charge . Therefore, we can neglect the effect of the charging energy.
(ii) Half-filling case. We have to consider the long-range Coulomb interaction.
On the other hand, the band energy is calculated as where we have used the density of states D (ε) = ε 2 2πv 2 F for the Weyl Hamiltonian. Therefore, E band is proportional to ξ as in the Fig. S2. If we use the parameter used in the calculation, we get α ≈ 4 × 10 −4 . In the Fig. S2, we obtain α 1 = 2 × 10 −2 for the Ising case and α 2 = 3 × 10 −3 for the Heisenberg case. In the evaluation, we use the numerical result. We obtain the optimized ξ by minimizing E band + E charge .
For the half-filling case, only the ∇ · n term contributes the charging, and the amplitude is less than 1/L y = 1/200, and ρ 0 ξ ≈ 1/200 × 10 = 0.05. Therefore, we get the optimized ξ as for the Heisenberg case. Therefore, we conclude that the effect of the charging energy is negligible for the half-filling case.
(iii) The case where the Fermi energy is in the 2D gap. In this case, the Jackiw-Rebbi solutions accumulate. We can estimate ρ 0 ξ ≈ N/L for the N electron doped system. On the other hand, the particle number is expressed as the function of the Fermi energy. Therefore, The Fermi energy which gives ξ opt ≈ 1 is µ ≈ 0.3eV. It is much larger than the surface gap (≈ 50 meV). Therefore, the charging energy is negligible for the arbitrary Fermi energy. Figure S6 shows the domain wall energy E DW for various depths of the magnetic layer. In the case of the second and middle layers, the energy differences are small because the amplitudes of surface states are small. The domain wall energy E DW for the top layer with ϕ is identical to that of the bottom layer with π − ϕ, while the domain wall energy of the middle layer is symmetric between ϕ and π − ϕ, because of the mirror symmetry at the middle layer.