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Probing domain wall dynamics in magnetic Weyl semimetals via the non-linear anomalous Hall effect

The magnetic textures of Weyl semimetals are embedded into their topological structure and interact dynamically with it. Here, we examine electric field-induced structural phase transitions in domain walls mediated by the spin transfer torque, and their footprint in charge transport. Remarkably, domain wall dynamics lead to a transient, local, non-linear anomalous Hall effect and non-linear anomalous drift current, which serve as direct probes of the magnetization dynamics and of the domain wall location. We discuss experimental observation in state-of-the-art samples.

In magnetic WSMs the magnetic texture is inherent to the topological structure and exhibits a dynamical interplay with it. The momentum space locations of the Weyl nodes k w are determined by the magnitude and direction of the magnetization, therefore any topological defects, such as domain walls, can move the nodes in momentum space, causing them to exhibit a space and time dependence of the formk w =M (r, t). The vital question in this context is understanding inhomogeneous current responses stemming from DW dynamics, which can detect magnetic texture dynamics in regions otherwise inaccessible. In this regard, we demonstrate that DW dynamics in magnetic WSMs lead to a transient, local, non-linear anomalous Hall effect (NAHE) and non-linear anomalous drift current (ADC). Such nonlinear responses are the only non-linear signals of the system and are enabled by inversion symmetry breaking by the inhomogeneous magnetization.
Whereas we focus here on magnetic WSMs, the physics we describe is general and the non-linear anomalous Hall effect stemming from domain wall motion has a straightforward dynamical explanation. To begin with, in an infinitesimal time interval δt an electron spin However, in this case the magnetization is also a function of time. Thus, in a time δt, while the electron travels from s(r) to r + δr, M (r + δr) has evolved in time, and the change in the magnetization is proportional to its gradient, and in turn to the electric field. This extra change in M induces an additional contribution to the Hall effect proportional to the square of the electric field: one factor for the electron acceleration and one factor for the change in magnetization.

II. THEORY AND MODEL
The low-energy Hamiltonian describing three-dimensional (3D) WSMs coupled with local magnetic moments through the exchange interaction J is given by where τ z is acting on the chirality with eigenvalues χ = ±1 and σ i are the Pauli matrices and M = SM is the 3D magnetic texture vector with constant magnitude S. The k-space coordinate of the Weyl nodes in the ferromagnetic domains with M = ±Sẑ is given by The presence of a spatially inhomogeneous magnetic texture, e.g., walls between ferromagnetic domains, only alters the Weyl node coordinate but never changes the Weyl node separation 2|k 0 | [ Fig. 1]. The spatio-temporal dependency of M (x, t) should be smooth in comparison to the typical length and time scale of the system, i.e., long-range and low-frequency magnetic texture. In magnetic WSMs, magnetic textures are mapped onto chiral electromagnetic fields coupled with opposite signs to the Weyl nodes [62][63][64][65][66]. Using the definition of the U (1) axial-vector field A 5 = J S ev FM [67] and the axial current J 5 = χ χJ χ = −ev F σ , the last term in H 0 can be written as J 5 · A 5 , which means that spin-density operator acts like an axial current density coupled to the magnetization M , and axial perturbations such as domain walls host an axial charge density.
The localized axial magnetic field B 5 , which is intrinsic to the spatially inhomogeneous magnetic configuration exists when the curl of the magnetic orientation is non-zero, i.e., [62]. The main consequence of this feature leads to the local STT, T e = k 0M × σ in which the non-equilibrium spin polarized electrons σ ∝ k 0 B 5 × E is induced by an external electric field [62,63]. Similarly, the time-dependence of A 5 , gives rise to an axial electric field E 5 = −dA 5 /dt. Since the averaged pseudofields vanishes over the whole sample, the domain walls are arranged in such a way as to cancel each other out after summing over the entire sample. It is a general statement and does not depend on the domain wall type or domain (wall) width. We will demonstrate how the STT-induced E 5 could contribute to the non-linear electronic responses, i.e., J y = σ yxx E 2 x , where σ yxx ∝ B 5 . Therefore, non-linear signals in magnetic texture (DWs) must have a non-zero real space curvature. This originates from breaking the local mirror symmetry.
We focus on magnetic WSMs with inversion symmetry e.g. Co 3 Sn 2 S 2 [5], HgCr 2 Se 4 [68] and PrAlSi [69,70], which allows us to neglect the Dzyaloshinskii-Moriya interaction, as well as shape and dipolar anisotropy. Such terms may be strong in e.g. layered materials [71][72][73], and may affect DW motion, but they do not change our fundamental argument that DW motion in WSM is accompanied by a non-linear electrical response, which can be used to probe the magnetic inhomogeneity. without structural deformation in response to E x and (c 1 ) shows that there is no rigid motion when E ŷ. In the limit of at|E| ≥ 1, the magnetic configuration shows a structural deformation of domain walls and in the higher limit when at|E| 1, there will be a full magnetic transition of the domain texture. (a 2 ): A NDW between two ferromagnetic domains with ±M z disappears under the action of E x as the magnetization is rotated toŷ. (b 2 (c 2 )): Applying an electric field along x-(y-) to a BDW leads to a structural transition to the head-to-head (tail-to-tail) DW alongM y (M x ). Here, X and L represent the DW center and ferromagnetic width, respectively.

A. Structural phase transition of magnetic texture
The electrically induced spin dynamics is described by the Landau-Lifshitz-Gilbert equa- where γ 0 is the gyromagnetic ratio and B ef f is an effective magnetic field. The coefficient α is the Gilbert or viscous damping parameter which is proportional to the rate of energy loss [75].
The torque T e is a spin-transfer torque describing the background electronic contribution to the magnetic texture dynamics, arising from the exchange interaction between conduction electrons and magnetic moments. The torque T e can be expressed in terms of the axial electric field E 5 , and Eq. 2 is solved recursively aŝ where E 5 = e k 0 (T e + αM × T e ) with the damping parameter α and the normalization where w is the DW width, and ρ s is the number of local magnetic moments per unit volume.
We consider two orientations of the electric field: parallel and perpendicular to the hard axis of each DW, that is, E x and E ŷ. The DWs at t = 0 are described by the vector 2(b 2 ) and (c 2 ), on the other hand, changes its structure to the head-to-head and tail-to-tail configurations, respectively. The steady-state condition for NDW and BDW, respectively, is established beyond aE x t 10 5 and aE x t 10 3 which means, interestingly, that the BDW approaches its steady-state much sooner than the NDW. The electric field alongŷ does not induce any dynamics for NDW since the torque T e | NDW Ey = 0.
Therefore, the major characteristic of the STT in WSMs that causes structural phase transitions (SPT) in both the ferromagnetic domains and domain walls is that, depending on the direction of an external electric field, the NDWs can either vanish simultaneously or even be resistant to structural deformation. When the electric field is parallel to or perpendicular to the hard axis of the DW, the BDW also modifies its structure to the head-to-head or tail-to-tail configurations, respectively.

III. ELECTRICAL SIGNALS OF DOMAIN WALL DYNAMICS
In previous sections, we have discussed the electric field-induced STT and how it is employed to change the magnetic structure. The STT, together with intrinsic B 5 across the domain wall, is the main origin of the presence of E 5 or magnetic texture dynamics. Such localized pseudo-electromagnetic fields like ordinary electromagnetic fields result in the electronic responses of intrinsic charge accumulation near the domain walls. Therefore, the temporal and spatial modulation of the ferromagnetic orders in magnetic texture induces anomalous electronic charge pumping from a macroscopic point of view [77]. Having determined the domain wall dynamics in the presence of an external electric field together with axial fields, we are in a position to calculate the localized inter-and intra-band electronic currents using the quantum kinetic equation.
Therefore, the spatially and temporally inhomogeneous magnetic configurations affect the dynamics of itinerant electrons, and are captured by the quantum kinetic equation [78] Here H 0 is the band Hamiltonian given by Eq. (1), ρ i is the density matrix of the i th node averaged over disorder, κ( ρ i ) is the scattering integral, the electric-field-induced driving [79] and the temporal and spatially inhomogeneous driving terms are given by [Appendix. C] The covariant derivative is defined as DO R a k e a is the (vector) Berry connection with components [R a k ] mn = i u m k |∂ ka u n k . The electron-texture coupling terms are embedded in driving terms in Eq. (5), which can be used to derive T e .
The density matrix ρ i can be decomposed into band-diagonal and band off-diagonal The electronic current densities induced by DW motion originate from the intra-(n → n) and inter-band (n → n ) excitations and can be obtained as Therefore, the non-linear current is found by taking the trace of the current operator with the density matrix, and takes the form [Appendix. E] where f E (x, t) is a normalization factor. This expression denotes the AHE in WSMs in which the node distance alongM z is replaced byM (x, t) in the dynamical regime leading to the nonlinear behavior in the anomalous Hall current (E 5 ∝ k 3 0 E). We note that the first and second terms in Eq. (7) contain the linear and non-linear AHE proportional to k 0 (= J S/ v f ) and k 3 0 , respectively. These two terms can be likely distinguished experimentally by imposing the mechanical force to reduce the Fermi velocity. The other non-linear terms are suppressed by inversion symmetry. Therefore, the SPT dynamical regime occurs when the magnetic component perpendicular to the hard axis,M SPT ⊥ , is generated, and the new DW configuration is constructed. In the steady-state the non-linear signal vanishes, while the surviving linear response corresponds to the ordinary static AHE due to the equilibrium magnetic structure, i.e., J (aEt ≫ 1) ∝ M New × E. Existing proposals for the NAHE rely on the Berry curvature dipole, an entirely different mechanism active in non-magnetic and non-centrosymmetric materials [80][81][82][83].
To estimate the non-linear Hall voltage, we use E y = j x ρ yx + j y ρ yy and the conventional relationship between the resistivity and conductivity tensors. After straightforward calculations and considering dominate conductivity elements, the second-order induced electric field alongŷ is given by E (7)) and σ yy = e 2 µτ 6π 3 v F . Using the prior parameters for J IB , the induced-electric field is E (2) y ≈ 7.15 × 10 6 Vm −1 which yields the non-linear Hall voltage V nm. This is straightforwardly measured.
Magnetization dynamics likewise leads to an intra-band drift current. The full expression is induced by magnetic dynamicsṀ is given by The drift current leads to the ADC in response to an axial electric field, i.e., J d [62,84]. In order to estimate σ (D) χ , we set µ − = µ + = 100 meV, v F = 10 6 m/s and τ d = 1 ps, hence the drift conductivity for each node in a WSM is −2 × 10 5 S/m. The total ADC after summing over two nodes is given by To estimate the average of axial magnetic field over a single DW, we take J S = 1 eV, In a future publication, the impact of electron-electron interactions will be examined.

V. ACKNOWLEDGMENTS
SH is grateful to Dr. Azadeh Faridi for useful discussions. DC is supported by the Australian Research Council Future Fellowship FT190100062.

Appendix A: Spin-Transfer Torque
A non-equilibrium spin polarization of electrons σ(r) leads to a chiral current in Weyl semimetals, i.e., J 5 = ev F σ(r) . Then the electrically induced STT is given by where ρ s is the number of local magnetic elements per unit volume. Therefore, in the absence of a magnetic field, we may write The first term is the precession term that makes the magnetic moments rotate around an axial current J 5 , while the second term opposes this dynamical spin rotation with damping coefficient α. Therefore, for the magnetization dynamics to be non-vanishing in the absence of a real magnetic field, the chiral current J 5 must be generated in the system. This chiral current can be induced by an external electric field through a conventional Hall effect in the presence of an axial magnetic field B 5 , i.e., J 5 = σ (H)B 5 × E leading to the electrically induced STT [63] where σ The full dynamical behavior of magnetic moments described in Eq. A1 can be written in a more compact form whereΘ =M ×. The above equation tells us the magnetic moment dynamics is provided by the STT of the background electrons, T e , and the other terms indicate the dissipation which opposes T e .
The magnetic-moment dynamics can also be understood in terms of an axial electric field E 5 (r, t) = −Ȧ 5 (r, t) = − J S ev FṀ (r, t) [62,63], which couples to the Weyl fermions with opposite sign in the same manner as an axial magnetic field B 5 . The axial electric field induced by magnetic dynamics is given by It is worth noting that since the Gilbert damping rate α is relatively small, the effective terms are up to the linear in α and then we neglected the terms higher than O(α 2 ). Accordingly, the E x -induced axial electric field E 5 is maximum at the domain wall center for both the whereM (x) = (α sech 4 ( x−X w ) sinh( x−X w ), 1, α sech 4 ( x−X w )) and the time-dependent spin moment isM In the steady state, atE x → ∞, and far from the center, the second term dominates, hencê M (x, t) | NDW Ex = (0, 1, 0). In addition, the structure arising from the components of E 5 near the center is given by α(y, 1/α, 1 − 2y 2 )/ √ 1 + α 2 where y = x − X tends to zero.
The presence of an external electric field E and intrinsic B 5 is crucial to generate spin dynamics or axial electric field E 5 . It is in contrast to the strain-induced E 5 in Weyl semimetals which may exist even in the absence of B 5 [92]. Furthermore, in our system the sign of As discussed in the paper, the various regimes describing the time evolution of the magnetic texture are determined by the electric field strength and its direction. For weak electric fields and short time-scales, domain walls start moving rigidly without structural deformation, which is confirmed by several experimental observations [13,14,28]. The numerical results in Fig. 6 (a 1 , b 1 , c 1 [63]. Above this threshold value, the domain wall changes its structure and then the ferromagnetic domains with ±M z (at t = 0) begin to undergo a magnetic transition to the new ferromagnetic configuration witĥ M x,(y) . Fig. 6 (a 2 , b 2 , c 2 ) presents the rate of magnetic evolution as a function of time and electric field (the parameter T in figure represents the product T = aEt). In the rigid motion regime (induced by E x in (a 2 ), (b 2 )), the magnetization components change with a constant  [14,15,28]. The rate of magnetic domain evolution would be a non-linear function of aEt and all cases approach a steady-state condition where dM d(aEt) → 0. The time evolution is needed to reach the structural deformation in a fixed electric field; the stronger the electric field, the sooner the Walker-breakdown regime is approached and vice-versa.
Structural transitions in the domain wall configuration have received considerable attention. The increase in domain wall mobility and width has also been attributed to increasing temperature [13,43,45,46]. Our theoretical study shows that the domain wall dynamical regime, which includes the rigid domain wall motion and the Walker-breakdown regime, depends on the domain wall type, which was also confirmed by experimental evidence [13][14][15].
It is worth stressing that the magnetic texture evolution is plotted on a logarithmic scale as a function of atE and it becomes extremely slow in the regime of large atE. The label L and R refer to the regions −L < x < 0 and 0 < x < L, respectively. Note that the labels BDW and NDW correspond to the initial configuration of domain walls, since both will bear the structural transition.

Appendix C: Contribution of magnetic textures to the kinetic theory
Now, we take the effect of magnetic texture into account. If the spatial and temporal variation inM (r, t) be slow enough, the interaction of electron's spin with local magnetic texture can be taken into account through the concept of the axial vector potential A 5 (r, t) = (J S/ev F )M (r, t). This axial vector potential leads the momentum p = −i ∇ to transform p → p − χA 5 = p − χ(J S/ev F )M (r, t), where χ = ±1 denotes chirality. The electric-fieldinduced driving term is given by D E ( ρ i ) = eE · D ρ i Dk , and following the same method in [79] and replacing χA 5 instead of eA, the driving force due to the domain-wall-dynamics would arises from the spatially inhomogeneous magnetic configuration in a ferromagnetic Weyl semimetal. Here we define the covariant derivative notation DO 1. The matrix elements of temporal magnetic driving term: The matrix elements of this driving term can be written as Using the fact that ∇ k n|n = ∇ k δ n,n = 0, we get According to the above result, the first term represents the diagonal components of the temporal magnetic driving term and the second term denotes the off-diagonal part. The real electric field plays the same role as an axial electric field in the driving term, so we can write 2. The matrix elements of spatial magnetic driving term: D Mr ( ρ i ) By making use of the magnetic-moment configuration at the domain wall asM (x) = , the corresponding axial magnetic field would be which changes sign with chirality. This axial magnetic field is localized at the domain wall and near the center (x → X) B 5 or the curl of the magnetization lying along the y-axis.
Therefore, the spatial magnetic driving term due to the non-vanishing curl of the local magnetization at the domain wall can be written as Here { , } represents the matrix anticommutator. In order to calculate spatial magnetic driving term originating from the diagonal part of the density matrix ξ , one can replace ξ → ρ i in the above equation and then obtain its matrix elements. In the eigenstate basis, we have Then we can find and where g nn zx = (R nn x R n n z + R n n x R nn z )( n k − n k )(ξ n k − ξ n k ), g n n zx = (R n n x R nn z + R nn x R n n z )( n k − n k )(ξ n k − ξ n k ), Finally, we get or equivalently where we have used the fact that n k = − n k = k .

Appendix D: Density Matrix Elements: Dynamical regime
Now, we assume that the external electric field makes the magnetic-moment of magnetic domain walls to be evolved in time so we have non-zero E 5 as well as a non-zero external electric filed E. We also assume that in the absence of B 5 , the diagonal and off-diagonal parts of the density matrix can be separated as ρ t = ξ t + S t , where ρ t is the deviation of density matrix from its equilibrium condition ρ 0 . The kinetic equation in this case would be In the steady-state condition the diagonal part of the kinetic equation is written as where the second term originally stems from the external electric field. The off-diagonal part of the density matrix satisfies the following equation where, for simplicity, we have assumed that the off-diagonal part of the density matrix is independent of the weak disorder, i.e., κ nn k ( S t ) = 0, and according to the previous arguments the term D nn t ( ρ 0 ) = D nn Mt ( ρ 0 ) + D nn E ( ρ 0 ) for n = n is obtained as where R nn k = i n|∇ k n . Following the same methods as before, we find that This time integral can be easily evaluated and and the answer is provided by S t and S E nn k , respectively, take into account the spin-dynamic-induced and external electric-field-induced off-diagonal density matrices.

Appendix E: Inter-band current and non-linear anomalous Hall effect
In order to calculate the current, we first define the velocity and density matrix tensor in the eigen-states basis (+ for conduction and − for valance bands) as the following and , where ξ nn k and S nn k are the diagonal and off-diagonal components of the density matrix. Then we can obtain where we have used V nn k = v n k . The current density as an observable quantity is exactly the trace of the above matrix, since we have J = Tr[ [Eq. D6] in response to an external electric field (labeled by subscript E) and magnetic texture dynamics (labeled by subscript t) is given by Having integrated the current J IB = −ie n,n ,χ [dk]( n k,χ − n k,χ )R nn χ,k S t n n χ,k , leads to the following inter-band current n,n ,χ [dk]R nn k (κ n n k ( ξ Mt )+κ n n k ( ξ E )), dt | 2 and we have used n=± k,χ = ± v F |k − χk w |. It is easy to show that the first two terms in Eq. E5 is exactly the anomalous Hall effect in WSMs, i.e., J ∝M E (x, t) × E, whereM E (x, t) is a function of E. Then, the first and second terms represent the linear and non-linear anomalous Hall effect (we note that |E 5 | ∝ |E| (Eq. A4)), and the third term stems from the role of scattering process in the current parallel to the Berry vector potential R nn k . The E 5 -induced off-diagonal density matrix, i.e., the first term of Eq. E4, does not contribute to the inter-band current, i.e., an anomalous Hall effect cannot be generated by an axial electric field.

Appendix F: Local chiral/charge non-conservation by domain wall motion
Having employed the quantum kinetic theory, we show that the STT at the domain wall can activate the localized axial anomaly in magnetic Weyl semimetals with domain walls. This axial anomaly generated by E 5 · B 5 = 0 causes an axial density n 5 to build up at the domain wall. The rate of electric-field-induced charge accumulation at each node is given by The first two terms are proportional to E 5 · B 5 and the last two terms are proportional to E · B 5 . These terms are the source of creation or annihilation of charges in each valley.
1. Electric-field-induced chiral anomaly: ∂ µ J µ 5 = 0 According to Eq. C10, the diagonal driving term D nn Mr [ ξ t ] is zero. Then for electric field parallel to x-axis we have We note that D nn Mr [ S E ] = 0 for E x. Then we get where we have used ξ r = χ|k 0 |f 0n (∇ ×M ) · Ω n χ,k . Then we have Combining Eq. F3 and F5, we get the chiral anomaly expression generated by the domain wall motion. The rate of charge pumping between valleys would be or in a more compact form we have The second term takes into account inter-valley scattering processes, where τ 2 is the intervalley relaxation time related to processes that can change the valley quasiparticle number, while the intra-valley process denoted by τ −1 1 preserves the particle number. As expected from the chiral anomaly, the rate of change of the particle number in each valley is proportional to the chirality χ meaning that chiral fermions are created in one node and annihilated in another one. In conclusion, in the case of E x and using the explicit expression for axial magnetic and electric fields for the Néel and Bloch domain walls we find the field-induced anomaly as and In the long-time limit t τ 2 , the steady state solution for each valley reads δn 2π 2 2 E 5 · B 5 . In the opposite limit t τ 2 where the time evolution of the magnetic texture is faster than the inter-valley scattering, the particle number density at each valley is given by δn (1) (x, t) = dt χ 2π 2 2 E 5 · B 5 (x, t ) leading to a shift in the chemical potential δµ (1) χ (x, t), where µ 0 is the initial location of the chemical potential and v F is the Fermi velocity. The STT-induced axial chemical potential, δµ 5 = δµ + − δµ − , at the domain walls decays to zero when the domain wall structures reach their new steady-state configuration. The axial anomaly is activated for any non-zero electric field strength, while it is strongly dependent on the field direction. Interestingly we conclude the axial anomaly generation across the NDW is an anisotropic phenomenon and only exists when E x. The axial anomaly can be also activated in magnetic Weyl semimetals by a magnetic field B beyond the Walker-breakdown field, i.e., |B| > |B t |, where the internal angle of the spin moments oscillates in time leading to a structural deformation of domain wall [93].
2. Electric-field-induced local charge non-conservation: ∂ µ J µ = 0 For an electric field along the y-axis, the non-zero terms on the right-hand site of Eq. F1 Then the rate of particle number change in each node when E ŷ is given by where we denote the relevant scattering time by τ . In contrast to Eq. F7, the rate of particle number change in each valley is independent of the chirality. Therefore, applying an external electric field along the y-axis induces the charge density in both nodes with the same sign, leads to a slight shift in the chemical potential δµ (2) (x, t) = 2π 2 3 v 3 F µ 2 0 dt E · B 5 (x, t ), in the limit of t τ . In the steady-state condition t τ , on the other hand, the time-dependent function in the integral behaves adiabatically, and then it can be pulled out of the integral. For a NDW, we have E 5 | NDW Ey = 0 and f (x, t) | NDW Ey = 1, therefore the E y -induced charge density in NDW is δn (2) | NDW = eτ 2π 2 2 E · B 5 (x, 0). For BDW, the charge density δn (2) (x, t) can be obtained from Eq. F12. Since the induced charge density δn (2) χ couples to each node with the same sign, both nodes receive the same amount of charge due to an external electric field along the y-axis. This term can be interpreted as localized at the BDW between domains with ±M z (atE y 1) and the highest panel is the headto-head (or tail-to-tail) walls between domains with ±M x (atE y ≫ 1).
the anomalous non-conservation of the local charge at a single domain wall. The violation of local charge conservation in Weyl semimetals has also been studied in strained Weyl semimetals, which leads to the propagation of novel plasmon modes [94,95] or even the modification of phonon modes [96]. Since the domain-wall-induced pseudo-magnetic field will vanish after summing over the entire system, the physical issue of non-conservation of charges at a single domain wall can also be solved by taking into account the contribution of all domain walls. Figure. 7 represents a simple schematic of the three snapshots of the domain wall (start from BDW at t = 0): (i) aEt 1, (ii) aEt 1, and (iii) aEt ≫ 1 (steady-state limit). Following the general statement that the pseudo-fields should average to zero over the whole sample, the charge density δn  χ is non-linear in the electric field for the BDW, it increases linearly for the NDW.