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Deterministic Néel vector switching of altermagnets via magnetic octupole torque

Seungyun Han,1 , 2 Insu Baek,1 , 2 Kyoung-Whan Kim,3 , ∗ Hyun-Woo Lee,1 , 2, † and Suik Cheon1 , 2, ‡

arXiv:2508.13585v2 [cond-mat.mtrl-sci] 14 May 2026

1 Department of Physics, Pohang University of Science and Technology, Pohang 37673, Korea 2 Center for Quantum Dynamics of Angular Momentum, Pohang University of Science and Technology, Pohang 37673, Korea 3 Department of Physics, Yonsei University, Seoul 03722, Korea

Department of Physics, Yonsei University, Seoul 03722, Korea (Dated: May 15, 2026) Altermagnets have recently emerged as promising materials for next-generation spintronic de- vices. For their device applications, realizing a single-domain configuration is essential but remains challenging. We theoretically consider injecting magnetic multipoles into altermagnets, which can be achieved by applying an in-plane current to an altermagnet/normal metal bilayer. We demon- strate for d-wave altermagnets that the torque generated by the magnetic octupole injection can achieve magnetic-field-free deterministic switching of the altermagnets’ N´eel vector and transform their multidomain configurations into a single domain. This method allows the switching in diverse altermagnets, thereby facilitating their device applications and fundamental studies. This work also exemplifies the usefulness of magnetic multipole currents. Introduction– Altermagnets (AMs) represent a new class of magnetism [1–8] that combines characteristics of ferromagnets (FMs) and antiferromagnets (AFMs). Whereas their spins are compensated like AFMs with opposite spins on the sublattices A and B, they exhibit FM-like features such as non-relativistic spin splitting [9– 15] and the anomalous Hall effect [16, 17]. These prop- erties lead to interesting effects such as the spin-splitter Hall effect and spin-splitter torque [12, 18–20], which are useful for device applications. However, experimentally measured effects are significantly weaker than theoreti- cal predictions, primarily due to the multidomain issue inherent in AMs [16, 21, 22]. To utilize the intriguing properties of AMs for device applications, it is desired to develop a method to con- trol the N´eel vector [15, 23] and force AMs into a single- domain configuration. However, due to their AFM na- ture, achieving this goal remains elusive. Aligning the N´eel vector using an external magnetic field is limited to AMs with net magnetization [24]. Moreover, this method is not suitable from a device application perspective. Re- cently, a few ideas to realize electrical control were re- ported. One method exploits broken inversion symmetry in AMs [25, 26]. However, this method is not applicable to centrosymmetric AMs. Another method utilizes N´eel spin currents [27, 28], which are effective only in AMs with special sublattice structures. We explore another method that utilizes a defining difference of AMs from conventional AFMs: magnetic multipoles [29]. Recent studies have identified magnetic multipoles as secondary order parameters in AMs. For instance, d-wave AMs exhibit ferroic magnetic octupole (MO) ordering, while g-wave AMs are characterized by magnetic triakontadipole ordering [30–32]. Additionally, magnetic multipoles couple to the N´eel vector of AMs ∗ [email protected] † [email protected] ‡ [email protected] in a manner analogous to the coupling in FMs between magnetic dipoles (spin) and local magnetizations. Due to the coupling, injecting a spin current into FMs gen- erates the spin-transfer torque (STT) or the spin-orbit torque (SOT) on the local magnetization in FMs [33– 35]. Some of us recently demonstrated that injecting a magnetic multipole current into AMs similarly generates a magnetic multipole torque on the N´eel vector [29] in AMs. In this Letter, we explore the analogy to the FM case further and examine the possibility that the magnetic multipole torque may open a novel paradigm of the N´eel vector control in AMs just as the STT or the SOT in- duces the deterministic switching in FMs [33–35]. As a specific example, we examine the N´eel vector dynamics in d-wave AMs induced by the MO torque (MOT). The electrically induced MOT generates staggered effective magnetic fields to the sublattices of d-wave AMs [29]. We show that the MOT can switch the N´eel vector deterministically without the assistance of an external magnetic field and transform AMs in multidomain mag- netic configurations into a single magnetic domain. This mechanism does not require the net magnetization, the Dzyaloshinskii-Moriya interaction, the inversion symme- try breaking, or special sublatice structures of the AMs. Schematic– We assume that MO is externally in- jected into a d-wave AM. This can be achieved in an AM/normal metal (NM) bilayer (Fig. 1) by applying an in-plane electric field to the NM so that the MO current generated by the MO Hall effect in the NM is injected into the AM (similar to the in-plane-current-induced spin current injection in the spin-orbit torque (SOT) geome- try). It was recently predicted [36] that many transition metals exhibit MO Hall effect with the MO Hall conduc- tivity comparable to the spin Hall conductivities of Pt, Ta, and W. The NM side in Fig. 1(a) illustrate schemat- ically what an MO current (brown arrow) is microscopi- cally: the electron velocity of conduction electrons (grey dashed arrows) is reversed when either electron spin mag- netic moment direction (yellow solid arrows) or the elec-

2

AM NM MO current x y z Electric field NM MO current Electric field AM A B A B (a) (b) FIG. 1. Schematic of the N´eel vector switching in an AM/NM bilayer for N along the z-direction (a) and −z-direction (b). An in-plane electric field (green arrows) along the x-direction generates an MO current (brown arrows) in the NM through the MO Hall effect [36] and injects it to the AM. The yellow arrows in the NM denote the spin magnetic moment direc- tion of conduction electrons, and the black arrows in the AM denote the magnetization direction of the local magnetic mo- ment at the sublattices A and B of the AM. The horizontal (red) and vertical (blue) ellipses attached to the yellow and black arrows represent the orbital elongation directions. The yellow arrows in the AM denote the effective field generated by the injected MO. When the field is parallel (a) [anti-parallel (b)] to the magnetization direction of the local magnetic mo- ments, the MO current does not [does] switch N. tron orbital elongation direction [horizontal (red) vs ver- tical (blue) elongation] is flipped. When they are injected into an AM, they interact preferentially with those local moments (black solid arrows) that share the same or- bital elongation direction (horizontal vs vertical). This orbital-elongation-direction dependence of the interac- tion becomes sublattice-selective interaction since the or- bital elongation direction is different between the sublat- tices A and B. The sublattice-selective interaction then results in a staggered effective magnetic field along the MO electron spin magnetic moment directions in the sub- lattices A and B. The resulting field-like MOT on the N´eel vector N of an AM is given by, TM OT = −N × i gi O i , (1) which may be regarded as a dynamical implication of the coupling eneregy −N · ( i gi O i ) in equilibrium [31]. Here Oi is the MO density in an AM induced by the injected MO current injection. The vector direction of Oi denotes the spin magnetic moment polarization axis and the subscript i specifies the electron orbital elonga- tion direction of the injected MO electrons. On the other hand, the coupling constant gi describes the orbital elon- gation direction of the local moments. In the schematic in Fig. 1, gi is nonzero only for i = x2 − z2 , since x2 − z2 distinguishes the horizontal and vertical orbital elonga- tion directions. In a rutile-type d-wave AM film, only gx y is non-zero for the film grown along the [001] direction, only gx 2 −z2 is non-zero in the film along the [110] direc- tion, and both gx y and gy z are non-zero for the film along the [101] direction [31, 37]. TM OT in Eq. (1) acts anal- ogously to a uniform field-like SOT that drives the N´eel vector switching in collinear AFMs with hidden Rashba coupling [38, 39]. However, it differs in two key aspects: (1) The MOT does not require the hidden Rashba cou- pling in AMs. (2) Instead, the MOT necessitates the MO ordering, and does not function in conventional collinear AFMs with gi = 0 for all i. In other words, the MOT is a unique intrinsic feature of d-wave AMs. By the way, we ignore the damping-like MOT in the subsequent dis- cussion since its effective field is not staggered in general and not crucial for deterministic switching of N. The schematic in Fig. 1 allows one to guess the MOT- induced N´eel vector dynamics in an AM. For simplic- ity, we assume only one gi is nonzero and positive, and consider the case when Oi for that i is either parallel or antiparallel to the easy axis ˆn of N. Then, TM OT is expected to set N along the effective field direction Oi . Thus, if N is initially antiparallel to Oi , TM OT will switch N to make it parallel to Oi [Fig. 1(b)]. On the other hand, if N is initially parallel to Oi , TM OT will not switch N [Fig. 1(a)]. As a result, the AM in a multi-domain mag- netic configuration initially will be converted to a single domain. Below, we check this expectation through quan- titative calculations. Uniform domain dynamics– We examine quantita- tively how each domain in an AM responds to the MOT injection. We take the spin Hamiltonian of an AM as follows, Hs pin = ℏωe xS A · SB − σ=A, B ℏωe asy 2S2σ (Sσ · ˆn)2 , (2) where Sσ (σ = A, B) is the unit vector of the sub- lattice magnetization, ℏωe x is the exchange interaction, ℏωe asy the magnetic easy-axis anisotropy. The possible anisotropy of exchange stiffness [40] is neglected due to its irrelevance in single-domain AMs. The sublattice mag- netization dynamics can be described by the Landau- Lifshitz-Gilbert equation, ˙Sσ = −Sσ × He ff σ + αSσ × ˙Sσ + TS OT σ + TM OT σ + Tt h σ , (3) where He ff σ = −∂Hs pin/ ℏ∂Sσ is the effective magnetic field acting on sublattice σ (= A, B), and α is the Gilbert damping parameter. We consider both SOT and MOT since an in-plane electric field applied to an NM gen- erates both of them. For the SOT, we consider only the damping-like SOT since its effective field is stag- gered, whereas the effective field for the field-like SOT is not. At the sublattice σ, the SOT and the MOT are given by TS OT σ = T SOT Sσ × (Sσ × ˆp), and TM OT σ = −Sσ × ησ ( i gi O i ) [29], where ˆp represents the spin po- larization direction of the spin current injected into an

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sha1_base64="FrdPg6xJuASp/RqfE2dZ1gUaQ18=">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</latexit> ˆu → ˆn, ˆu → ˆp, ˆn → ˆp <latexit sha1_base64="IB0iJ12wPDc4LnTrWOOcc039USQ=">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</latexit> ˆu → ˆn, ˆu → ˆp, ˆn → ˆp <latexit sha1_base64="I0bRuSQNg+/kpyL4vViwp33R6XA=">AAACg3icbVFLa9tAGFwrTeuqjzjpsZePmEKhxUghpL20BHrpMYH6AZYxq9Una8nuatlHqTHq/+tP6K/INb1104gGOZ7TMN8My87kWnDrkuR3L9p7tP/4Sf9p/Oz5i5cHg8Ojia29YThmtajNLKcWBVc4dtwJnGmDVOYCp/nVl9v79Dsay2v1za01LiRdKV5yRl2QlgOWVdRtsrwE30CmqaFCoID/qmrew0/omtDoe0V3DWqHYTkYJqPkH+AhSVsyJC0uloe9flbUzEtUjglq7TxNtFtsqHGcCWzizFvUlF3RFW6otJK6qoFt1a5l3gB09Vx20ytDdcXZjwbegEIssICyNmBriVDylTdoO/4iVO6l2mV3NJRuIe6+OPeu/LjYcKW9Q8VCMNxKL8DVcDsIFNwgc2IdCGWGhx8Cq8IOzIXZ4tBdut3UQzI5GaVno7PL0+H557bFPnlNjslbkpIP5Jx8JRdkTBj5Ra7JDfkT7UfvopPo9M4a9drMK9JB9OkvK93CIQ==</latexit> ˆu → ˆn, ˆu ↑ ˆp, ˆn ↑ ˆp (d) (c) (a) (b) ◆ ◆ ★ ★ <latexit 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sha1_base64="jpjj5ym5mIHJ1KvIAAMPGbQiXz8=">AAACMnicbZC7TsMwGIUdrqXcWhhZLCokpipBqLCAkFjYANGbREvlOH9aCzuJ7D+oVZRXYYUH4GVgQ6w8BAa6hHKmo/OfI8ufn0hh0HVfnbn5hcWl5dJKeXVtfWOzUt1qmzjVHFo8lrHu+syAFBG0UKCEbqKBKV9Cx78//753HkAbEUdNnCTQV2wYiVBwhjYaVLaad1kPYYzZzWUzz+kJdQeVmlt3f0RnjTc1NTLV1aDqlHpBzFMFEXLJjLn13AT7GdMouIS83EsNJIzfsyFkTBnFcJTTv6mZKD+ntJj7qrgeapaMBB/ndI9GAAEENIw1NbECGophqsEU+oEllKrovzoyy8jQcvHF2xTD434moiRFiLgd2luYSoox/eZHA6GBo5xYw7gW9oeUj5hmHC3lsmXn/SU1a9oHda9Rb1wf1s5OpxRLZIfskn3ikSNyRi7IFWkRTsbkkTyRZ+fFeXPenY/f6pwz3WyTgpzPL9edplE=</latexit> T SOT = 0 <latexit sha1_base64="rVhnbUJbzplybOyjv4WmxfwmPro=">AAACNXicbZC9TsMwFIUdfkv5a0FMLBYVElOVIARMCImFDRAtINFSOc5Na2E7wb5BraI8DCs8AM/CwIZYeQUMdAlwpqNzz5HlL0ylsOj7L97E5NT0zGxlrjq/sLi0XKuvXNgkMxzaPJGJuQqZBSk0tFGghKvUAFOhhMvw9ujrfnkPxopEt3CUQlexvhax4Axd1KuttW7yDsIQ8/OTVlHQjoY76vdqDb/pf4v+NcHYNMhYp726V+lECc8UaOSSWXsd+Cl2c2ZQcAlFtZNZSBm/ZX3ImbKK4aCgv1M7UmFBaTkPVXndNywdCD4s6CbVABFENE4MtYkCGot+ZsCW+pGDlCn9Xx2Zw2RptfzidYbxfjcXOs0QNHdDd4szSTGhXwhpJAxwlCNnGDfC/ZDyATOMowNddeyC36T+movtZrDb3D3baRwejClWyDrZIFskIHvkkByTU9ImnOTkgTySJ+/Ze/XevPef6oQ33qySkryPT8Tbp9I=</latexit> T SOT →= 0 <latexit sha1_base64="uOV1jJhWB34N2eR5O22IiGz6PSw=">AAACKXicbZDNSgMxFIUz9a/Wv1aXbi4WwVWZEaluFEEElwrWFtpSMpk7bWgyGZKMWMo8hlt9AJ/Gnbr1RYzazVTP6nDuOYR8YSq4sb7/7pUWFpeWV8qrlbX1jc2tam37zqhMM2wxJZTuhNSg4Am2LLcCO6lGKkOB7XB88X1v36M2XCW3dpJiX9JhwmPOqHVRd9oLY7jM4RT8QbXuN/wfwV8TzEydzHQ9qHnlXqRYJjGxTFBjuoGf2v6UasuZwLzSywymlI3pEKdUGkntKIf51ExkmAMU81AW10NN0xFnDznsQ4IYYQSx0mCURIj5MNNoCv3Igclk8l/dUofGQKX4Yjez8Ul/ypM0s5gwN3S3OBNgFXxjg4hrZFZMnKFMc/dDYCOqKbMObsWxC+ZJ/TV3h42g2WjeHNXPz2YUy2SX7JEDEpBjck6uyDVpEUYUeSRP5Nl78V69N+/jt1ryZpsdUpD3+QWYnaKi</latexit> E =0 FIG. 2. (a) Top panel: The N´eel vector dynamics induced by the MOT with T SOT = 0 when the initial direction of N is along ˆn = −ˆz. Here, the MOT strength T MOT is set to 2 × 10− 3 ωe x. Middle (bottom) panel: The N´eel vector dynamics induced jointly by the MOT and the SOT with ˆu ∥ ˆn ⊥ ˆp (with ˆu, ˆn, ˆp all perpendicular to each other). T MOT and T SOT are set to 2×10− 3 ωe x and 0.5×10− 3 ωe x for the diamond symbol in (b), and 2.3×10− 3 ωe x and 0.1×10− 3 ωe x for the star symbol in (d), respectively. In the bottom panel, the yellow transparent region shows the dynamics when both the MOT and the SOT are turned off. (b)–(d) Switching phase diagram for the 180◦ switching of N when ˆn ∥ ˆu ⊥ ˆp (b), ˆn ∥ ˆu ∥ ˆp (c), and ˆn ⊥ ˆu ⊥ ˆp ⊥ ˆn (d). In the orange-colored regions, the switching occurs from −ˆn to +ˆn, whereas in the dark blue- colored regions, the switching occurs the other way around. The deterministic switching is not possible in the grey-colored regions. The toggle switching [47] occurs in the narrow green- colored region in (d). AM and ηA /B = ±1. For simplicity, we assume only one component of gi is nonzero and set i gi O i = T MOT ˆu, where ˆu is the unit vector of MO. Furthermore, we incor- porate thermal fluctuations as Tt h σ = −Sσ × Ht h σ [41, 42], with the random field defined by Ht h σ = N x σ (0, u)ˆx + N y σ (0, u)ˆy + N z σ (0, u)ˆz. Here, N i= x,y,z σ (0, u) denotes a normal distribution with zero mean and a standard de- viation u = 2γαkB T /VA MM s ( 1 + α2 )∆t [41, 42], with kB denoting the Boltzmann constant, γ the gyromagnetic ratio, VA M = 1000 × 1000 × 50 nm3 the volume of the al- termagnet layer, Ms = Ms ,A = Ms ,B = 1.5 × 105 A/m the saturation magnetization, ∆t = 1 ps the fluctuation time interval, and T = 300 K the temperature. Using the Landau-Lifshitz-Gilbert equation [Eq. (3)], we numerically calculate the response of N = (SA − SB ) /2 to the MOT and the SOT. We first consider the case where T SOT = 0 and only T MOT is activated at time t = 0 with ˆu = ˆn and T MOT > 0. We find that if N(t = 0) = −ˆn, the MOT reverses N to +ˆn, pro- vided T MOT is larger than a critical value [top panel in Fig. 2(a)]. On the other hand, if N(t = 0) = +ˆn, the MOT leaves N essentially unchanged (not shown). This confirms the schematic in Fig. 1. Before N is reversed in the top panel, a long incubation time appears. This is due to the fact that the MOT vanishes when N is perfectly anti-aligned with ˆu. Once the thermal fluctuations tilt N sufficiently away from −ˆu, the MOT reverses N quickly. This behavior is analogous to the magnetization reversal in an FM induced by the STT [35, 43, 44]. When T SOT is also activated at t = 0 together with T MOT (ˆu = ˆn), it may modify the MOT-induced dy- namics. Figures 2(b) and 2(c) show the switching phase diagrams for this case with the spin polarization ˆp of the SOT perpendicular to and (anti)parallel with ˆn, re- spectively. In the orange-colored regions, the joint ap- plication of T MOT and T SOT achieves the determinis- tic switching of N if N(t = 0) = −ˆn and does not switch N if N(t = 0) = ˆn. In the dark blue-colored re- gions, the deterministic switching occurs the other way around. In the grey-colored regions, the deterministic switching does not occur. Regardless of the direction of ˆp, we find that a moderate strength of T SOT assists the MOT-driven switching and reduces the critical strength of T MOT for switching. In particular, when ˆp is perpen- dicular to ˆn, T SOT also shortens the incubation time since the SOT tilts N away from the easy axis ±ˆn, thereby making the MOT more effective. Compared to the case with T SOT = 0 [top panel in Fig. 2(a)], the case with T SOT = 0.5 × 10− 3 ωe x reduces the critical MOT strength by 62% and the switching time by 84% [middle panel in Fig. 2(a)] [45]. On the other hand, very strong T SOT makes the deterministric switching difficult and increases the critical MOT strength regardless of the direction of ˆp since the strong SOT leads to precessional dynamics and probabilistic switching. Next, we consider the case where the MOT ˆu is per- pendicular to ˆn. From Eq. (3), we find that the MOT alone can achieve the probabilistic switching but the de- terministic switching is not possible except for a narrow range of T MOT [green-colored region in Fig. 2(d)], where a toggle switching occurs, similarly to the SOT-induced toggle switching in FMs [46]. Details of toggle switching are presented in Ref. [47]. Interestingly, we find that de- terministic switching can be restored when the SOT is applied together with the MOT in a wide range of the phase diagram [Fig. 2(d)] if the spin polarization ˆp of the SOT is perpendicular to both ˆu and ˆn. Thus, although either the MOT or the SOT alone can achieve only the probabilistic switching, their joint action sets a preferred direction of the switching and converts the probabilistic switching into the deterministic switching. The deterministric switching in this case is analogous to the SOT-induced deterministic magnetization switch- ing in FMs via the joint action of the SOT and an ex- ternal magnetic field [35] when the SOT spin polariza- tion, the field, and the easy axis of the magnetization are all perpendicular to each other. This analogy allows

4 one to understand a few interesting differences between the deterministic switchings in the two cases: ˆu ⊥ ˆn [Fig. 2(d)] and ˆu ∥ ˆn [Figs. 2(b) and 2(c)]. First, the switching dynamics in the former case is very different from the dynamics in the latter case. The joint action of T MOT and T SOT in the former case tilts N to a direction roughly perpendicular to ˆn and the deterministic switch- ing is achieved only after T MOT and T SOT are turned off [bottom panel in Fig. 2(a)], whereas T MOT in the latter case completes the deterministic switching while T MOT is turned on [top and middle panels in Fig. 2(a)]. The former dynamics is analogous to the SOT-induced deterministic magnetization switching dynamics in FMs, whereas the latter dynamics to the STT-induced deter- ministic magnetization switching dynamics in FMs [35]. The second difference appears in the behavior of the N switching direction upon the in-plane electric field re- versal. When an in-plane electric field is reversed, both T MOT and T SOT reverse their signs. Under this electric field reversal, the deterministic switching in the latter case (ˆu ∥ ˆn) reverses its switching direction, that is, the (−ˆn)-to-(+ˆn) switching is reversed to the (+ˆn)-to- (−ˆn) switching or vice versa [Figs. 2(b) and 2(c)]. In contrast, the in-plane electric field reversal in the former case (ˆu ⊥ ˆn) leaves the deterministic switching direction unchanged [Fig. 2(d)]. This unidirectional switching in the former case can be understood by resorting to the analogy to the SOT-induced deterministic switching in FMs [35]: Since the deterministic switching direction in FMs is reversed by the SOT sign reversal and also by the external magnetic field reversal, the joint reversal of the SOT and the field leaves the deterministic switching direction in FMs unchanged. Next, we discuss the relative directions of ˆu, ˆp, and ˆn for various AM/NM bilayer systems. When an in-plane electric field is applied to the bilayer along the x(y)- direction, the MO Hall effect injects into the AM var- ious components of MO currents, including Oxx y (Oyx y ), Ozy z (Ozx z ), Oyx 2 (Oxx 2 ), Oyy 2 (Oxy 2 ), and Oyz 2 (Oxz 2 ) [36]. Out of these components, only one (or at best two) com- ponent matches the orbital elongation direction of the equilibrium MO ordering in the AM and generates the MOT. On the other hand, the SHE injects into the AM spin currents with the spin polarization along the y(x)- direction. In the FeSb2 / NM bilayer, the easy axis ˆn lies along the y-direction [48] with the equilibrium MO or- dering of Oyx y . Then, an electric field application along the y-direction generates the MO current of Oyx y , thereby realizing the situation with ˆn (ˆy) ∥ ˆu (Oyx y ) ⊥ p (ˆx) [Fig. 2(b)]. In the RuO2 [ 110]/NM bilayer, ˆn lies along the y-direction [31, 49] with the equilibrium MO ordering of Oyx 2 −z2 . Then, an electric field along the x-direction gen- erates the MO current of Oyx 2 −y2 , thereby realizing the situation with ˆn (ˆy) ∥ ˆu (Oyx 2 −y2 ) ∥ p (ˆy) [Fig. 2(c)]. In the RuO2 [ 001]/NM or MnF2 / NM bilayer, ˆn is typically along the z-direction [30, 31, 49] with the equilibrium MO ordering of Ozx y . Then, an electric field along the x- direction realizes the situation with ˆn (ˆz), ˆu (Oxx y ), ˆp (ˆy) all perpendicular to each other [Fig. 2(d)]. Therefore, the deterministic N´eel vector switching is possible in all these bilayers. Domain wall dynamics– The domain wall (DW) mo- tion is an important element of the magnetization dy- namics. Here, we examine the DW motion in AMs in- duced by the application of the MOT and/or the SOT. For this study, we use the collective coordinate ap- proach [50, 51], which describes the DW dynamics in terms of the DW center position r(t) and the DW center angle ϕ(t). In the presence of the MOT and the SOT, the time evolution of the two collective coordinates are given by ¨r + aγlα˙r ± π 2 aγ2 lλT SOT ⊥ cos ϕ ∓ aλγT MOT ∥ = 0, (4) ¨ϕ + aγlα˙ϕ − aγ2 lT SOT ∥ + π 2 aγT MOT ⊥ sin ϕ = 0, (5) where l represents magnitude of the N´eel vector, γ is the gyromagnetic ratio, a denote the homogeneous exchange constants, and λ is the domain wall width. T MOT ∥ and T MOT ⊥ amount to the field-like MOT T MOT for the cases with ˆu ∥ ˆn and ˆu ⊥ ˆn, respectively. T SOT ∥ and T SOT ⊥ amount to the damping-like SOT T SOT for the cases with ˆp ∥ ˆn and ˆp ⊥ ˆn, respectively. The upper and lower signs in Eq. (4) apply to the light blue-to-light pink DWs and the light pink-to-light blue DWs [Fig. 3(a)], respectively. Here, the light blue and the light pink domains have N along the +ˆn and −ˆn directions, respectively. First, we consider the case with T MOT ∥ ̸= 0 and T SOT = 0 with ˆu ∥ ˆn. In this case, Eq. (4) predicts the steady state DW velocity of ˙rs teady = ±λT MOT ∥ /(αl), which has opposite signs for the light blue-to-light pink and the light pink-to-light blue DWs. Thus, for a posi- tive T MOT ∥ , the light blue (light pink) domains expand (shrink) [Fig. 3(b)]. For a negative T MOT ∥ , it is the other way around [Fig. 3(c)]. Therefore, the DW motion in- duced by T MOT tends to convert a multi-domain con- figuration into a single domain [light blue (light pink) domain for positive (negative) T MOT ∥ ]. Note that the re- sulting single domain agrees with the single domain en- forced by the uniform domain switching considered above [Fig. 2(b) and 2(c)]. The DW velocity does not depend on DW type (Bloch vs N´eel). Next, we consider the joint application of T MOT ⊥ ̸= 0 and T SOT ⊥ ̸= 0 with ˆu, ˆp, and ˆn all perpendicular to each other. In this case, Eq. (4) predicts the steady state DW velocity of ˙rs teady = ∓(π/2)(γλ/α)T SOT ⊥ cos ϕs teady , where the steady state DW angle ϕ steady is determined by Eq. (5) to be cos ϕs teady = sgn(T MOT ⊥ ). Here sgn(· · · ) denotes the sign function. We then obtain ˙rs teady = ∓(π/2)(γλ/α)T SOT ⊥ sgn(T MOT ⊥ ), which has opposite signs for the light blue-to-light pink and light pink-to-light blue DWs. Thus, for positive T MOT ⊥ T SOT ⊥ , the light pink (light blue) domains expand (shrhink) [similar to Fig. 3(c)]. For negative T MOT ⊥ T SOT ⊥ , it is the other way around [similar

5

AM NM Electric field Electric field (b) (c) (a) xy z FIG. 3. Schematics of the DW dynamics in AMs driven bythe MOT in AM/NM bilayers. (a) Schematic of an AM/NM bi- layer with the AM in a multidomain configuration. Light blue and light pink regions represent domains with N´eel vectors aligned along ˆn and −ˆn, respectively. (b), (c) When an elec- tric field (green arrow) is applied, the resulting MOT induces the DW motion. For ˆu ∥ ˆn, positive T MOT ∥ drives the light blue domains to expand and the light pink domains to shrink (b). Negative T MOT ∥ drives the domain expansion and shrink- age the other way around (c). to Fig. 3(b)]. Therefore, the DW motion induced jointly by T MOT ⊥ and T SOT ⊥ tends to convert a multi-domain con- figuration into a single domain [light pink (light blue) domain for positive (negative) T MOT ⊥ T SOT ⊥ ]. Note that the resulting single domain again agrees with the sin- gle domain enforced by the uniform domain switching in Fig. 2(d). Note also that an in-plane electric field reversal, which reverses the signs of both T MOT ⊥ and T SOT ⊥ , does not reverse the DW motion direction, which is consistent with the unidirectional switching in Fig. 2(d). Discussion– Our switching mechanism offers several key advantages. First, the MO Hall effect in NMs respon- sible for the MOT in AMs does not require any symmetry breaking and thus can remain operative even in polycrys- talline NM layers. Second, our mechanism is stable with respect to at least moderate amount of the AM/NM in- terface roughness. For instance, in the A-type AFM or- dering in AMs, when the interface layer on the AM side fluctuates between the sublattices A and B region by re- gion, our switching mechanism remains active since the staggered effective magnetic field of the MOT arises from the orbital-elongation-direction dependence rather than the sublattice location. Third, since the MO Hall con- ductivity and spin Hall conductivity vary across different NM materials [36], this variation provides an additional degree of freedom to engineer the electrical deterministic switching behavior of AMs. Our switching mechanism applies also to g-wave AMs. Due to the coupling in g-wave AMs between their N´eel vector and the rank-5 magnetic triakontadipole (MT) [31, 32], the injection of the MT into g-wave AMs generates the MT torque, which generates the staggered effective fields on the sublattices of the AMs and allows for the deterministic switching of the N´eel vector. The MT injec- tion can be achieved in g-wave AM/NM bilayers through the MT Hall effect in NMs. For instance, the MT Hall effect is sizable in Pt [47]. A recent experiment [24] on g-wave AM/Pt bilayers reported field-free switching in g- wave AMs and attributed it to the joint action of the SOT and the Dzyaloshinskii-Moriya interaction. We speculate that the MT torque may be partially responsible for the reported switching. Our switching mechanism based on magnetic multi- poles may be combined with other switching mecha- nisms [52, 53]. For an AM1/NM/AM2 trilayer, suppose a single domain configuration can be achieved in the AM1 by other switching mechanisms (magnetic field [24] for instance). Then, an out-of-plane-direction electron flow from the AM1 to the AM2 can generate a magnetic multi- pole torque on the AM2 [54], which can be utilized for the deterministic switching of the AM2. This switching re- sembles the switching in FM1/NM/FM2 trilayers caused by out-of-plane-current-induced STT [52, 53]. An NM1/AM/NM2 trilayer may allow for additional functionalities. Suppose the in-plane current I1 and I2 to the NM1 and NM2 are controlled separately by a word line and bit line, respectively, and the NM1 and NM2 generates mainly the MOT and SOT, respectively, with their directions depicted in Fig. 2(d). Then, reversing the sign of I2 reverses the switching behavior caused by I1, realizing a reconfigurable switching device. To conclude, we demonstrated that the N´eel vector in AMs can be deterministically switched via magnetic mul- tipole torque, without requiring any external magnetic field or Dzyaloshinskii-Moriya interaction. By harnessing magnetic multipoles as active control parameters, our study introduces a new paradigm for manipulating the N´eel vector in AMs. This approach facilitates domain alignment, which is essential for realizing the intrinsic physical properties of AMs—making our findings of both technological and fundamental significance. ACKNOWLEDGMENTS We thank Changyoung Kim, Sang-Wook Cheong, and Daegeun Jo for the fruitful discussions. This work was financially supported by the National Research Foun- dation of Korea (NRF) grant funded by the Korean government (MSIT) (No. RS-2024-00356270, RS-2024- 00334933, RS-2024-00410027) [1] S. Hayami, Y. Yanagi, and H. Kusunose, Momentum- Dependent Spin Splitting by Collinear Antiferromagnetic Ordering, JPSJ 88, 123702 (2019).

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