# Electric manipulation of domain walls in magnetic Weyl semimetals via the axial anomaly

### Submission summary

 As Contributors: Jens H Bardarson · Yago Ferreiros · Julia Hannukainen Preprint link: scipost_202101_00004v1 Date submitted: 2021-01-08 11:06 Submitted by: Hannukainen, Julia Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Approach: Theoretical

### Abstract

We show how the axial (chiral) anomaly induces a spin torque on the magnetization in magnetic Weyl semimetals. The anomaly produces an imbalance in left- and right-handed chirality carriers when non-orthogonal electric and magnetic fields are applied. Such imbalance generates a spin density which exerts a torque on the magnetization, the strength of which can be controlled by the intensity of the applied electric field. We show how this results in an electric control of the chirality of domain walls, as well as in an improvement of the domain wall dynamics, by delaying the onset of the Walker breakdown. The measurement of the electric field mediated changes in the domain wall chirality would constitute a direct proof of the axial anomaly. Additionally, we show how quantum fluctuations of electronic Fermi arc states bound to the domain wall naturally induce an effective magnetic anisotropy, allowing for high domain wall velocities even if the intrinsic anisotropy of the magnetic Weyl semimetal is small.

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission scipost_202101_00004v1 on 8 January 2021

## Reports on this Submission

### Strengths

1 - To my knowledge, this is the first theoretical work to treat the characteristics of domain wall chirality in the context of magnetic Weyl semimetal. This is in a clear contrast with the situation for the domain wall motion, which was discussed to some extent in Refs.17 and 18. In the context of magnetism and spintronics, chiralities of magnetic textures are attracting a great interest, in connection with the emergent electromagnetism and the spin-wave dispersion. The authors' findings may provide new way in manipulating chiralities of magnetic textures more efficiently in magnetic Weyl semimetals. This point meets one of the acceptance criteria, "3. Open a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work."

2 -The analysis is performed in a systematic way. The authors derive the effective action for the domain wall by the path integral formalism with respect to the electron fields, which is a well-defined and controlled approach. Once the order of perturbative expansion is specified, this approach is capable in deriving all the possible terms in equilibrium. The effective action derived by this approach can be systematically incorporated in the Euler-Lagrange formalism of the domain wall.

3 - The scientific background and the process of analysis are presented in a self-contained manner. In Section 2, the authors introduce the collective coordinate parametrization and its Euler-Lagrange formalism for conventional magnetic domain walls as the scientific background. Such explanation would be helpful for readers who are not familiar with the theory of magnetic textures.

4 - The physical quantities influenced by their findings are properly estimated. The authors suggest that the axial anomaly effect on the domain wall chirality can be achieved with a magnetic field at the order of 0.01-1T and an electric field up to the order of a few MV/m (namely a few volts per micrometer), which are well available in experiments. The estimated domain wall velocity up to 2-3km/s is also in a realistic range that can be measured with the Hall transport or the magneto-optical imaging techniques.

### Weaknesses

1 - The derivation of the effective action for magnetic textures and electromagnetic field, using the path-integral formalism, is systematic and straightforward. However, it is capable in capturing only the equilibrium effect, and cannot fully derive the effect from out-of-equilibrium perturbations. For example, the spin-transfer torque (namely the torque from the transport current) would also be present, as pointed out in Ref. 17 and mentioned in Eq.(4), this torque is not captured in the effective action derived in Section 3. To show that the path-integral approach is good enough for their calculation, the authors should clarify their motivation in employing this approach.

2 - I suspect that there are some other equilibrium effects that are missing in the effective Lagrangian. For example, while the authors derive the axial separation effect contribution in Eq.(28), the contribution from the chiral separation effect, namely the axial current $\boldsymbol{J}_5 \propto \mu \boldsymbol{B}$, is missing. Since they are taking into account the external magnetic field, the spin torque induced by this axial current would also be present. The authors should clarify what kind of condition and approximation they have used to achieve Eq.(25).

3 - In Section 4A, there is no intuitive picture why the anomaly effect favors the Bloch-type chirality rather than the Neel-type. Since most of the readers interested in magnetic domain walls may not be familiar with the physics of Weyl fermions, an intuitive picture in parallel with conventional Heisenberg spin systems would be helpful for those readers to understand this important result. The Dzyaloshinskii-Moriya interaction (DMI), namely the noncollinear spin interaction due to the breaking of inversion symmetry, is often responsible for chiralities of magnetic textures. Since $\mu_5$ in this model breaks inversion symmetry, perhaps the chirality effect may be understood as the effective DMI.

4 - The setups employed in Section 4 seem ambiguous. While the magnetic field in Section 4A is fixed in x- and y-directions, it is not specified in Section 4B. Since Eq.(42) is derived from Eq.(31), perhaps B-field is pointing in z-direction, which is the situation quite different from Section 4A. The authors should specify the directions of $\boldsymbol{E}$ and $\boldsymbol{B}$, and $\boldsymbol{B}_5$ corresponding to the domain wall structure, in each section.

5 - In Appendix A1, they identify the bound states at the domain wall, and derive the effective action corresponding to them. While I find this method reliable, I am not confident of the choice of bound states used for the path integral. The bound states shown in Eqs.(A6)-(A7) are the "Fermi arc" states with zero energy, while there are usually many other bound states with finite energies, as shown in Ref.55. The authors should comment why these finite-energy bound states do not contribute to the effective action. For instance, if the domain wall is thin enough, the finite-energy states are energetically well separated from the zero-energy Fermi arc states, and hence the treatment with only the Fermi arc states would be rationalized.

### Report

In the present manuscript, the authors focus on the dynamics of domain walls in magnetic Weyl semimetals, and theoretically show how the axial anomaly of the Weyl fermions influences the domain wall dynamics. Starting from the low-energy effective model of Weyl electrons coupled with a domain wall, they derive an effective Lagrangian for the domain wall by integrating out the electron degrees of freedom, and obtain the equations of motion in the collective coordinate formalism. With the obtained equations of motion, the authors mainly find two effects arising from the axial anomaly, which are present if the electric and magnetic fields are applied in parallel: (i) The anomaly leads to the shift of the chirality in the domain wall structure in equilibrium. (ii) In the motion of the domain wall driven by the magnetic field, the anomaly tends to suppress the Walker breakdown, namely the saturation of the domain wall velocity due to the dynamics in the domain wall chirality, and enhances the maximum velocity of the domain wall. The authors expect that these new features would be useful in application of magnetic domain walls to logic gate designs, and also in detecting the axial anomaly directly in experiments.

Throughout this manuscript, I have no doubt that the authors employ the scientifically valid procedure, present their results in a clear way, and cite the previous literatures appropriately. I also find several advantages of this manuscript, as listed in the "Strengths" section. From these points, I consider that this manuscript almost meets the Acceptance criteria of SciPost Physics. On the other hand, I would raise several comments and questions, regarding the validity of their theoretical setups and the significance of their findings, as I list up in the "Weaknesses" section. I would encourage the authors to improve these points before publication.

### Requested changes

1 - In Section 4A (left column of page 7), there are statements
"For electric fields larger than this critical value, $E_y > E_c$ ..."
"If the electric field instead is smaller than the critical value ..."
However, I am afraid that the words "large" and "small" are sometimes confusing in this context, since $E_y$ also takes a negative value with large magnitude in their calculations. I would encourage the authors to use some other words.

• validity: high
• significance: ok
• originality: top
• clarity: good
• formatting: excellent
• grammar: perfect