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Metallic p-wave magnet with commensurate spin helix
Rinsuke Yamada 1 ,¶ , ∗ Max T. Birch 2 ,¶ , Priya R. Baral 1 ,¶ , Shun Okumura 1 , Ryota Nakano 1 , Shang Gao 2 ,§ , Motohiko Ezawa 1 , Takuya Nomoto 3 , Jan Masell 2 , 4 , Yuki Ishihara 1 , Kamil K. Kolincio 2 , 5 , Ilya Belopolski 2 , Hajime Sagayama 6 , Hironori Nakao 6 , Kazuki Ohishi 7 , Takashi Ohhara 8 , Ryoji Kiyanagi 8 , Taro Nakajima 9 , Yoshinori Tokura 1 , 2 , 10 , Taka-hisa
arXiv:2502.10386v2 [cond-mat.str-el]
9
Sep 2025
Arima 2 , 11 , Yukitoshi Motome 1 , Moritz M. Hirschmann 2 , and Max Hirschberger 1 , 2†
1 Department of Applied Physics and Quantum-Phase Electronics Center,
The University of Tokyo, Bunkyo, Tokyo 113-8656, Japan
2 RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
3 Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan
4 Institute of Theoretical Solid State Physics,
Karlsruhe Institute of Technology (KIT), 76049 Karlsruhe, Germany
5 Faculty of Applied Physics and Mathematics,
Gdańsk University of Technology, Narutowicza 11/12, 80-233 Gdańsk, Poland
6 Institute of Materials Structure Science,
High Energy Accelerator Research Organization, Tsukuba, Ibaraki 305-0801, Japan
7 Neutron Science and Technology Center,
Comprehensive Research Organization for Science and
Society (CROSS), Tokai, Ibaraki 319-1106, Japan
8 J-PARC Center, Japan Atomic Energy Agency,
2-4 Shirakata, Tokai, Ibaraki 319-1195 (Japan)
9 The Institute for Solid State Physics,
University of Tokyo, Kashiwa, Chiba 277-8561, Japan
10 Tokyo College, University of Tokyo, Bunkyo, Tokyo 113-8656, Japan
11 Department of Advanced Materials Science,
The University of Tokyo, Kashiwa, Chiba 277-8561, Japan
§ Current address: Department of Physics,
University of Science and Technology of China, Hefei 230026, China and
¶ Equal contribution
2 Abstract
Antiferromagnetic states with spin-split electronic structure give rise to novel
spintronic, magnonic, and electronic phenomena despite (near-) zero net mag-
netization [1–7]. The simplest odd-parity spin splitting – p-wave – was originally
proposed to emerge from a collective instability in interacting electron systems
[8–12]. Recent theory identifies a distinct route to realise p-wave spin-split elec-
tronic bands without strong correlations [13, 14], termed p-wave magnetism.
Here we demonstrate an experimental realisation of a metallic p-wave magnet.
The odd-parity spin splitting of delocalised conduction electrons arises from
their coupling to an antiferromagnetic texture of localised magnetic moments:
a coplanar spin helix whose magnetic period is an even multiple of the chemi-
cal unit cell, as revealed by X-ray scattering experiments. This texture breaks
space inversion symmetry but preserves time-reversal (T ) symmetry up to a
half-unit-cell translation – thereby fulfilling the symmetry conditions for p-wave
magnetism. Consistent with theoretical predictions, our p-wave magnet exhibits
a characteristic anisotropy in the electronic conductivity [13–15]. Relativistic
spin-orbit coupling and a tiny spontaneous net magnetization further break T
symmetry, resulting in a giant anomalous Hall effect (AHE, σ xy > 600 S/cm, Hall angle > 3 %), for an antiferromagnet. Our model calculations show that the spin
nodal planes found in the electronic structure of p-wave magnets are readily
gapped by a small perturbation to induce the AHE.
∗ [email protected] † [email protected]
3 Main Text The spin splitting of electronic band states in momentum space is deeply intertwined with the symmetries of magnetic order in direct space [1–5, 13]. Uniform ferromagnetic order results in different energies for spin-up ( ↑ ) and spin-down ( ↓ ) bands, producing a net spin polarisation in momentum space that is isotropic across the Fermi surface, as depicted in Fig. 1a. In the language of spherical harmonics, this corresponds to an s -wave state with even parity. In contrast, p -wave spin splitting is characterised in momentum space by an anisotropic spin-split Fermi surface with collinear ↑ and ↓ states exhibiting odd parity, shown in Fig. 1b. In solids, p -wave spin splitting has long been sought as the electronic analogue to the A -phase of superfluid helium-3. An initial theoretical route to p -wave spin splitting was pro- posed in the form of a spin-channel Pomeranchuk instability — a spontaneous Fermi surface distortion driven by electronic correlations that breaks inversion symmetry but preserves the translational symmetry of the system (i.e., magnetic modulation vector k m ag = 0) [8, 9, 12]. However, recent theory suggests that p -wave spin splitting also arises when materials form a magnetic superlattice with k m ag ̸ = 0, where the spin polarisation of electronic bands is now considered in the magnetic Brillouin zone [13, 14]. This scenario, termed p -wave magnet, can be realized without strong electronic corre- lations and arises when conduction electrons are coupled to an antiferromagnetic texture of localized magnetic moments obeying certain symmetries: Time reversal symmetry T paired with a half-unit cell translation t 1 / 2 constrains the spin splitting to have odd parity or to vanish, while a 180 degree spin rotation C 2 ⊥ combined with t 1 / 2 ensures a unique axis for the spin expectation value of electronic states in momentum space – say, having only ⟨S x ⟩ [4, 5, 14]. To render ⟨S x ⟩ finite, we require noncollinear magnetism and inver- sion symmetry breaking in practice, since collinear magnets only allow for even parity spin splitting [5]. Strictly speaking, the symmetry requirements related to t 1 / 2 are satisfied only by magnetic orders whose period is an even, integer multiple of the lattice constant: for example, the commensurate spin helix illustrated in Fig. 1b, where the p -wave spin splitting is described by a spin polarisation vector α (gray arrow) perpendicular to the rotation plane of magnetic moments (violet disk with looped arrow).
4 Low-energy electronic structure of a p-wave magnet
To show that a spin helix generically leads to an electronic structure with p -wave character, we use its symmetries T ∥ t 1 / 2 and C 2 ⊥ ∥ t 1 / 2 to constrain a single-orbital, low-energy model around k = 0. In the nonrelativistic limit, the only allowed terms are H 0 ( k ) = t 0 k 2 σ 0 + pk x σ x where σ 0 , σ x , t 0 and p are the identity matrix, a Pauli matrix, the transfer integral, and the magnitude of spin splitting, respectively (see Methods). This yields a spin-dependent band splitting with spin expectation value along k x , described by the spin polarisation vector α shown in Fig. 1c. Here, the spin-nodal plane is depicted as the line of states, indicated by the black parabola, on which ↑ (red) and ↓ (blue) bands touch. Figure 1d shows the model’s momentum-resolved expectation value for the spin component ⟨S x ⟩ at the chemical potential. Anisotropic charge flow is naturally expected in this model of a p -wave magnet due to the difference in electronic structure along the k x and k y axes [6, 7, 13–17]. The degeneracy of bands on the spin-nodal plane can be lifted by breaking of T symmetry using m z and explicitly introducing a relativistic spin-orbit coupling λ allowed by inversion symmetry breaking in the p -wave phase (Methods), H ( k ) = t 0 k 2 σ 0 + pk x σ x + m z σ z + λk y σ y . (1) The band hybridization induced by m z ̸ = 0, λ ̸ = 0 results in an anomalous Hall effect (AHE) due to the Berry curvature for the two bands shown in Fig. 1e, f respectively (see Methods). We stress that – in contrast to other spin-split antiferromagnets of recent interest, such as d - or g -wave altermagnets [3–5] – the p -wave spin splitting does not, perforce, violate T . Instead, the AHE emerges only in presence of a T -breaking m z .
Search for a candidate p-wave magnet
To realize a p -wave magnet based on an evenly commensurate spin helix [13, 14], we focus on the intermetallic compound Gd 3 R u 4 A l 1 2 , which is known to host coplanar spin helices which are incommensurate to the underlying atomic lattice [18–21], and where the exchange interaction between the localised Gd moments is mediated by the conduction electrons. Re- placing Rh for Ru and tuning the band filling in Gd 3 ( Ru 1 −δ Rh δ ) 4 A l 1 2 , we shift the Fermi level – and thus the period of the magnetic spin helix (see Methods and Supplementary Fig. S2). This allows us to approach an N = 6 commensurate spin helix, where both T ∥ t 1 / 2 and C 2 ⊥ ∥ t 1 / 2 symmetries are satisfied. Around δ ≈ 0 . 05, we identify a distinct
5 magnetic ground state, with signatures of p -wave magnetism, as shown in the composi- tion–temperature phase diagram of Fig. 2b. To confirm that the magnetic order in the δ ≈ 0 . 05 phase is consistent with the symme- try requirements, we perform resonant elastic X-ray scattering (REXS) measurements on a series of single crystals with different Rh content δ (Methods). For each sample, we deter- mine the magnetic propagation vector k m ag and its orientation relative to the crystal axes. The position of k m ag in the Brillouin zone is shown schematically in the inset of Fig. 2b, where blue, red, and black symbols indicate k m ag for δ = 0, 0 . 05, and 0 . 20, respectively. In a narrow composition window around δ ≈ 0 . 05, we find that the magnetic satellites are indexed as (1 / 6 , 1 / 6 , 0) in Miller notation – indicating that the magnetic period is evenly commensurate with the crystal lattice ( N = 6). The reconstructed unit cell, spanning six crystallographic unit cells, is outlined in orange in Fig. 2d. Supporting neutron experiments are shown in Extended Data Fig. E3.
Coplanar antiferromagnetism from resonant X-ray scattering
To satisfy C 2 ⊥ ∥ t 1 / 2 symmetry, we specifically require coplanar magnetic order; that is, all magnetic moments should be perpendicular to a single axis. This is probed by X-ray polarisation analysis measurements in the experimental geometry of Fig. 2c, where the incoming beam k i and the outgoing beam k f span the scattering plane (gray shaded, semi- circular plane; see Methods). The polarisation of incident X-rays is in the scattering plane ( π -polarised), while we detect both horizontal ( π ′ ) and vertical ( σ ′ ) components of the scattered beam. The corresponding intensities are I π −π ′ and I π −σ ′ , respectively. The π − π ′ channel probes the magnetic moment perpendicular to the scattering plane, I π −π ′ ∝ m 2 z , whereas the π − σ ′ channel is I π −σ ′ ∝ ( k i · m i p ) 2 [22] (see Methods). Here, m i p is the pro- jection of the magnetic moment onto the scattering plane. For each of the three magnetic reflections illustrated in Fig. 2e – corresponding to three p -wave magnetic domains allowed by the hexagonal symmetry – we measured I π −π ′ and I π −σ ′ , shown in Fig. 2f-h. The finite I π −π ′ intensity for all three reflections demonstrates that m z is nonzero. In Fig. 2h, I π −σ ′ nearly vanishes, which we attribute to the relative angle between k i and m i p approaching 90°, consistent with a spin helix. These X-ray measurements identify the magnetic ground state as a coplanar spin helix, where every spin rotates within a common m z - m i p plane nor- mal to the spin polarisation vector α (shown in Fig. 2i). This magnetic structure preserves
6 the combined spin rotation and translation operation [ C 2 ⊥ ∥ t 1 / 2 ] as well as [ T ∥ t 1 / 2 ] , and therefore satisfies the symmetry requirements for p -wave magnetism. The spin polarisation vector α for electronic states in momentum space, introduced in Fig. 1b, is then orthogonal to all local magnetic moments in direct space.
Electronic anisotropy of a p-wave magnet
One of the predicted consequences of the symmetry-protected p -wave state is in the elec- tronic anisotropy, where the directions parallel and perpendicular to the spin polarisation vector α are not equivalent, and we expect to see a resistance which depends on the angle of the applied current relative to the direction of α [13–15]. To probe this, we utilised fo- cused ion beam milling to fabricate a freestanding, circular transport device from a single crystal with δ = 0 . 05. The meandering contacts relieve strain caused by thermal expansion mismatch with the substrate (Fig. 3a, Methods), helping to suppress extrinsic pinning of α . This device enables simultaneous measurement of the resistance along three symmetry- equivalent directions of the hexagonal crystal structure, which become inequivalent upon p -wave magnetic ordering (Fig. 3b). Resistance R x x measured along the crystallographic a -axis as a function of temperature is shown in Fig. 3c, revealing magnetic phase transitions consistent with the bulk magnetization (red). Applying a finite magnetic field perpendicular to the c -axis and rotating the sample by the angle ϕ , we observe a characteristic two-fold anisotropic resistivity in Fig. 3d-f below the magnetic transition – in contrast to the weak, conventional anisotropic resistivity [23] ob- served in the paramagnetic state (Fig. 3d, white circles). For each magnetic field direction, one of the three current directions I a , I b , or I c exhibits a higher resistance, while the other two have lower resistance. This demonstrates that, for each sextant of the magnetic field angle ϕ , one of the three magnetic domains characterised by distinct orientations of the spin polarisation vector α is selected. Following the conventions of Fig. 1a, the insets in Fig. 3d show the evolution of α in discrete steps during the field rotation, and confirm that current flows more easily along the direction perpendicular to α . In the present compound, the spin nodal plane is fixed at k x = 0 by symmetry, so the two-fold anisotropic resistivity is directly related to the two-fold symmetry of the p -wave magnet’s spin-split electronic structure (see Methods).
7 Zero-field, giant anomalous Hall effect induced by time reversal breaking
Our p -wave magnet exhibits a tiny spontaneous magnetisation, resulting in a T -breaking distortion of the p -wave electronic structure – a scenario anticipated by models of spin helices of localized magnetic moments coupled to itinerant electrons [24]. Returning to the bulk crystal with δ = 0 . 05, we set the magnetic field B along the crystallographic c -axis and find that the magnetization increases roughly linearly and approaches the saturation value M s at expected for Gd 3 + moments (Methods), as shown in Fig. 4a. Around zero magnetic field, we observe a small spontaneous magnetization m z of about 2 % of M s at with a coercive field of about 0 . 15 T. While the p -wave state is slightly distorted when m z becomes finite, the character of p -wave splitting – or ’ p -waveness’ [25] – remains robust in the regime of small m z (see Supplementary Fig. S11). The hysteretic behaviour is discussed in terms of weak exchange anisotropies in Supplementary Note S8. As outlined above, we argue that finite spin-orbit coupling in the real material, combined with slight T -breaking of the p -wave bands in momentum space, can generate an intrinsic anomalous Hall effect driven by Berry curvature. Consistent with this expectation, a strik- ing transport response appears when B is parallel to the c -axis: with the current in the ab plane, a sharp, giant anomaly is observed in the Hall conductivity σ x y around B = 0, which is smoothly suppressed as B approaches the critical field B c 1 . At higher magnetic field, in the incommensurate fan-type phase II and in the field-aligned ferromagnetic state, the Hall effect returns to conventional behaviour, being composed of a magnetization-dependent term and a term linear in B : σ x y = S n B + S m µ 0 M with constants S n and S m (Methods, Supplementary Fig. S7) [26]. The strongly B -dependent contribution ∆ σ x y in the p -wave state is isolated by subtracting these two terms, and the resulting traces of ∆ σ x y are shown in Fig. 4c,d. Returning to the minimal low energy model presented in Eq. (1), we can discuss the experimentally observed B -field dependence of the AHE. The result is shown in Figure 4f, which demonstrates a sharp anomaly in the anomalous Hall conductivity σ x y around zero magnetic field or around zero m z , when |λ| < |p| (see Supplementary Note S13). Finally, Fig. 4e compares the zero-field AHE in various bulk antiferromagnets, highlighting the large ∆ σ x y response in our p -wave candidate, which exceeds 600 S/cm – comparable to the intrinsic AHE of ferromagnets. Unlike other antiferromagnets with large AHE and small net magnetization [27–32], M ( H ) here saturates at modest, accessible fields, enabling practical switching between the p -wave and field-aligned ferromagnetic states.
8 Conclusions
The experimentally observed, commensurate spin helix with N = 6 period in direct space satisfies the symmetry requirements for a p -wave magnet up to a tiny, uniform spin splitting m z . This p -wave magnet exhibits anisotropic electronic properties and a surprisingly large anomalous Hall effect in zero magnetic field. The Hall response is attributed to Berry curvature in momentum space arising when m z opens a gap at the nodal planes of a p -wave magnet. Recent theoretical studies predict the generation of large linear and non-linear spin currents in p -wave magnets as well as a large tunneling magnetoresistance [6, 7, 15, 33– 35]. In particular, it is believed that the low number of nodal planes in spin-split p -wave antiferromagnets is advantageous for functionality as a spin injector [33, 34]. Expanding on the literature of proximitized spin helices as a route to unconventional superconductiv- ity [36–39], superconducting proximity effects have also been recently discussed for p -wave magnets [17, 40]. Our findings establish metallic p -wave magnets as an ideal platform to explore the interplay of spin-split electronic states in magnets, spintronics, superconductiv- ity and electronic band topology. Note added: We became aware of the realization of an insulating odd-parity wave magnet after submission of this work [41].
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14 Methods Crystal growth and characterization. Single crystals of Gd 3 ( Ru 1 −δ Rh δ ) 4 A l 1 2 were synthesized under Ar gas flow in a high-vacuum floating zone (FZ) furnace. The sample quality is confirmed by powder X-ray diffraction (XRD), energy-dispersive X-ray spec- troscopy (EDX), and by examination under an optical polarisation microscope. We used Laue XRD and a diamond saw to precisely orient and align the sample surfaces.
Magnetization and electrical transport measurements on bulk samples. Magne-
tization was measured in a commercial Quantum Design MPMS3 (DC-mode) and Quantum Design PPMS-14T (VSM, vibrating sample magnetometer). The magnetic field is applied along the crystallographic c -axis of the crystal. Electrical transport measurements were performed on polished single crystal plates with electrical contacts made of 30 µ m gold wires, using silver paste (Dupont). We set the sample surface perpendicular to the c -axis ( B ∥ c ) and applied the electric current along the a -axis to measure the data in Fig. 3. We perform a demagnetization correction by assuming that the sample shape is an ellipsis to compare field-dependent physical properties of samples with different shapes [42].
Focused ion beam device fabrication and measurements. Using a Thermo Fischer
Helios 5UX focused ion beam (FIB) system, a lamella was carved and extracted from the bulk crystal, thinned down, and shaped into a circle. Using an in-situ micro-manipulator, this circle was placed on to a Al 2 O 3 substrate patterned with Au contacts, and attached with ion-beam induced Pt deposition. The meander-shaped bonds and final sample geometry were created with a final ion milling step. The entire device was coated in a capping layer of 5 nm of Al 2 O 3 using atomic layer deposition. The anisotropic resistance of the finished device was measured in a Quantum Design PPMS-14T as described in the previous Methods paragraph, using the as-provided rotator probe option. Resonant elastic X-ray scattering (REXS). Magnetic resonant elastic X-ray scat- tering (REXS) experiments were conducted at BL-3A (KEK, Photon Factory, Japan) using the Gd- L 2 absorption edge. The scattering plane is ( H, K, 0), so that the incoming and outgoing beam wave vectors k i and k f are perpendicular to the crystallographic c -axis. We
15 used the (006) reflection of a pyrolytic graphite plate (2 θ ∼ 88 ◦ at the Gd- L 2 edge) to perform polarisation analysis of the scattered X-ray beam (see Fig. 2c and Supplementary Note S7 for more details). We define the orientation of the detector with respect to the scattering plane by an angle γ such that γ = 0 ◦ detects the π −σ ′ component of the scattered X-ray beam, whereas the π − π ′ component is detected in the out-of-plane orientation of the detector at γ = 90 ◦ . p -wave spin splitting. In this work, we consider collinear spin splitting in the mag- netic Brillouin zone of momentum space, i.e., spin-split electronic states bands with a single non-zero component of the spin expectation value ⟨S x ⟩ ̸ = 0, ⟨S y ⟩ = ⟨S z ⟩ = 0. ⟨S x ⟩ > 0 and ⟨S x ⟩ < 0 are labeled as ↑ and ↓ , respectively, although |⟨S x ⟩ | < ℏ / 2 is allowed. In the low-energy limit, the electronic band structure has an energy gap ∼ cos( lθ q ) between ↑ , ↓ bands in momentum space, with l = 0, 1, 2 for s , p , d wave spin splitting. The azimuthal angle θ q is the angle of q relative to the x -axis [9]; q = k − k T RIM is measured from the position of the TRIM around which the low-energy model is expanded (Supplementary Note S15).
Collinear spin expectation value in momentum space. We use the symmetries
of the present material with six-fold expanded magnetic unit cell ( N = 6) to discuss the spin polarisation of conduction electron states in momentum space. Here, spin-space group symmetries are denoted as [ S ∥ R ], where S acts on spin-space and R applies to real space, while operators without brackets are used to refer to specific (matrix) representations of symmetry operations. In the present material, the symmetry [ C 2 ⊥ ∥ t 1 / 2 ] is equivalent to [ C 2 x ∥ t 1 / 2 ] and all bands and eigenstates can be labelled using its symmetry eigenvalues λ s . We introduce the composite operator U s t 1 / 6 with the intention of defining the λ s : a sixfold spin-rotation around the x -axis, U s = exp i 2 π 6 S x ℏ , combined with a translation t 1 / 6 = a o / 6 + b o / 2. Here, S x is the spin-1 / 2 operator component. We use the lattice vectors a o , and b o of the magnetic unit cell (orthorhombic), with lattice constants a o , b o – as in Fig. 2d. Let a Hamiltonian of conduction electrons be exchange-coupled to the magnetic state discussed in the Main Text. Without SOC, this Hamiltonian commutes with ( U s t 1 / 6 ) 3 = i 2 S x ℏ t 1 / 2 ≡ ˜ S x . The energy eigenstates |E, λ s ⟩ are also eigenstates of ˜ S x with eigenvalue λ s .
16 Using that ˜ S x is unitary and ˜ S x S y = −S y ˜ S x , the spin expectation value of S y vanishes, ⟨S y ⟩ = ⟨E, λ s | ˜ S † x ˜ S x S y |E, λ s ⟩ = ⟨E, λ s | λ ∗ s ( −S y ) λ s |E, λ s ⟩ = − ⟨S y ⟩ ⇒ ⟨S y ⟩ = 0, which holds at any momentum k . By the same argument ⟨S z ⟩ = 0, so only ⟨S x ⟩ ̸ = 0 is possible. Low-energy model from symmetries. Starting from the point-group symmetries of p -wave magnets, we construct a low-energy model describing an electronic band in two dimensions, coupled to the magnetic texture. The model represents the low-energy limit of the electronic structure of a p -wave magnet in the magnetic Brillouin zone. For such an expansion around k = 0, the symmetries [ C 2 x ∥ t 1 / 2 ] and [ T ∥ t 1 / 2 ] are represented by C 2 x = iσ x and T = iσ y K , respectively, with the complex conjugation K . Without SOC, the Hamiltonian is required to commute with C 2 x , and further, is required to be consistent with p -wave time-reversal symmetry T − 1 H ( k ) T = H ( − k ). To linear order, all allowed terms are equivalent to pk x σ x . As for spin-orbit coupling (SOC), we consider leading-order terms that are allowed due to broken inversion symmetry and breaking of all mirror planes in the p -wave state. The spin rotation C 2 x can now be broken and any term of the form k i σ j with i ∈ {x, y} and j ∈ {x, y, z} is consistent with T . However, SOC does not break a combined rotation [ C 2 x ∥ C 2 x t 1 / 2 ] in spin and direct spaces, which is characteristic for the magnetic state depicted in Fig. 2i. Among the remaining symmetry-allowed terms, k x σ y , k y σ y , k x σ z , k y σ z , the k x - dependent terms are expected to be negligible compared to the p -wave splitting. As k y σ z and k y σ y are equivalent in their effect on spin-splitting, we include only k y σ y , which is needed for a non-zero anomalous Hall effect. Finally, the kinetic energy of a parabolic band is k 2 σ 0 and the Zeeman-type energy in an exchange field along the z -axis is ∝ σ z . These considerations lead to Eq. (1), which can be rewritten in terms of the vector d as H ( k ) = t 0 k 2 σ 0 + pk x σ x + m z σ z + λk y σ y = t 0 k 2 σ 0 + i d i σ i . (2) with eigenenergies E ± = k 2 ± | d | and Berry curvature Ω ± x y = ∓ϵ a bc d a ∂ k x d b ∂ k y d c / (2 | d | 3 ), where ϵ a bc is the Levi-Civita symbol. We note that H 0 ( k ) = t 0 k 2 σ 0 + pk x σ x can also be obtained explicitly as the low-energy limit of a tight-binding model considering conduction electrons coupled to a commensurate spin helix composed of 4 f local magnetic moments. This calculation is shown in Supplementary Notes S9, S10.
17 Spin-nodal plane in momentum space. In the following, we make some more general remarks on the electronic structure of the p -wave magnet. We can argue for the existence of degenerate nodal planes in momentum space by invoking T symmetry on the eigenstates of ˜ S x = U s t 1 / 6 3 with eigenvalues ±λ s . The T operation comprises complex conjugation of wavefunctions and a spin flip, so we have T ˜ S x = ˜ S x T . This yields E ( k ) = E ( − k ), since two eigenstates of the Hamiltonian are related by T |E, k , λ s ⟩ ∝ |E, − k , −λ s ⟩ with T λ s = −λ s T . The λ s is a good quantum number for all crystal momenta k ; hence, along any line connecting the momenta k and − k , there must be an odd number of exchanges between eigenstates labelled by + λ s and −λ s . This implies an odd number of nodal planes on which the energy bands of the Hamiltonian are two-fold degenerate in absence of SOC. Pinning of nodal plane to high symmetry directions. We discuss symmetry con- straints that force the nodal planes to be flat and spanned by high-symmetry directions in k -space. Again invoking the [ C 2 x ∥ C 2 x t 1 / 2 ] symmetry and combining with [ T ∥ t 1 / 2 ] , we define their product G = [ C 2 x T ∥ C 2 x ] . This G commutes with the Hamiltonian and with [ C 2 x ∥ t 1 / 2 ] up to a full lattice translation. In addition, G acts on crystal momenta like a mirror operation, ( k x , k y , k z ) → ( −k x , k y , k z ) and exchanges the sign of λ s by complex conjugation. Thus, G enforces a nodal plane at k x = 0, where the eigenvalues ±λ s label states of degenerate energy. Extended Data Fig. E4 and Supplementary Fig. S3 discuss the two-fold symmetric anisotropic magnetoresistance attributed to electronic anisotropy and the pinned spin-nodal plane in our p -wave magnet. Electronic structure calculations. We performed spin density functional theory (DFT) calculations for Gd 3 R u 4 A l 1 2 . The calculations were carried out using the projector augmented-wave method as implemented in the Vienna ab initio simulation package [43], together with the generalized gradient approximation for the exchange-correlation functional proposed by Perdew, Burke, and Ernzerhof [44]. The crystal structure data were adopted from Ref. [18]. In the electronic structure calculations, we employed a plane-wave energy cutoff of 400 eV and a 9 × 9 × 9 Monkhorst-Pack k -point grid. All calculations were per- formed assuming a ferromagnetically ordered state. To account for the strong correlation effects in the Gd 4 f orbitals, we used the DFT+ U approach, setting the on-site Coulomb interaction U to 6 . 7 eV and the Hund’s coupling J to 0 . 7 eV[45, 46].
18 Acknowledgments
We acknowledge Akiko Kikkawa for help in crystal growth. We also acknowledge support from the Japan Society for the Promotion of Science (JSPS) under Grant Nos. JP22H04463, JP23H05431, JP22F22742, JP22K20348, JP23K13057, JP24H01607, JP23H00171, 25K17336, and JP24H01604, as well as from the Murata Science Foundation, Yamada Science Foun- dation, Hattori Hokokai Foundation, Mazda Foundation, Casio Science Promotion Foun- dation, Inamori Foundation, and Kenjiro Takayanagi Foundation. This work was partially supported by the Japan Science and Technology Agency via JST CREST Grant Num- bers JPMJCR1874, JPMJCR20T1, and JPMJCR20T2 (Japan) and JST FOREST (JP- MJFR2238). It was also supported by Japan Science and Technology Agency (JST) as part of Adopting Sustainable Partnerships for Innovative Research Ecosystem (ASPIRE), Grant Number JPMJAP2426. P.R.B. acknowledges Swiss National Science Foundation (SNSF) Postdoc.Mobility grant P500PT 217697 for financial assistance. Mo. H. is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - project number 518238332. J. M. acknowledges funding from the DFG under the Project No. 547968854. Resonant X-ray scattering at Photon Factory (KEK) was carried out under proposal num- bers 2022G551 and 2023G611. The neutron experiments at the Materials and Life Science Experimental Facility of J-PARC were performed under a user program (Proposal No. 2020B0347, TAIKAN, and No. 2020A0282, SENJU).
Data and code availability
The raw data and code used to produce the figures and conclusions presented in this study have been deposited, with detailed comments, on the Publication Data Repository System of RIKEN Center for Emergent Matter Science (Wako, Japan). These data are available from the corresponding authors upon reasonable request.
Author contributions
Ma.H., Y.Mo., T.A., and Y.To. conceived the project. R.Y. and Ma.H. grew and charac- terized the single crystals. R.Y., R.N., and Ma.H. performed magnetization and transport measurements on bulk samples. M.T.B. prepared focused ion beam devices and performed all device measurements. P.R.B, R.N., S.G., H.S., H.N., and Ma.H. performed resonant
19 X-ray scattering experiments and analyzed data with guidance from T.-h.A.. T.Na., K.O., K.K., T.O., R.K. and Ma.H. performed neutron scattering experiments, while the data was analyzed by P.R.B. and Y.I. S.O., Mo.H., M.E., J.M., and Y.Mo. performed symmetry analysis and model calculations. T.No. performed ab-initio calculations. R.Y., M.T.B., P.R.B., and Ma.H. wrote the manuscript with help from I.B.; all authors discussed the results and commented on the manuscript.
Competing interests
The authors declare no competing interests.
20 Main Text Figures
21
m z = 0 k y k x E c d e f nodal line k y k x m
z
≠
0
[a.u.] E m z = 0.01 𝐶 2 ⊥ ∥ 𝒕 1 /2 𝒯 ∥ 𝒕 1 /2 α α y x z k y k x k z b a nodal line 〈 S x 〉
FIG. 1. Magnetic structure of a p-wave magnet in direct space and spin-split electronic
bands in momentum space. a , Ferromagnets have spins along the z -axis in direct space and a uniform spin polarisation S z ( k ) in k -space (right). b , p -wave magnetism (conceptual): Non- collinear spin modulation in direct space and alternating spin polarisation in momentum ( k -)space. Red and blue bars: momentum-resolved expectation value of S x ( k ) at the chemical potential. For commensurate order with wavelength λ m ag = N (even integer N ), this state is unchanged under 180 ◦ spin rotation followed by translation of half a lattice constant, termed [ C 2 ⊥ ∥ t 1 / 2 ] symmetry. The spin polarisation vector is α [15]. Violet disk and circular arrow: spin rotation plane in direct space. c , Minimal electronic structure model for a p -wave magnet depicted in the magnetic Brillouin zone according to Eq. (1). The spin-splitting of bands causes two-fold electronic anisotropy. Red (blue) indicate spin parallel (antiparallel) to k x . d , Spin polarisation ⟨S x ⟩ when m z = λ = 0 in Eq. (1), which represents a sum of all occupied states – i.e., of the two bands in ( c ). e , Electronic structure in presence of a tiny net magnetization m z . f , Anomalous Hall conductivity σ x y induced by Berry curvature Ω x y appears in the k x - k y plane when m z ̸ = 0 and λ ̸ = 0, due to lifting the band degeneracy of the nodal plane.
22
m i p m i p m i p k i a h * b h * G 2 50 k ( 1) mag k ( 2) mag k ( 3) mag 𝛌 m ag = 2.6 nm γ = 90 0 γ = 0 0 k i k f π ’ 𝜎 ’ Detector π Analyzer m i p m z c a b c e f g h i d Gd Ru/Rh Al c b h a h b h a o b o a h k i k i G 2 50 + k ( 1) mag G 2 50 - k ( 2) mag G 2 50 + k ( 3) mag T N 2 T N 1 p
-
w
a
v
e
M K Γ K’ δ , E F I π - π’ I π - σ’ I π - π’ I π - σ’ I π - π’ I π - σ’ 𝐶 2 ⊥ ∥ 𝒕 1 /2 𝒯 ∥ 𝒕 1 /2
FIG. 2. Commensurate spin helix in direct space, consistent with p-wave magnetism,
revealed by resonant elastic X-ray scattering (REXS). a , Crystal structure of hexagonal Gd 3 ( Ru 1 −δ Rh δ ) 4 A l 1 2 . b , Tuning the magnetic propagation vector k m ag by chemical substitution of Rh for Ru. T N 1 (dotted line) is the magnetic transition temperature from the paramagnetic state, while T N 2 (solid line) corresponds to the transition into the p -wave state. Inset: Evolution of the magnetic ordering vector k m ag with a shift of the Fermi energy E F due to Rh-doping δ : k m ag = (1 / 6 , 1 / 6 , 0) appears for δ = 0 . 04 − 0 . 07, c.f. shaded region in the main panel. For δ = 0 and δ = 0 . 2 the magnetic wavevectors in the ground state are (0 . 272 , 0 , 0) and (0 . 5 , 0 , 0), respectively. Momenta are expressed in reciprocal lattice units (r.l.u.) and only half of the first hexagonal Brillouin zone is depicted, sliced at k z = 0. c , Single crystal REXS experiments in reflection geometry, with incoming and outgoing beams k i , k f spanning the scattering plane (light gray). In polarisation analysis, I π −π ′ and I π −σ ′ components of k f are separated by rotating the detector around an analyzer plate by the angle γ (Methods). d , Crystallographic unit cells (hexagonal, black grid) and reconstructed magnetic unit cell (orthorhombic, black box). The requirements for p -wave magnetism, [ C 2 ⊥ ∥ t 1 / 2 ] and [ T ∥ t 1 / 2 ] symmetry, are satisfied by the black arrow. e , Due to the formation of three p -wave magnetic domains, six magnetic reflections appear in a given Brillouin zone. f - h , Polarisation analysis using magnetic satellites around the fundamental (2 , 5 , 0) reflection. Insets: Geometry of k i relative to the modulated magnetization projected into the scattering plane, m i p . i , Illustration: Commensurate spin helix with short wavelength λ m ag in direct space.
23 d e f 𝑅 ! ! " (Ω) 𝑅 ! ! # 𝑅 ! ! $ 0.1450 0.1465 𝝰 𝝰 𝝰 𝐼 " 2 K, 0.9 T 30 K, 0.9 T a b 5 μm c a c 𝜙 B 𝐼 % 𝐼 & 𝐼 ' 𝑉 % 𝑉 & 𝑉 ' 100 mT Au Pt 𝐼 # & 𝐼 # ' 𝐼 $ & 𝐼 $ ' 𝐼 " & 𝐼 " ' T N 2 T N 1 B ∥ c 𝐼 $ 𝐼 # Improved Fig. 3 a B ∥∥ c 𝐼 % 𝑉 ( B ∥∥ c 𝐼 ∥ a 𝑉 ( FIG. 3. Anisotropic electronic transport properties. a , Scanning electron micrograph of the rotationally symmetric focused ion beam device. The meander-shaped arms release the strain on the device, which arises due to thermal expansion mismatch of device and substrate. The false colour regions indicate the target material (blue) and the electrical contacts (orange), which are composed of evaporated gold (Au) wires and ion beam-deposited platinum (Pt), respectively. The contacts utilised in the directional transport measurements are labelled and colour coded. The scale bar is 5 µm. b , Device geometry with crystal axes a , c and three pairs of current contacts I a , I b , I c ; the corresponding longitudinal voltage drops V a , V b , V c are measured simultaneously. The magnetic field B is rotated by an angle ϕ to the a -axis. c , Temperature-dependent traces of longitudinal resistance (left) and magnetization M (right), with two antiferromagnetic transitions T N 1 and T N 2 . d , Change in longitudinal resistance R a x x when switching between different p -wave domains by a small magnetic field of 0 . 9 T. Insets: deduced direction of the spin polarisation vector α in momentum space, and the periodic spin texture in direct space. The gray wedge indicates the sextant of the magnetic field angle over which the depicted p -wave domain is stable. Two- fold anisotropy emerges, with higher resistance for current I a (center arrow) parallel to α . Open circles: Data at 30 K > T N 1 plotted for comparison. e , f Simultaneous measurements for other contact pairs, d emonstrating the anisotropic resistance is due to the relative alignment between current and α .
24
a b c Phase II (IC) p -wave (C) d B || c B || c B || c B || c B ∥∥ c I a ∥∥ a V y -m z domain +m z domain -500 0 500 B c 1 B c oerc B c 1 5 1 2 3 4 0 1000 100 10 1000 100 10 This work Mn 3 S n Mn 3 G e Mn 3 G a Mn 3 S b Mn 3 P t CoNb 3 S 6 CoTa 3 S 6 Mn 5 S i 3 NbMnP TaMnP e Ce 2 C uGe 6 f μ = 0.2 μ = 0.4 λ / p = 0.02 Model calculation
FIG. 4. Giant anomalous Hall effect (AHE) induced by a tiny net magnetization m z and
spin-orbit coupling (SOC). a , Comparison of net magnetization M ( B ) measured at T = 2 K and 20 K. The magnetic field is applied along the c -axis. Black triangles correspond to a transition field ( B c 1 ) from the p -wave magnetic phase (commensurate, C) to the high-temperature Phase-II (incommensurate, IC). Inset: small spontaneous magnetization m z appears around zero field below B c oerc (red triangles) at low temperature. b , Hall conductivity with large zero-field (anomalous) value σ A x y reaching up to 600 Ω − 1 cm − 1 , measured on a bulk single crystal. c , d , Hall conductivity ∆ σ x y after subtraction of components proportional to the magnetic field and to the magnetization (Supplementary Fig. S5). ∆ σ x y is enhanced in the p -wave magnetic Phase-I. Open circles marking the phase boundaries in panel c are obtained from magnetization measurements in Extended Data Fig. E7. Red arrows in panel d indicate the direction of the field sweep. e , As compared to other bulk antiferromagnets with AHE, our slightly distorted p -wave state in Gd 3 R u 4 A l 1 2 has large σ A x y induced by m z , large Hall angle σ A x y /σ x x , and a low saturation magnetic field B s at . Raw data for the bar plot are provided in Extended Table E1. f , Calculated anomalous Hall conductivity from the low-energy model Eq. (1) plotted against m z / p . We consider the breaking of p -wave state around m z /p ∼ 1 . 5, where anomalous Hall conductivity is suppressed to zero (see Supplementary Note S13 for details).
25 Extended Data k y k x E a b c E - E + λ = 0 λ = 0.4 λ = 0.4 m z = 0.01 FIG. E1. Fermi surfaces of a p -wave band. a , Band dispersion with p -wave splitting, spin- orbit coupling (SOC), and time-reversal breaking m z = 0 . 01. Here, we set p = 1 . 2 and λ = 0 . 4, where p -wave exchange splitting is larger than the size of SOC and the net magnetization is small but non-zero. b , c , Fermi surfaces of a p -wave split band with ( m z = 0, λ = 0) and without ( m z = 0 . 01, λ = 0 . 4) time-reversal symmetry, respectively. The two bands are degenerate on the k x = 0 plane for m z = 0. However, the degeneracy is lifted due to band hybridization when m z is finite in Eq. (1). The equal energy surface is calculated at E = 10 in panels ( b , c ). Contrary to the low-energy model for a p -wave Pomeranchuk instability, |⟨S x ⟩ | < ℏ / 2 for the p -wave magnet of conduction electrons coupled to a spin helix (Methods, Supplementary Note S15).
26 6 7 8 9 r
b
.
u
n
i
t
s
)
10 4 15 20 25
(
a
r
b
.
u
n
i
t
s
)
k M = G 4 40 - k ( 1) m ag (HH0) (HH0) 2 θ i f a b ω
(
)
single crystal k Г a * b * G 4 40 ( 1) mag k
(
1
)
m
a
g
G 4 40 k ( 1) mag k ( 2) mag k ( 3) mag k k –
0 7.92 7.94
E (keV)
7.96 0 7.98 0 0.2 0.4 0.6
2 1 - cos ()
0.8 1 FIG. E2. Resonant elastic X-ray scattering of magnetic satellites in Phase-I. a, The
magnetic reflection intensity at k M ( ≡ G 4 40 − k ( 1) m ag ) and total fluorescence signal obtained at T = 4 K strongly depend on the energy E of the incoming synchrotron X-ray beam. The maximum intensity due to magnetic scattering appears at E = 7 . 935 keV, in resonance with the Gd- L 2 absorption edge. Inset: schematic of momentum space and magnetic satellites around the (440) fundamental (structural) reflection. We indicate the center of momentum space, Γ, and lattice vectors of momentum space. b , Measurement geometry for detecting the k M reflection: k i and k f are the wavevectors of the incoming and outgoing X-ray beams, respectively. The angles ω , 2 θ defined here are referred to in Eq. (S5) and following. c , polarisation analysis of magnetic satellites close to (440), (250), and (700) fundamental reflections using Eq. (S6) reveals good agreement with a distorted spin helix model. Dashed green (yellow) lines represent expected behaviour for a harmonic cycloid (harmonic helix). The data are consistent with a distorted helix, elliptically squeezed into the hexagonal basal plane.
27
high angle detector bank small angle detector banks mid angle detector banks magnet a* c B sample a b 0 .0 0 .5 1 .0 1 .5 0 1 2 3 4 5 6 (x 10 - 3 ) I
n
t
e
n
s
i
t
y
(
a
r
b
.
u
n
i
t
s
)
I
n
t
e
n
s
i
t
y
(
a
r
b
.
u
n
i
t
s
)
c d 5.66 4.9 9 4.48 0 5 1 0 15 T (K) -0.05 0.00 0.05 0 .10 0 .15 0.20 0.25 I
n
t
e
n
s
i
t
y
(
a
r
b
.
u
n
i
t
s
)
e 1.62 1.42 1.34 | Q | (Å - 1 ) -0.05 0.00 0.05 0.10 0.15 0.20 0.25 f 2 K 8 K 15 K 2 T 1 T 0 T -6 00 - 200 + (1) mag 2 K 8 K 15 K - 600 M = - 200 + (1) mag G G G G Q k M = - 200 + (1) mag G Q k k k i 1.51 | Q | (Å - 1 ) 1.62 1.42 1.34 1.51 | Q | (Å - 1 ) a * b * k ( 1) mag k ( 2) mag k ( 3) mag T = 2 K G - 210 G - 2-10 G - 310 G - 1-10 G - 200 -0.3 -0.3 0 .0 -2.25 -2 -1.75 H (r.l.u.) (
-
K
,
2
K
,
0
)
(
r
.
l
.
u
.
)
L = 0
FIG. E3. Magnetic order in Phase-I , probed by elastic neutron scattering. a , Experimental geometry. A rectangular cuboid shaped crystal with Gd in natural abundance was installed inside a 4 T superconducting magnet. The magnet with sample assembly was rotated around the vertical axis by 29 ◦ in order to access reflections on the high-angle detector. The beamline is equipped with five detector banks; only the high-angle detector (shown in blue) was used to collect scattered neutrons. b , Reciprocal space map at L = 0, obtained at T = 2 K and B = 0 T. Data measured on an empty can at 15 K are used for background correction. Nuclear reflections are labelled. The red and green dashed lines represent the coordinate axes in the hexagonal system. Inset: six-fold symmetric magnetic satellite reflections in the region around G − 200 , marked by a white dashed box in the main panel. k m ag -vectors are labelled according to Main Text Fig. 2e. (Caption continues on next page.)
28 FIG. E3. (Caption continues from previous page.) c , d , Temperature and magnetic field evolution of Q M ( ≡ G − 200 + k ( 1) m ag ) are shown in panels c and d , respectively. Solid lines represent back-to- back exponential fits as defined in Eq. (S1). The curves are shifted by a constant value for better visibility. e , Line scan through the G − 600 nuclear reflection as a function of total momentum transfer | Q | , tracked across the two T -induced transitions. f , Integrated intensities of G − 600 and Q M obtained from fitting a back-to-back exponential function. For the fits shown in panels c , d , and e , width and center position were kept constant while background and amplitude of the peaks were treated as free parameters. Experiments were performed at beamline BL15 (TAIKAN) of MLF, J-PARC (Tokai, Japan).
29
a B a B a B Configuration 3 𝑅 𝑥 𝑥 a (mΩ) 𝐼 a 2 K, 0.9 T 30 K, 0.9 T c d a c 𝐼 a V a B 𝜙 e f i g h j Configuration 2 b Configuration 1 a
FIG. E4. Step-like anisotropic magnetoresistance due to switching of p-wave domains,
corresponding to switching the spin polarisation vector α. a-c, Three types of α-domain
configurations that can be controlled by rotating the magnetic field in the ab -plane. The relative angle between the a -axis and B is ϕ . The gray shaded wedges indicate the range of angles where a given domain is stable: ϕ = 0 ◦ ∼ 60 ◦ (180 ◦ ∼ 240 ◦ ), 60 ◦ ∼ 120 ◦ (240 ◦ ∼ 300 ◦ ), and 120 ◦ ∼ 180 ◦ (300 ◦ ∼ 360 ◦ ), respectively, for Configuration 1, 2, and 3. (Caption continues on next page.)
30 FIG. E4. (Caption continues from previous page.) d , Polar plot of AMR with illustrations of α - domain configurations. Step-like AMR observed in the p -wave phase originating from the switching of α -domains. For current I a ∥ a , a higher resistance R a x x appears in Configuration 2 (red region), where I is nearly parallel to α . We also show the measurement configuration of anisotropic magnetoresistance (AMR). The current I a is applied along the a -axis, while the magnetic field B is rotated within the ab -plane. e , f , AMR of a fabricated device with strain relief at T = 2 K and 30 K. The second y -axis on the right-hand side shows the relative magnitude of the AMR. g , h , AMR of a bulk sample at T = 2 K and 30 K. Bulk and device curves are in good qualitative agreement, although the AMR’s relative magnitude is three times larger in the suspended device. We note that the AMR measurement on devices allows us to measure the resistivity along several current directions simultaneously (see Fig. 3, Main Text). i , j , Separation of AMR into (i) step- type anomaly related to resistivity anisotropy in the p -wave magnet (green line, right axis) and (ii) six-fold patterns with the periodicity of the underlying crystal lattice (blue symbols, left axis). As the relative magnitude of the p -wave AMR is larger in the device, the six-fold patterns are harder to distinguish. Arrows in e , g indicate the sweep direction of the magnetic field angle ϕ .
31 a 85 b 20 80 75 4000 65
< xy (+ -1 cm -1 )
70
10 5 60 6000 5000 15 ;
yx (µ+cm)
;
xx (µ+cm)
c3000 2000 1000
22 K
55 " T = 2 K 0 0
50
-5 -2.5 0 2.5 5
B (T)
-5 -2.5 0
B (T)
2.5 5 -1000
-5 -2.5 0 2.5 5
B (T)
FIG. E5. Electrical resistivity and Hall resistivity data for a bulk single crystal of
Gd 3 ( Ru 0 . 95 R h 0 . 05 ) 4 A l 1 2 . a - c , Longitudinal resistivity ρ x x , Hall resistivity ρ y x , and Hall conduc- tivity σ x y at various temperatures. In support of Fig. 3 of the Main Text, here we show raw data of each quantity. Direction of magnetic field ramping is indicated by black arrows, with solid lines for ∂B/∂t < 0 and dashed lines for increasing magnetic field. The magnetic field is applied along the c -axis of the crystal. The offset between curves is set to be 2 µ Ωcm, 2 µ Ωcm, and 500 Ω − 1 cm − 1 in panels a , b , and c , respectively.
32
FIG. E6. Comparing transport properties of bulk single crystal and suspended device at
T = 2 K. The electric current and magnetic field are applied along the a -axis and c -axis, respectively. a Longitudinal resistivity ρ x x for a device fabricated by the focused ion beam technique. b , c Magnetic field dependence of Hall resistivity ρ y x and longitudinal conductivity σ x x . d Hall conductivity σ x y calculated for the device (red/blue) and for the bulk single crystal (gray line). The data are in good qualitative agreement. Inset: schematic measurement geometry. The hysteresis in ρ x x and σ x x is attributed to the texture of p -wave domains, as discussed further in Supplementary Fig. S4. To calculate the conductivities and resistivities of the device, we approximate the width and length of the central circular patch as a rectangular bar, implying significant systematic errors due to the geometry of the device. Note: The Hall conductivity data presented in the Main Text was obtained from bulk single crystals to suppress systematic measurement errors for the absolute value of σ x y .
33 2 0 -2 CW 0 100 200
T (K) d 20 5 0 0 2 4
B (T)
5 T = 2 K -2 12 2 6 10
B (T)
B coerc 0 14 B c1 10 -1 -1 10 3.5 K T = 1 K
6.5 K 7 K T = 0.5 K
10 K T = 1 K
14 K 16 K 10
-5 dM/d B (µ B f.u. -1 M (µ B f.u.
) 15 15 -15 300 T )
bT 6 -1 T )
0 0.08
0.4 T -0.1 dM/d T 20 0.06 dM/d T (emu mol -1 K -1 )
e 25 -1 Moment 1/
4 6 8 10 m
o
l
e
m
u
-
1
)
B = 0.1 T 5 15 25 n
t
(
B
f
.
u
.
-
1
)
a c B c 1
dM/d B ( B f.u.
-0.2 -0.3 0.04 0 0.2 T 10 20 T (K)
0.6 T 0.02
B = 0.2 T 4.0 T 0
8
-0.02 6 4
0 1
B (T)
2 3 -0.04 5 10 15
T (K)
20 25
FIG. E7. Bulk magnetization data with B ∥ c . a , Curie-Weiss fit of magnetic susceptibility χ = M/H , measured with 0 . 1 T external magnetic field applied along the c -axis. b , M ( H ) curves obtained at constant temperatures T as indicated in the panel. c , Magnetization isotherm and its corresponding field derivative at T = 2 K. The dashed vertical line, labelled as B c 1 , represents the hysteretic boundary between Phase-I and Phase-II. Around this region, ramping up and ramping down direction of external magnetic field is indicated by solid and dashed arrow, respectively. d , Derivative of magnetization with respect to applied field at various T , showing peak-like features at B c oerc (red arrow) and B c 1 (black arrows) defined in Fig. 4a. Consecutive curves are shifted by 0 . 2 µ B f . u . − 1 T − 1 for better visibility. Similar to Panel-c, solid and dashed lines denote the direction of magnetic field ramping. e , Temperature dependent derivative dM/dT of magnetization data measured at constant magnetic field, showing two clear transitions, T N 1 and T N 2 as defined in Fig. 3. Each consecutive curve is shifted by 0 . 002 emu mol − 1 K − 1 for better visibility. Inset: Zoom-in of data at low magnetic field.
34 /p = 0.01 = 0.2 = 0.4 = 1.2 0 1 y
/
m
a
x
x
y
/p = 0.02 b = 0.2 = 0.4 = 1.2
0.4 0.0 0.4 0.8 S z k = 0 / S 0 0 1 /p = 0.1 = 0.2 = 0.4 = 1.2 0 1 y
/
m
a
x
x
y
d
1
0.4 0.0 0.4 0.8 S z k = 0 / S 0 /p = 1 = 0.2 = 0.4 = 1.2
1 0.8
0.4 0.0 z
S k = 0 / S 0 0.4 0.8 0.8 0.4 0.0 z
S k = 0 / S 0 0.4 0.8
FIG. E8. Anomalous Hall conductivity of a distorted p -wave magnet. Anomalous Hall conductivity as a function of the uniform net magnetization S z k =0 . Here, S 0 is the total length of the magnetic moment (saturation moment). In the low-energy model, only m z is varied while p = 0 . 1, k B T = 0 . 04 – as well as spin-orbit coupling strength λ and chemical potential µ , written in each panel – are fixed. We calculate S z k =0 from m z /p and the parameters J , t of a tight-binding Hamiltonian, as described in Supplementary Note S12. A sharp anomaly emerges for small S z k =0 when λ is weak, λ/p < 1. In the gray region, the spin helix collapses as discussed in Supplementary Note S13.
35
FIG. E9. Solitonic spin states with a net magnetization from biquadratic magnetic
interactions, following Supplementary Note S8 . To explain the observed distorted spin helix state with a weak net magnetization, we develop a simple 1D two-parameter model in Eq. (S8). In this model, we introduce the exchange anisotropy ∆ > 0 and ∆ ′ > 0 to favour a spin helix with net magnetization along the z -axis; we also use the RKKY interaction J = 1, and the four spin interaction K . We optimize the energy to obtain the ground state of the model by simulated annealing. This figure shows the calculation results obtained for ∆ = 0 . 1. Magnetic states with higher harmonics and spontaneous magnetization appear when the four-spin interaction is sufficiently large, K > 0 . 25.
36 Material Spin texture q -vector B e st s at σ A x y σ A x y /σ x x T m ax Ref. (r.l.u.) (T) (Ω − 1 cm − 1 ) (%) (K) Gd 3 ( Ru 0 . 95 R h 0 . 05 ) 4 A l 1 2 non-collinear ( 1 6 , 1 6 , 0) 5 . 5 600 4 . 2 - 4 . 7 2 This work Mn 3 S n non-collinear (0 , 0 , 0) 660 20 - 140 0 . 4 - 2 . 4 2 [27, 47] Mn 3 G e non-collinear (0 , 0 , 0) 1500 15 - 350 0 . 4 - 3 . 6 2 [28, 48] Mn 3 G a non-collinear (0 , 0 , 0) 180 18 0 . 5 300 [49] Mn 3 S b non-collinear (0 , 0 , 0) 650 310 1 . 1 - 1 . 5 150 [50] Mn 3 P t non-collinear (0 , 0 , 0) 2445 100 0 . 8 120 [51] CoNb 3 S 6 non-coplanar ( 1 2 , 0 , 0) 250 27 0 . 7 23 [29] CoTa 3 S 6 non-coplanar ( 1 2 , 0 , 0) 170 75 - 135 0 . 9 - 1 . 6 6 [30, 31] Mn 5 S i 3 non-coplanar ( 1 2 , 0 , 0) 180 160 2 . 3 25 [52] NbMnP non-collinear (0 , 0 , 0) 1100 230 1 . 9 - 3 . 2 2 [53] TaMnP non-ollinear (0 , 0 , 0) 1200 370 1 . 4 - 2 . 7 2 [54] Ce 2 C uGe 6 collinear (0 , 0 , 0) 90 550 0 . 2 1 . 3 [55]
A in zero-field and sat- TABLE E1. Comparison of anomalous Hall conductivity σ
uration magnetic field B e st s at among bulk single crystalline magnets with small net magnetization. Gd 3 ( Ru 0 . 95 R h 0 . 05 ) 4 A l 1 2 shows a commensurate non-collinear spin texture, a giant anomalous Hall conductivity, and a large Hall angle. It also has a low magnetic field for transition into the ferromagnetic state, simplifying the data analysis. T m ax is a temperature where σ A x y takes a maximum value. Error bar of σ A x y /σ x x in our material is estimated from the variation of the values between a bulk sample and a device. As for Mn 3 S n and Mn 3 G e, error bars for σ A x y and σ A x y /σ x x indicate variation of the values for different field directions. Since σ A x y /σ x x of Mn 3 S b, NbMnP, and TaMnP takes a maximum at temperatures different from T m ax of σ A x y , we added error bars for these compounds. As for CoTa 3 S 6 , error bars are added because there is a variation in reported values of σ A x y . In Fig. 4e, we use σ A x y and σ A x y /σ x x at T m ax , and also add error bars for σ A x y /σ x x based on this table.
