Perfectly harmonic spin cycloid and multi-Q textures in the Weyl semimetal GdAlSi

Domain: physics
Published: 2026-02-23
Editorial outcome: major_revision

Abstract
A fundamental question in condensed matter physics concerns how topological electronic states are influenced by many-body correlations, and magnetic Weyl semimetals represent an important material platform to address this problem. However, the magnetic structures realized in these materials are limited, and in particular, no clear example of an undistorted helimagnetic state has been definitively identified. Here, we report clear evidence of a harmonic helimagnetic cycloid with an incommensurate magnetic propagation vector Q in the Weyl semimetal GdAlSi via resonant elastic X-ray scattering, including rigorous polarization analysis. This cycloidal structure is consistent with the Dzyaloshinskii–Moriya interaction prescribed by the polar crystal structure of GdAlSi. Upon applying a magnetic field, the cycloid undergoes a transition to a novel multi-Q state. This field-induced, noncoplanar texture is consistent with our numerical spin model, which incorporates the Dzyaloshinskii–Moriya interaction and, crucially, anisotropic exchange interactions. The perfectly harmonic Weyl helimagnet GdAlSi serves as a prototypical platform to study electronic correlation effects in periodically modulated Weyl semimetals.

Introduction
Inversion-breaking Weyl semimetals host two-fold degenerate band crossings with a quasi-relativistic, linear energy-momentum dispersion and protected surface Fermi arcs1. As such, inversion breaking Weyl semimetals provide a platform to study the interplay between many-body correlations and topologically nontrivial electronic states. Relevant proposals include the formation of an axion insulator2–4 and a three-dimensional fractional quantum Hall effect from disordered domains in a Weyl charge-density-wave5–7. Such states are realized by band-folding of Weyl fermions into a small Brillouin zone upon formation of periodically modulated order, and conditions for the hybridization of Weyl points have been derived and phrased in the language of special relativity8. In this context, topics of interest have been not only charge order but also the formation of periodically modulated helimagnetic states in inversion broken Weyl semimetals, and the question of the relevant magnetic interactions in such Weyl semimetals9–15. However, an ideal platform for studying the interplay of Weyl fermions with helimagnetism—a perfect and undistorted helimagnetic structure in an inversion breaking Weyl semimetal—has never been reported by quantitative polarization analysis in diffraction experiments16,17.

Here, we report the undistorted, cycloidal helimagnetic ground state of GdAlSi with perfectly isotropic Gd3+, a material in which Fermi arcs from Weyl nodes have been identified, and a possibly related anomalous Hall effect has been observed under a magnetic field; the experiments supersedes a prior numerical prediction of collinear antiferromagnetism (altermagnetism) in this material18,19. We further demonstrate a field-induced transition to a complex staggered cone structure, which represents a superposition of two magnetic waves, i.e., a multi-Q state. Our numerical spin calculations successfully reproduce the ground state and field-induced phase transitions of GdAlSi, emphasizing the role of symmetry-allowed Dzyaloshinskii-Moriya (DM) and anisotropic exchange interactions.

Results
Electronic and magnetic state of GdAlSi
GdAlSi crystallizes in the polar tetragonal I4md structure with neither inversion center nor M z mirror plane, leading to a Weyl semimetallic state that is strongly coupled to the magnetic texture, see Fig. 1a. Large

The period and propagation direction of the helimagnetic “wave” in GdAlSi are readily understood from the viewpoint of a minimal exchange Hamiltonian with isotropic nearest-neighbor interaction J 1 < 0 and inter-layer coupling J c < 0, illustrated in Fig. 1c. Consistent with the experiment, this natural choice of interactions (i) selects only propagation vectors of the type Q 1,2 = (±q, ±q, 0), constrained by tetragonal symmetry and (ii) reproduces the experimentally relevant range of q for a reasonable J c/J 1 ratio (see Fig. 1d). Details of the calculation are given in Supplementary Note S5.

Importantly, the Q-vector observed experimentally in GdAlSi is well matched to the distance between sets of Weyl points in the electronic structure, as shown in our calculations of Fig. 1e. Prior work on NdAlSi has demonstrated a modification of quantum oscillations at the Weyl nodes below the onset of magnetic order12. The reciprocal effect of relativistic fermions on the magnetic structure, a strong contribution to magnetic interactions by Weyl electrons via modified Ruderman-Kittel-Kasuya-Yosida interactions, has not yet been conclusively demonstrated9–11. Indeed, electronic structure calculations on RAlSi (R = rare earth) show that states far from the Fermi energy predominantly fix the value of Q31. Therefore, Weyl nodes and helimagnetic order both exist in GdAlSi, but helimagnetism is not predominantly caused by the Weyl electrons and the matching of Q to the separation of Weyl points in momentum space is accidental, see also Supplementary Note S4.

Zero-field cycloid from resonant elastic X-ray scattering (REXS)
From magnetic susceptibility measurements and the resistivity ρ xx in GdAlSi, we demonstrate a conducting ground state with a magnetic Néel point T N = 32 K (Fig. 2a, b). The magnetic susceptibility has very weakanisotropy both above and below T N (see Supplementary Fig. S3); ρ xx appears affected by magnetic fluctuations above and around T N, but fluctuations freeze out upon cooling. We employ resonant elastic X-ray scattering (REXS) measurements to reveal the helimagnetic order of GdAlSi (Methods). Figure 2c illustrates the experimental geometry of the sample with respect to the X-ray beam. Satellite magnetic peaks for two magnetic domains D1, D2 in vicinity of the fundamental Bragg reflection (2, 0, 0) are observed with the propagation vector along the 〈110〉 directions: QD1 = (q, −q, 0) or QD2 = (q, q, 0). Figure 2d indicates that q is not perfectly matched to the lattice, i.e., incommensurate, in GdAlSi: q = 2/3 + δ with δ = 0.006.

Polarization analysis of the scattered X-rays reveal that the zero-field state of GdAlSi is the only known, undistorted helimagnetic spin texture in a magnetic Weyl semimetal12,13. In general, the Fourier transformed magnetic moment m(Q) is separated into three mutually orthogonal components:

m(Q) = m c (Q)e c + m k (Q)e Q + m ? (Q)e c × e Q (2)

where e c and e Q are unit vectors along the [001] direction of the tetragonal structure and along Q, respectively. In our experiment, the incident X-rays are linearly polarized with their electric field component within the scattering plane (π-plane). The scattered X-rays can have two polarization components: parallel (π0) and perpendicular (σ0) to the scattering plane. These two polarization components are separated at the detector (Methods). In our geometry, the π → π0 scattered intensity always detects m c(Q); we further select two representative magnetic reflections for which the intensity of the π → σ0 channel corresponds to m ∥(Q) and m ⊥(Q) in Eq. (2), respectively. Figure 2e, f suggests that GdAlSi has m c(Q) and m ∥(Q) of comparable magnitude, but not m ⊥(Q): this corresponds to a harmonic helimagnetic cycloidal structure at zero magnetic field, which is further confirmed by quantitative analysis with m ∥(Q)/m c(Q) = 0.966(7) in Fig. 2g. This incommensurate magnetic cycloid has face-centered orthorhombic symmetry (magnetic space group Fdd2. When approximated as a commensurate structure with q = 2/3, the magnetic space group is Fd0d0 2.).

Multi-Q spin textures of GdAlSi induced by a magnetic field
We apply a magnetic field to the cycloidal helimagnetic ground state of GdAlSi (phase I), B ∥ [110], and observe two field-induced transitions in Fig. 3a, b. Strictly speaking, the critical field for the transition into the field-aligned ferromagnetic state, B c ≈ 75 T, is beyond our experimental range. We estimate B c by linear extrapolation of M(B) to the saturation magnetization of Gd3+, 7 μ B. Using REXS with polarization analysis in an applied field of 7 T along the [110] direction, we determine the magnetic structure of phase II (Supplementary Fig. S1). As summarized by the illustration in Fig. 3c, the phase II is a superposition of two magnetic waves, i.e., an incommensurate magnetic cycloid along Q 1 = (q, −q, 0) with q ≈ 0.673(2), whose spin components are in the [1 1 10]-c plane, perpendicular to B. This is paired with a subdominant up-up-down pattern along the commensurate Q 2 = (2/3, 2/3, 0) for the spin component parallel to B, m ∥(Q). The combination of incommensurate and commensurate modulations can be depicted as a staggered cone structure, as in Fig. 3c.

Spin model calculations and field-induced phase transition
GdAlSi’s simple helimagnetic cycloidal order in zero magnetic field can be explained from a simple model of frustrated isotropic exchange interactions, as in Fig. 1c, d, when adding Dzyaloshinskii-Moriyainteractions (DMI) that are allowed by the symmetry of the polar crystal structure31. While this real-space spin Hamiltonian provides intuitive understanding, it suffers from an inherent frustration of model parameters. Therefore, to reproduce the helimagnetic multi-Q textures under a magnetic field, we fix the direction and amplitude of Q and introduce a Fourier-space model that includes both DMI and anisotropic exchange interactions. This model is discussed in the following. Figure 3d illustrates the relative alignment of the ordering vector Q, the polarization P imposed by the I4md crystal structure, the Dzyaloshinskii-Moriya (DM) vector D, and the anisotropic exchange J anis. Consider the example of Q ∥ [110]: Here, the relevant D couples the magnetization components m [110] (Q 1) and m c(Q 1), whereas J anis couples m [110] (Q 1) and m [1 1 10] (Q 1).

We use an effective Fourier-space Hamiltonian on a square lattice, which includes both DMI and anisotropic exchange interactions in the matrix Γαβ32, viz.

H = − 1/2 ∑ ν,α,β Γα Q β ν m α (Q ν)m β (Q ν) − ∑ i B · m(r i) (3)

where m α(Q ν) with α = x, y, z and ν = a-h are Fourier components of the localized magnetic moment m(r i); ν = a, b are incommensurate with the lattice, ν = c, d are commensurate, and ν = e-h represent higher harmonics formed by superposition of ν = a-d (Methods). The second term represents the Zeeman energy with B = (B, B, 0) = √ 2.

In Fig. 3e, f, we show the calculated magnetic scattering intensities, i.e., the square of the magnetic structure factor, for phases I and II. The model reproduces the cycloidal helimagnetic ground state, with two incommensurate Fourier components ±Q 1. Moreover, the model also features the field-induced phase II, with two strong and incommensurate reflections at ±Q 1 of cycloidal spin habit, combined with two weaker commensurate reflections at ±Q 2 with collinear spin habit (spins parallel to B). In other words, the calculation is well consistent with the experimental magnetic structure in zero field (Fig. 1b) and in finite magnetic field (Supplementary Fig. S1). This good alignment between theoretical and experimental results, as a function of magnetic field, support the accuracy of our magnetic structure analysis in Figs. 2, 3. It also indicates that the Hamiltonian of Eq. (3) is suitable to describe magnetic order in GdAlSi. In Supplementary Note S5, we further truncated the Fourier-space model of Eq. (3) after a few near-neighbors and recover the intuitive real-space Hamiltonian in Fig. 1c. The present model describes effective magnetic interactions between local magnetic moments, which are implicitly mediated by conduction electrons. However, models of itinerant magnetism may provide a complementary perspective on the spin cycloid and multi-Q order in GdAlSi33–35.

Methods
Crystal growth and characterization
Single crystals of GdAlSi were synthesized by the Al flux method. High purity chunks of Gd, Al, and Si were loaded into alumina crucibles with molar ratio Gd:Al:Si = 1:20:2, sealed in an evacuated quartz tube, heated to 1175 °C, held at this temperature for 12 h, cooled to 700 °C at 0.7 °C/h, kept at 700 °C for 12 h, and centrifuged to remove residual Al flux. The samples were characterized by powder X-ray diffraction (XRD) of crushed single crystals and by Laue X-ray diffraction of single crystals.

Magnetization and electrical transport measurements
Magnetization was measured with a Magnetic Property Measurement System (MPMS, Quantum Design) and with the vibrating sample magnetometer (VSM) option of a Physical Property Measurement System (PPMS, Quantum Design). The M-H curve up to 60 T in Fig. 3a was measured in a pulsed magnetic field using an induction method with coaxial pickup coils. Electric transport measurements with rectangular shaped samples were performed using the PPMS. A standard four-probe method was applied to enable precise transport measurements irrespective of contact resistances.

Resonant elastic X-ray scattering (REXS)
REXS measurements were performed at beamline BL19LXU of SPring-8 and BL-3A of Photon Factory, KEK. The photon energy of the incident X-rays was tuned to the Gd L absorption edge (~7.936 keV), corresponding to the 2p to 5d transition (Supplementary Figs. S13 and S14). For the zero field measurements at SPring-8, a single crystal of GdAlSi with a polished (100) surface was set in a cryostat so that the scattering plane was (H, K, 0). For the finite field measurements at KEK, we set the sample in a vertical-field superconducting magnet with a scattering plane close to (H, H, L), so that the magnetic field was applied almost along the [110] direction.

At both SPring-8 and KEK, the incident X-rays had linearly polarized electric field in the scattering plane (π-polarization). The (006) reflection of a pyrolytic graphite (PG) plate with 2θ = 88.6° was used to detect the polarization component of the scattered X-rays in the scattering plane (π0) and out of the scattering plane (σ0), by rotating the PG plate around the X-ray beam. Leakage of unwanted polarization components is only 0.6%. Inserting the analyzer plate changes the absolute scattering intensity by almost two orders of magnitude. However, in the present study we quantitatively analyze the relative strength of the π-π0 and π-σ0 channels. Therefore, neither the difference in absolute intensity nor the momentum-transfer dependence significantly affect our conclusions. The PG plate had a size of ~3 cm × 3 cm, sufficiently larger than the incident beam size (~1 mm × 1 mm), ensuring that the entire beam was covered and that geometrical effects are negligible. During the resonant process, the X-ray photons couple to the 5d conduction electrons, which in turn are coupled to the localized 4f magnetic moments. The resonant scattering amplitude f res can be expressed as

f res = C 0 ε* · ε + iC 1 (ε* × ε) · m(Q) + C 2 ε y O ε (4)

where C i, ε i, ε f, and m(Q) are constants, the polarization vectors of the incident and scattered X-rays, and the Fourier component of the magnetic moment, respectively. Among these terms, only the second term depends explicitly on m(Q), and thus encodes information about the magnetic structure. Consequently, the resonantly enhanced scattering intensity I = f res · f res* is given as follows37;

I ∝ |(ε i × ε f) · m(Q)| 2 (5)

In our experiments, the intensity I π→π0 for the π→π0 channel is proportional to |m out(Q)| 2 and the intensity I π→σ0 for the π→σ0 channel is proportional to (m in(Q) · k) 2, where m in(Q) and m out(Q) are magnetic moments in the scattering plane and perpendicular to it. In our experimental geometry, m out(Q) corresponds to m c(Q), and m in(Q) is separated into m ∥(Q) and m ⊥(Q) as described in the Main Text. The harmonic cycloidal structure at zero field obtained in Fig. 2g is supported by the absence of higher-harmonic reflections (Supplementary Note S7 and Supplementary Figs. S5, S15) and by neutron diffraction measurements (Supplementary Note S2 and Supplementary Fig. S2).

Fourier-space model calculations
The data in Fig. 3e, f and Supplementary Fig. S16 are obtained by simulated annealing for an effective spin Hamiltonian on a two-dimensional square lattice,

H = − 1/2 ∑ ν,α,β Γα Q β ν m α (Q ν)m β (Q ν) − ∑ i B · m(r i) (6)

where m α(Q ν) with α = x, y, z, ν = a-h are Fourier components of the localized magnetic moment m(r i), and Γα Q β ν is a matrix containing the DMI and anisotropic exchange interaction. The second term represents the Zeeman energy with B = (B, B, 0) = √ 2. N = 60 2 is the total number of spins and the amplitude of each spin is fixed to unity. To obtain the low-energy spin configuration, the temperature is gradually reduced from T = 1 to T = 0.001 with a ratio T n+1 = αT n, where T n is the nth-step temperature and α is set between 0.999990 and 0.999999. At each temperature, a local spin update is performed in real space based on the standard Metropolis algorithm. At the final temperature, 10 5–10 6 Monte Carlo sweeps are performed for the measurement.

The explicit propagation vectors in Eq. (6) are: Q a = (q, q, 0), Q b = (−q, q, 0), Q c = (q, −q, 0), and Q d = (−q, −q, 0) with q = 0.35 and q0 = 1/3. Here, Q a and Q b are the incommensurate wave vectors, Q c and Q d are the commensurate wave vectors, and Q e-h represent higher harmonics formed by superposition of the commensurate and incommensurate wave vectors, and Q −Q are the field-induced higher-order terms for the field-induced distorted helimagnetic state consisting of several spin density waves.

To map the three-dimensional spin structures onto the two-dimensional square lattice, q = 0.35 and q0 = 1/3 are incorporated, instead of the experimentally observed q = 2/3 + δ (δ = 0 and 0.006). The model parameters of the exchange matrix in Eq. (3) are set to satisfy the effective 4-fold rotational symmetry of the crystal structure of GdAlSi as follows38: The isotropic exchange interaction is given by Γxx a = Γyy a = Γzz a = Γxx b = Γyy b = Γzz b = J, Γxx c = Γyy c = Γzz c = Γxx d = Γyy d = Γzz d = α 1 J, the DMI is given by Γzx a = −Γxz a = Γzy a = −Γyz a = −Γzx b = Γxz b = Γzy b = −Γyz b = D and Γzx c = −Γxz c = Γzy c = −Γyz c = Γzx d = −Γxz d = Γzy d = −Γyz d = α 1 D, and the symmetric anisotropic exchange interaction is given by Γxy a = Γyx a = −Γxy b = −Γyx b = E and Γxy c = Γyx c = −Γxy d = −Γyx d = α 1 E0 with J = 1, α 1 = 0.98, α 2 = 0.95, D = 0.01, E = 0.045, and E0 = 0.05. With these parameters, the harmonic spin cycloid at zero field and multi-Q order under a high magnetic field are reproduced, see also Supplementary Notes S3, S8 and Supplementary Figs. S16, S17, and S18.

Electronic structure calculations
First-principles calculations of the electronic structure were carried out using the all-electron, full-potential Korringa-Kohn-Rostoker Green’s function method39. Relativistic spin-orbit coupling was included self-consistently, and the Gd atoms were treated with the generalized gradient approximation (GGA) exchange-correlation functional with a Hubbard U (GGA+U)40. U was chosen to be 6 eV. The angular momentum cutoff for the Green’s function orbital expansion was set to l max = 3, and the complex energy contour was defined using 51 integration points. To reduce computational cost, the calculations are carried out based on the primitive crystallographic unit cell, not the conventional unit cell (Supplementary Fig. S9). Self-consistent calculations for the primitive cell were performed with a k-mesh of 30 × 30 × 30. For Fig. 1e and Supplementary Fig. S8a, b, Fermi surface cuts were computed from the quasiparticle density of states (QDOS) in the fully polarized ferromagnetic state41. For the primitive helimagnetic supercell, we considered a k-mesh of 4 × 4 × 4 and the Gd magnetic moments were constrained to the helimagnetic configuration with propagation vector Q = (2/3, 2/3, 0) (Supplementary Fig. S8). Effective magnetic interactions between the Gd 4f magnetic atoms are computed using the infinitesimal rotation method (Supplementary Fig. S6)31,42–44. The Berry curvature integrated over all the occupied states in the ferromagnetic state is shown in Supplementary Fig. S7.

Second harmonic generation (SHG)
SHG measurements were performed to show inversion-symmetry breaking of bulk GdAlSi in the paramagnetic phase at room temperature. A pulsed laser with a pulse duration of 100 fs, 80 MHz, and center wavelength of 800 nm. The laser pulse is focused onto the c-plane surface of GdAlSi in 45-degree oblique incidence and the reflected SHG is detected by a spectrometer equipped with a liquid-nitrogen-cooled charge coupled device. Supplementary Fig. S4a, b respectively show the incident light-polarization dependence of horizontally polarized (P out) and vertically polarized (S out) SHG intensity at room temperature; the output polariser is fixed with polarization parallel to the [101]-axis (P out) and [010] axis (S out). We observed a clear polarization anisotropy of SHG in both cases, in accord with the inversion breaking crystal symmetry. In the paramagnetic phase with point group 4mm, the allowed terms for the horizontally polarized (P out) and vertically polarized (S out) SHG intensities, I P(ϕ) and I S(ϕ), are,

I P(ϕ) ∝ |(aχ xxz + bχ zxx + cχ zzz) cos 2ϕ + 2dχ zyy sin 2ϕ| 2 (7)

I S(ϕ) ∝ |eχ yyz cos ϕ sin ϕ| 2 (8)

Here, χ ijk(2ω) and ϕ represent a tensor element of the second-order nonlinear optical susceptibility and the polarization angle for the fundamental light, respectively; a, b, c, d, and e denote prefactors including the Fresnel coefficient, beam conditions of the incident light, and so on. As shown in Supplementary Fig. S4b, c, the model well reproduces the observed anisotropy of SHG polarization, which highlights inversion breaking of the crystal, in particular the absence of a horizontal mirror plane.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This manuscript presents a comprehensive experimental and theoretical study of GdAlSi, a non-centrosymmetric rare-earth-based Weyl semimetal, combining polarized resonant elastic X-ray scattering (REXS), magnetometry, transport measurements, SHG, and Monte Carlo spin simulations. The authors report the discovery of a harmonic incommensurate cycloidal magnetic ground state and a magnetic field-induced multi-Q spin texture. These magnetic structures are analyzed through symmetry considerations and modeled via both reciprocal and real-space Hamiltonians.\n\nWhile the experiments are technically impressive, a central conceptual motivation—the interplay between topological electronic structure and magnetic order—remains insufficiently addressed. I recommend a major revision before this manuscript can be considered for publication.\n\nKey Comments and Questions\n1. The authors note that the magnetic propagation vector Q happens to connect Weyl nodes but suggest this is accidental. Can the authors support this claim with more detailed electronic structure analysis? Could band folding, Berry curvature, or RKKY interactions favor this specific Q vector?\n2. Are there experimental signatures (e.g., anomalous Hall effect, magnetoresistance, or ARPES under magnetic field) that support a change in the electronic topology upon magnetic ordering or field-induced phase transitions? This clarification would help justify the relevance of GdAlSi as a material platform for exploring topological-magnetic coupling.\n3. Is the multi-Q state truly a coherent superposition of Q1 and Q2 components within a single domain, or might it reflect phase-separated magnetic domains?\n4. How sensitive is the cycloid-to-up-up-down (UUD) transition to the ratio Jc/J1? Is this transition first-order or continuous? Providing a phase diagram or energy comparison as a function of Jc/J1 or magnetic field would greatly enhance the reader’s understanding of the robustness and tunability of this magnetic phase competition.\n5. The screw-type DMI is claimed to cancel out due to sign oscillations. How robust is this cancellation against structural distortions or stacking faults that might locally break symmetry?\n6. If anisotropic exchange favors screw-type or distorted cycloids, why is the observed cycloid strictly harmonic? Is this an indication of fine-tuning or weak anisotropy?\n7. Can the authors provide more direct justification or references confirming the Weyl nature of GdAlSi? Are there supporting ARPES, quantum oscillation, or transport measurements (e.g., Fermi arcs, negative MR) that reinforce this classification?\n8. The manuscript presents detailed polarization-resolved REXS data to characterize the magnetic structure. However, REXS is not sensitive to the absolute magnitude of the magnetic moment without additional modeling. Can the authors clarify whether they attempted to extract moment magnitudes quantitatively, or whether the measurements are intended to provide only relative component analysis?\n9. Can the authors comment on the energy and momentum resolution in the REXS measurements, and how these influence their ability to distinguish between harmonic and potentially weak anharmonic or elliptical components of the cycloid? Are higher harmonics within the detection limits of the current setup?", "reviewer_2": "The authors have used several experimental techniques including the resonant-elastic x-ray scattering (REXS) together with model calculations to study the nature of magnetic order in the Weyl semimetal GdAlSi. Previous band-structure calculations predicted a collinear magnetic order arising due to momentum-dependent spin splitting associated with the so-called altermagnet instead of a conventional collinear magnetic order. The current work represents a significant progress in this direction as it establishes the nature of magnetic order in the weyl semimetallic GdAlSi, where it finds perfectly harmonic spin cycloid in the absence and multi-Q textures in the presence of magnetic field. The manuscript deserves publication provided that the authors can address the following issues:\n(i) The Q vector, the author refers to, does not only appear as modulation vector for helimagnetic order but also seems to connect the Weyl points. The authors mention that this is accidental. This point needs further clarification because it may be possible that correlation between the different Weyl fermions is responsible for the magnetic order. In that case, an important relevant question will be whether it would be possible for such correlation to introduce an ordering for magnetic moments without destroying or affecting the topologically protected Weyl points.\n(ii) The authors perform a model calculation to describe the origin of the harmonic spin cycloid where they do not consider isotropic exchange interaction. On the other hand, to describe multi-Q phase in the presence of magnetic field, they introduce anisotropic exchange interaction. I believe authors should justify the consideration of different models to describe different states in the same material system.\n(iii) The authors used linearly polarized x-rays in REXS to detect the orientation of magnetic moments in a particular ordering if I correctly understand. I think they should have briefly mentioned how the linearly polarized x-rays can couple to the magnetic moments in order to detect the underlying magnetic structure.\n(iv) Localized spin model with Dzyaloshinskii-Moriya interaction is used to describe the different magnetic ordering. However, has a microscopic model such as the Rasbha-Hubbard model been used earlier to study topological semimetallic state with such a complex magnetic ordering? (See e.g. in simpler cases of magnetic order or without magnetic order, Phys. Rev. B 107, 045123 (2023), Phys. Rev. B 110, 075134 (2024), Phys. Rev. B 111, 075166 (2025))."}, "author_response": "We would like to first thank two Referees for devoting their time to carefully reviewing our manuscript. Referee 1 judges that “the experiments are technically impressive”, but raises technical questions about our resonant X-ray scattering measurements. We address this point by introducing supporting neutron scattering data, using it for a refinement of the magnetic structure of GdAlSi—rare for Gd-based compounds with Gd in natural abundance, which strongly absorbs neutrons. The neutron data independently confirm the results of our magnetic structure study. Both Referees 1 and 2 request a clearer conceptual motivation regarding the interplay between topological electronic structure and magnetic order, and we respond to this in detail below. Nevertheless, Referee 2 points out that “the current work represents a significant progress” and believes that “the manuscript deserves publication provided that the authors can address [several issues]” in the Main Text and Figures.\nAs discussed below, we have added substantial experimental data to the Supplementary Information to respond to these and all other questions from the two Referees. The text provided by the Referees is reproduced in its entirety in black, and our response is printed in blue. Changes to the Main Text or to the Supplementary Information are emphasised in red.\nReferee Comment1.1 — This manuscript presents a comprehensive experimental and theoretical study of GdAlSi, a non-centrosymmetric rare-earth-based Weyl semimetal, combining polarized resonant elastic X-ray scattering (REXS), magnetometry, transport measurements, SHG, and Monte Carlo spin simulations. The authors report the discovery of a harmonic incommensurate cycloidal magnetic ground state and a magnetic field-induced multi-Q spin texture. These magnetic structures are analyzed through symmetry considerations and modeled via both reciprocal and real-space Hamiltonians.\nReply: We thank Referee 1 for studying our work in detail, including the Supplementary Information, and for detailed comments that have improved the manuscript.\nReferee Comment1.2 — While the experiments are technically impressive, a central conceptual motivation—the interplay between topological electronic structure and magnetic order—remains insufficiently addressed. I recommend a major revision before this manuscript can be considered for publication.\nReply: We thank Referee 1 for this important and constructive comment. Regarding the conceptual motivation of our work, the term “interplay” applies in two directions: on the one hand, the topological electronic structure influences magnetism; on the other hand, magnetic order reshapes the topological bands. In the first part of our response, we clarify that Weyl electrons do not play a primary role in determining the modulation vector Q, by performing extensive additional electronic band structure calculations. However, there remains a possibility that Weyls impact Dzyaloshinskii-Moriya interactions and anisotropic exchange. In the second part, our additional calculations suggest that several Weyl points are likely gapped by the helimagnetic order, while others remain intact.\nTopological electronic structure → Magnetism\nAn important issue was raised by both Referees: Do the Weyl points close to the Fermi energy have an impact on magnetic interactions in RAl(Si/Ge)? Providing more detailed calculations to respond to the question, we argue that the period of the magnetic order (i.e., the magnitude and direction of Q) is dominated by states far from E . In this sense – although Q seems to connect between regions with Weyl points in the Brillouin zone [Nat. Mater. 20, 1650 (2021)] – we call this ’accidental’. On the other hand, our calculation and analysis cannot rule out that anisotropic exchange and Dzyaloshinskii-Moriya interactions, which are clearly important for our spin models, have a contribution from Weyl points. The relative contribution of Weyls to DMI or anisotropic exchange remains a question for future research and can be addressed, e.g., by detailed E tuning using a series of doped single crystals.\nWe demonstrate these points in four stages, adding extensive additional band structure calculations.\na. Short-range exchange interactions in GdAlSi.\nWe calculated the isotropic exchange interactions of GdAlSi and found them to be mostly short-ranged and antiferromagnetic regardless of bond distance, cf. Fig. R1a. This suggests that superexchange due to states far from E dominates over ’true’ Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions, which should be long-ranged and oscillatory in form [Phys. Rev. B 109, L201108 (2024)]. The short-range interactions in Fig. R1 are also well consistent with our real-space spin Hamiltonian calculations (Supplementary Note S8), which show that magnetic order with propagation vector Q = (2/3,2/3,0) can be approximated by only two short-range, antiferromagnetic, and isotropic interactions, J and J (cf.\n1 c\nFig. 1d of the Main Text).\nb. Electronic structure calculation reproduces observed propagation vector.\nImportantly, the magnetic interactions from the electronic structure calculation successfully reproduce the experimentally observed propagation vector Q : As shown in Fig. R1b, exp the incommensurate state is energetically favoured over the ferromagnetic state at Q′ = (0, 0, 0), with the maximum occurring at Q′ = (q′, q′, 0) with q′ = 0.29 ≈ 1/3. This is in excellent agreement with Q = (q, q, 0) where q = 0.67 ≈ 2/3.\nexp\nThe difference of 1/3 vs. 2/3 is merely a matter of the convention used for ab-initio calculations [cf. Phys. Rev. B 109, L201108 (2024) and its SI]: To account for two magnetic Gd sites in the (primitive paramagnetic) unit cell of GdAlSi, the Fourier-transform of the magnetic interactions are defined in matrix notation and the maximum eigenvalue J(Q′) is extracted from this matrix. Whether this Q′ corresponds to an ordering vector Q in the first Brillouin zone, e.g., (1/3,1/3,0), or in a higher zone, e.g., (2/3,2/3,0), exp can be examined by considering the real-space magnetic structures in Fig. 3 of Phys. Rev. B 109, L201108 (2024). In summary, the theoretical study clearly demonstrates that the maximum of J(Q′) at Q′ ≈ (1/3,1/3,0) corresponds to ordering with antiferromagnetic coupling between the two Gd sites in the primitive cell, consistent with experiment.\nPhys. Rev. B 109, L201108 (2024) also discusses how the short-range interactions in RAlSi (R =rare earth) can be understood in terms of superexchange via the p orbitals of nonmagnetic Si and Al atoms; this is fundamentally different from RKKY-type interactions involving only Weyl electrons. We note that recent theoretical studies on SmAlSi have also questioned the nesting scenario based on a Lindhard susceptibility analysis [Commun. Phys. 6, 134 (2023); arXiv:2508.11244 (2025)].\nc. Improved band structure calculations and Weyl points in GdAlSi\nIn the process of addressing this question, and aiming to obtain a more accurate description of electronic states near the Fermi level where Weyl nodes emerge, we introduced a Hubbard U into the electronic structure calculations. We then searched for Weyl points in the Brillouin zone (BZ) by constructing a wannierised Hamiltonian. Fig. R2b shows the calculated electronic structure with U = 6eV and 24 Weyl points close to the k = 0 plane. The c Fermi surface of GdAlSi is consistent with a semimetallic electronic character. Note that a RKKY scenario generally requires a high density of states near the Fermi level, or at least, if Weyl points are to dominate, an absence of trivial electronic states at E . Nevertheless, F and consistent with many previous studies on this material class, our calculations clearly show that Q connects between Weyl points in the electronic structure.\nc\nd. Dzyaloshinskii-Moriya (DM) interactions and anisotropic exchange.\nThe isotropic interactions in Fig. R1 fix the period and direction of the magnetic ’wave’, but they do not fix the spin direction. This means that there remains a degree of freedom between cycloid or proper-screw (or a mixed character, as in certain multiferroics). The spin model calculations in the Main Text and SI show that DM interactions and anisotropic exchange play an important role in stabilizing the observed cycloid and multi-Q states. Therefore, the careful study of a possible impact of Weyl nodes on these anisotropic interactions remains an important subject for future investigation. We assert that our experimental evidence for the spin cycloid and multi-Q state in GdAlSi, combined with numerical evidence for Weyl fermions that is consistent with the recent, peer-reviewed experimental literature [Phys. Rev. B 109, 035120 (2024)], warrants publication of our work even without a detailed calculation of (weak) DMI and anisotropic exchange density functional theory, and their explicit attribution to Weyl fermions.\nWe added this discussion in Supplementary Note S2 of the revised manuscript. We also updated Fig. 1e to show the band structure calculations with Hubbard U and Weyl points.\n“helimagnetism is not predominantly caused by the Weyl electrons and the matching of Q to the separation of Weyl points in momentum space is accidental , see also Supplementary Note S2.”\nNext, we revised Methods as follows:\n“the Gd atoms were treated with the generalised gradient approximation exchange-correlation functional with a Hubbard U (GGA+U). U was chosen to be 6eV.”\n“Effective magnetic interactions between the 4f magnetic atoms are computed using the infinitesimal rotation method.\nMagnetism → Topological electronic structure\nOn the other hand, the formation of magnetic order affects the topological electronic states. In short, our additional electronic structure calculations suggest that some Weyl points are likely to remain intact in the helimagnetic state, while others are gapped. This is consistent with a recent ARPES study, which provides some evidence for Fermi arcs in GdAlSi at 24K < T [Phys. Rev. B 109, 035120 (2024)].\nN\nAs shown in Fig. R3a, some Weyl nodes in the ferromagnetic state of GdAlSi are located near the orange helimagnetic Brillouin zone (BZ) boundary. Figure R3b,c shows the ferromagnetic and cycloidal electronic structures, respectively, calculated in the folded helimagnetic BZ. When the periodic magnetic order expands the unit cell and thus folds the BZ, partial charge gaps open at the new BZ boundary. Significant modifications in the band structure are thus observed in the helimagnetic state. In GdAlSi, the change of the electronic structure due to the magnetic order is rather strong, and it is hard to track the fate of individual Weyl points. Moreover, we note that the search for Weyl nodes in the\nhelimagnetic state remains numerically challenging. However, Weyl points close to the BZ boundary in Fig. R3a are very likely to be gapped, while others remain intact.\nModification of Weyl nodes by magnetism has previously been studied in the RAlSi/Ge (R=rareearth)compoundswithanetmagnetization, boththeoreticallyandexperimentally [Phys. Rev. B 97, 041104(R) (2018); Nat. Commun. 14, 7185 (2023)]. Such a net magnetization may also shift Weyls or generate additional Weyl nodes. Our study reveals the harmonic cycloidal magnetic order of GdAlSi, which will serve as a material platform for studying the modification of Weyl nodes by Q in presence of zero net magnetization. In the Main Text, we have already noted that:\n“Such states are realised by band-folding of Weyl fermions into a small Brillouin zone upon formation of periodically modulated order, and conditions for the hybridization of Weyl points have been derived and phrased in the language of special relativity.”\n8\n“Indeed, controlling these Q-domains in magnetic Weyl semimetals, by magnetic field or current, provides a valley-selective way to open a (partial) charge gap around the Weyl cones, if they have an energy-like causal connection in the folded Brillouin zone.”\nIn response to this comment, we added Supplementary Fig. S7 and modified the Discussion & Conclusion to clearly describe our central motivation, as follows:\n“GdAlSi will serve as a prototypical material for studying the modification of Weyl nodes byaperfecthelimagneticcylcoid, incontrasttothepreviouslyreportedmaterialsthatexhibit a net magnetization (NdAlSi/Ge) ”\nReferee Comment1.3 — 1. The authors note that the magnetic propagation vector Q happens to connect Weyl nodes but suggest this is accidental. Can the authors supportthisclaimwithmoredetailedelectronicstructureanalysis? Couldbandfolding, Berry curvature, or RKKY interactions favour this specific Q vector?\nReply: As noted in the reply to Comment1.2, our additional electronic structure calculations demonstrate that RKKY interactions and the resulting band folding do not play a dominant role in stabilizing Q, although they may have an impact on the spin rotation plane by affecting the magnitude of the Dzyaloshinskii-Moriya interaction and of anisotropic exchange.\nMotivated by Referee 1’s comment, we further calculate the Berry curvature of this material. Figure R4 shows the Berry curvature integrated over all the occupied states, assuming a field-aligned ferromagnetic state for GdAlSi. Indeed, the c-component of the Berry curvature vector, Ω , has contributions that are large in vicinity to the Weyl points c around the k = 0 plane. While this result is related to the anomalous Hall effect previously c observed in GdAlSi [Phys. Rev. B 109, 035120 (2024)], we cannot see a justification to regard the Berry curvature as the origin of the propagation vector Q, but we remain open to comments from Referee 1.\nIn response to this comment, we have added Fig. R4 as Supplementary Fig. S6.\nReferee Comment1.4 — 2. Are there experimental signatures (e.g., anomalous Hall effect, magnetoresistance, or ARPES under magnetic field) that support a change in the electronic topology upon magnetic ordering or field-induced phase transitions? This clarification would help justify the relevance of GdAlSi as a material platform for exploring topological-magnetic coupling.\nReply: We thank Referee 1 for an in-depth comment, but would like to stress that our work is a magnetic structure paper, whereas the electronic transport properties of GdAlSi have been reported recently on single crystals of comparable quality to ours [Phys. Rev. B 109, 035120 (2024)]. In that work, high transport mobility and an anomalous Hall effect, possibly arising from Weyl fermions, were discussed. We feel that further detailed transport measurements will distract from the core focus of our paper, which is the magnetic structure – including zero-field perfect cylcoid and field-induced multi-Q states – of GdAlSi.\nIn response to the comment, we added the following sentence in the introduction.\n10\n“Here, we report the undistorted, cycloidal helimagnetic ground state of GdAlSi with perfectly isotropic Gd3+ ions, a material in which Fermi arcs from Weyl nodes have been identified, and a possibly related anomalous Hall effect has been observed under a magnetic field...”\nReferee Comment1.5 — 3. Is the multi-Q state truly a coherent superposition of Q1 and Q2 components within a single domain, or might it reflect phase-separated magnetic domains?\nReply: We thank Referee 1 for this important question. Based on our experimental results and model calculations, we conclude that phase II corresponds to a coherent multi-Q state within a single magnetic domain, rather than a phase-separated state with domains of different Q-vectors. Our reasoning is as follows:\nFigureR5: Field induced multi-Q textures reproduced from spin model calculations. a, Real-space multi-Q spin texture reproduced from spin model calculations in the Main Text. This is a coherent superposition of a harmonic cycloid along Q and 1 an up-up-down texture along Q . b, Calculated magnetic scattering intensity I(Q) in 2 reciprocal-space, which corresponds to this state.\na. Multi-Q texture in spin model calculations.\nOur spin model calculations clearly show that phase II corresponds to a coherent superposition of Q and Q within a single domain, when using model parameters consistent with 1 2 our physical understanding of magnetic interactions in this material. To illustrate this, we have added Supplementary Fig. S8 in the revised manuscript, which shows the calculated multi-Q spin textures in real-space (see Fig. R5).\nb. Negligible chemical inhomogeneity.\nThere is no experimental indication of chemical inhomogeneity that could promote phase separation. The elemental composition of our crystal was confirmed using scanning electron microscopy with energy-dispersive X-ray spectroscopy (SEM-EDX; JEOL and Bruker). As shown in Fig. R6, the homogeneous distribution of Gd, Al, and Si highlights the chemical uniformity of our sample.\nFigure R6: Chemical homogeneity of GdAlSi. a, Scanning electron microscopy (SEM) image of a GdAlSi single crystal. b-d, Corresponding elemental maps of Gd, Al, and Si obtained by energy-dispersive X-ray spectroscopy (EDX). All elements are uniformly distributed throughout the crystal.\ndom\nc. Domain evolution consistent with multi-Q state.\nReferee 1 proposes a scenario where the zero-field ground state is not phase separated, but the field-induced regime (phase II) suffers from phase separation. However, the domain\n13\nevolution associated with phase II is consistent with the behaviour expected for a single-domain, multi-Q state. If phase separation were essential to the problem, the magnetization process is expected to be more complex with at least three field-induced transitions, as illustrated in Fig. R7d-g. In contrast, the experimentally observed magnetization curve M(H) can be explained rather simply by the transition from a cycloidal helimagnetic order in B = 0 to a multi-Q state in a magnetic field (Fig. R7h–j).\nIn the tetragonal structure, the zero-field cycloidal ground state forms Q-direction domainsalongequivalent⟨110⟩directions(Fig.R7a). Whenasufficientlystrongmagneticfield is applied, the system first undergoes a transition to phase II. Only after this transition, the Q-direction domains align in our sample, as illustrated in Fig. R7b,c. This differs from the phase separation scenario, which would lead to a more complicated situation where various multi-domain states switch in response to the applied magnetic field (Fig. R7d-g). Fig- ure R7i,j compares precise magnetization measurements immediately after zero-field cooling (ZFC) from above T and after applying a high magnetic field. The resulting two transi- N tions, shown as open and filled triangles in Fig. R7i,j, are consistent with the – relatively speaking – simpler multi-Q scenario.\nWe argue that our observations of phase transitions in the bulk magnetization strongly support that phase II is a coherent single-domain multi-Q state, rather than arising from phase separation.\nWe added Supplementary Note S3 with the discussion above.\nReferee Comment1.6 — 4. How sensitive is the cycloid-to-up-up-down (UUD) transition to the ratio Jc/J1? Is this transition first-order or continuous? Providing a phase diagram or energy comparison as a function of Jc/J1 or magnetic field would greatly enhance the reader’s understanding of the robustness and tunability of this magnetic phase competition.\nHowever, we would like to stress that the real-space Hamiltonian is introduced in our work merely for the purpose of physical intuition, whereas the Fourier-space Hamiltonian is emphasized in the Main Text and used to describe the sequence of magnetic transitions in the real material GdAlSi. Our reply to Comment1.5 shows that the pure UUD state is not realised in this material.\na. Robustness of the UUD plateau.\nWe address the question of the robustness of the UUD state in a magnetic field by performing calculations with the real-space spin Hamiltonian on a finer parameter grid. For the phase diagram in Supplementary Fig. S17d, we narrowed down the target region to observe the transition more precisely. The updated results are shown in Fig. R8, and we have accordingly replaced Fig. 1e and Supplementary Fig. S17c,d. With these denser calculations, we demonstrate that the value q jumps abruptly from 30/47 to 2/3 to 30/43 as a function of J /J in a lattice of size 30×30 unit cells. While this may appear indicative of a finite size c 1 effect arising from commensurability of the UUD phase, our additional simulations with lat- ticesizes28×28and29×29stillexhibittheUUDplateau, albeitslightlynarrower(Fig.R9).\nb. Phase coexistence and first order transition.\nWe address the question of first or second order transition by checking for phase coexistence in MC. For every ratio J /J we have simulated 128 independent samples and evaluated c 1 which phase, Q ∥ [110] (cycloidal structure) or Q ∥ [110] (UUD structure), yields the 1 2 largest peak in the spin structure factor (SSF). The resulting average over all 128 samples is shown in Fig. R9a. While the result is unique in the middle of the UUD plateau, J /J = 2, c 1 as well as in the cycloidal phase, it varies smoothly at the phase transitions J /J ≈ 1.7 and c 1 2.4. Weinterpretthisresultasfollows: nearthephasetransitions, somesamplesdevelopthe cycloidal phase while others develop the UUD phase. This interpretation is consistent with images of real-space spin textures that we extracted from the simulations. The formation\n16\nof large domains of either phase, extending over the simulated system, indicates phase coexistence and, thus, a first order phase transition.\nFigure R9: Phase transition as a function of J /J in real-space spin Hamil- c 1 tonian calculations. We plot the dominant state, i.e., the state with maximal spin structure factor (SSF) obtained from real-space model calculations as a function of J /J with a magnetic field B = 15T, averaged over 128 independent simulations. c 1 110 a, Blue and red points represent the fraction of samples which were found to be in the states Q ∥ [110] (coned spin cycloid) and Q ∥ [110] (UUD), respectively, calculated 1 2 on a 30×30 lattice. Black and grey points for Q ∥ [110] are calculated on 28×28 and 2 29×29 lattices, respectively. Independent of the model size, the plateau for the UUD phase appears; this indicates that UUD is a true thermodynamic ground state. The calculations demonstrate phase coexistence and thus a first order transition between UUD and the spin-cycloid: At the boundary between the two states, the curve changes continuously, which means that both states are represented among the 128 independent simulations. b, The plateau for the UUD phase depends on temperature, indicating that thermal fluctuations are vital for its existence.\n17\nc. Analytical ansatz and energy competition.\nWe further note that it is rather straightforward to calculate the energy density of a coned cycloidal state and of the UUD state analytically with a suitable ansatz, if there are only the isotropic interactions J , J :\n1 c\n(cid:2) (cid:3)\nm (r) = cos(θ) cos(r ·Q+ϕ)e +sin(r ·Q+ϕ)e +sin(θ)(e ×e ) (R1)\ncyc z Q Q z\n(cid:20)\n1 4\n(cid:18)√\n4π\n(cid:19)(cid:21)\nm (r) = − cos 2 e ·r (e ×e ) (R2)\nuud Q Q z\n3 3 3\n√\nwhere we assume Q = Qe with e = (1,−1,0)T/ 2. We have computed the energy Q Q\ndensity, optimising the Q-value and cone angle θ of the cycloidal phase for a given external √ magnetic field B along e = (1,1,0)T/ 2 = e ×e . We assumed that the UUD phase B Q z\nhas no parameters to optimise. As a result, the energy densities of the two phases are:\nJ2 J (MB)2\nE = −2J − c − 1 (R3)\ncyc 1 4J (4J +J )2\n1 1 c\nuud 1 c\n3\nsuch that E < E except for J /J = 2 where they briefly touch at MB/J = 6. How- cyc uud c 1 1 ever, apart from this single point in the (J ,B)-space, the cycloidal phase in our calculation c\nalways has lower energy density than the UUD state.\nd. UUD state stabilized by thermal fluctuations in real-space Hamiltonian.\nWe confirmed that thermal fluctuations promote the stability of UUD over the coned cy- cloidal state in an applied magnetic field and create a finite window of J /J where UUD c 1 is the ground state – although this is not the case for the analytical ansatz. This means the free energy of the UUD phase at T = 1K, instead of the energy itself, is relevant.\nTo show this, we have performed additional MC simulations at both higher and lower temperatures, from T = 0.5K to T = 2K. The results are shown in Fig. R9b. Here, the plateau of the UUD phase shrinks at lower temperatures and widens at larger temperatures.\nIn summary, within the real-space Hamiltonian, the magnetic structure in an applied magnetic field depends sensitively on fine parameter tuning, and reproducing the experimentally observed multi-Q state ultimately requires the Fourier-space model presented in the Main Text. Nevertheless, the real-space model provides important intuitive physical insight into the magnetic ordering of GdAlSi.\nWe added the discussion of the phase transition to Supplementary Note S8. We reiterate that the multi-Q state in GdAlSi is suitably described by the Fourier-space Hamiltonian, not by the simplified real-space Hamiltonian that we discussed extensively in the response to this comment.\nReferee Comment1.7 — 5. The screw-type DMI is claimed to cancel out due to sign oscillations. How robust is this cancellation against structural distortions or stacking faults that might locally break symmetry?\nReply: As Referee 1 correctly states, local structural distortions or stacking faults may break all mirrors, and various magnetic patterns (not only helimagnetic) could be stabilised close to such defects. However, to realise a screw-type helimagnetic structure extending throughout the bulk sample, as detected by REXS or neutron experiments, it is necessary to lower the point group symmetry to break all mirrors. In the present case, this means (at least) lowering the space group symmetry from I4 md to the chiral I4 .\n1 1\nWe have carried out additional synchrotron single-crystal X-ray diffraction (XRD) experiments at SPring-8 beamline BL02B1 to check for (glide) mirror breaking. The data in Fig. R10 shows that the extinction rule associated with the d-glide symmetry is conserved to a very high precision: the (1, 1, 4) reflection is absent, even down to an intensity three\norders of magnitude weaker than the observed peaks. Thus, we conclude that there is no evidence of global symmetry lowering to chiral symmetry and insist on the discussion of screw-type DMI in the Main Text: Local distortions can locally induce DMI, but a screw structure that extends through the bulk cannot be stabilised by such distortions, as the (global) point group symmetry would have to be lowered.\nWe added Supplementary Note S4 for the discussion on the screw-type DMI.\nReferee Comment1.8 — 6. If anisotropic exchange favors screw-type or distorted cycloids, why is the observed cycloid strictly harmonic? Is this an indication of fine-tuning or weak anisotropy?\nReply: We appreciate Referee 1’s profound question regarding the energetics of magnetic\n20\nphases in GdAlSi. In brief summary, the very weak magnetic anisotropy of GdAlSi is sufficient to favour multi-Q orders in high field, but it causes only tiny anharmonicities in zero field in our calculations. These are too weak to be detected by REXS or neutron scattering in zero field.\na. Definition of harmonic cycloid.\nWe first revisit our definition of “a harmonic cycloidal structure”, beginning with the general description of Eq. (1) of the Main Text:\nm(r) = m (Q)e cos(Q·r)+m (Q)e sin(Q·r). (R5)\n∥ ∥ c c\nHere, e and e are unit vectors along Q and along the crystallographic c-axis, respectively.\n∥ c\nA harmonic cycloid corresponds to m (Q) = m (Q). When m (Q) ̸= m (Q), we call this\n∥ c ∥ c\nan elliptically distorted cycloid, as shown in Fig. R11. This definition is distinct from the classification between cycloid and proper-screw structures, or between single-Q and multi-Q states. As shown in Fig. 2g of the Main Text, our cycloid is perfectly harmonic with m (Q)/m (Q) = 0.966(7).\n∥ c\nIn the previous version of the manuscript, the term “distorted” had been used in a potentially misleading way to describe the multi-Q state. In response, we revised the relevant expressions to avoid confusion in Supplementary Note S1:\n“the distorted multi-Q helimagnetic cycloidal state”\nWealsoaddedthevaluem (Q)/m (Q) = 0.966(7)totheMainTextandtothecaption\n∥ c\nof Fig. 2.\nb. Consideration of anharmonicity in numerical calculations.\nThe experiment shows a perfectly harmonic cycloid in zero magnetic field, and a field-\n110 c\ninduced multi-Q state. Our model calculations using a Hamiltonian in Fourier space are well consistent with this observation. To address Referee 1’s question, we introduce the following two quantitative indicators, based on the Fourier-transformed magnetic moments:\n1. Harmonicity, defined by m (Q)/m (Q) as described above.\n∥ c\n2. Screw-cycloidratio, definedasr = m (Q)/m (Q), wherem (Q)isthecomponent\nsc ⊥ ∥ ⊥\nperpendicular to both Q and c. A value of zero indicates a purely cycloidal structure, while a finite r indicates a screw-type contribution.\nsc\nIn our Fourier-space model calculations, anisotropic exchange stabilises the multi-Q texture in a magnetic field, while only weakly perturbing the zero-field ground state. This is shown in Fig. 3e,f in the Main Text. In phase I near zero-field, the numerical calculation shows m (Q)/m (Q) = 1.05, indicating a harmonic cycloid. The screw–cycloid ratio r is\n∥ c sc\nzero, confirming a purely cycloidal character.\nUpon applying a magnetic field, the harmonic cycloid transitions into the multi-Q state, consisting of the cycloid and the up-up-down structures. The anharmonic distortion of the\n22\ncycloidal component is moderate, with m (Q)/m (Q) = 1.32; there is still no screw-type\n∥ c\ncontribution (r = 0), in agreement with our experimental findings. These observations\nsc\nare well consistent with a weakly anisotropic behaviour of the Gd 4f moments in GdAlSi.\nWe added Supplementary Note S5 for the discussion of the harmonicity.\nReferee Comment1.9 — 7. Can the authors provide more direct justification or references confirming the Weyl nature of GdAlSi? Are there supporting ARPES, quantum oscillation, or transport measurements (e.g., Fermi arcs, negative MR) that reinforce this classification?\nReply: We appreciate Referee 1’s question regarding the presence of Weyl fermions in GdAlSi. Above, in our response to Comment1.2, we have calculated the positions of Weyl nodes in GdAlSi. We have also calculated the Berry curvature in our response to Comment1.3.\nThe main focus of our study is the identification of the helimagnetic ground state, while the electronic structure of GdAlSi has recently been investigated by two independent studies [Phys. Rev. B 109, 035120 (2024); Phys. Rev. B 110, 224436 (2024)]. These authors report surface Fermi arcs consistent with their first-principles calculations via angle- resolvedphotoemissionspectroscopy(ARPES),indicatingtheemergenceoftopologicalWeyl fermions. In Phys. Rev. B 109, 035120 (2024), electronic transport properties show low carrier density (∼ 1020cm−3) and high transport mobility (∼ 1000cm2/Vs), consistent with a Weyl semimetal state. As Referee 1 knows, similar Weyl fermions and Fermi arcs have been reported in isostructural compounds, including both non-magnetic and magnetic systems: LaAlGe, PrAlGe, CeAlSi, NdAlSi, and SmAlSi [Ref. 20 and 24 of the Main Text; Nat. Commun. 11, 3356 (2020); Nat. Commun. 15, 1467 (2024); Commun. Phys. 6, 5849 (2023)]. Future studies should address how the Fermi-arc structure is altered by the helimangetic order in GdAlSi.\n“Here, we report the undistorted, cycloidal helimagnetic ground state of GdAlSi with perfectly isotropic Gd3+ ions, a material in which Fermi arcs from Weyl nodes have been identified, and a possibly related anomalous Hall effect has been observed under a magnetic field;”\nReferee Comment1.10 — 8. Themanuscriptpresentsdetailedpolarization-resolved REXS data to characterize the magnetic structure. However, REXS is not sensitive to the absolute magnitude of the magnetic moment without additional modeling. Can the authors clarify whether they attempted to extract moment magnitudes quantitatively, or whether the measurements are intended to provide only relative component analysis?\nReply: We thank Referee 1 for raising this insightful point regarding technique. To answer, we have carried out additional neutron scattering experiments on GdAlSi with hot neutrons and determine that the ordered magnetic moment is 6.2µ /Gd. We also use this data to B independently verify our magnetic structure model in zero field.\nIt is reasonable to assume that the localised magnetic moment of the Gd 4f shell is fixed to 7µ /Gd, as the orbital angular momentum of the Gd3+ ion is quenched. However, B as Referee 1 correctly states, the absolute magnitude of the magnetic moment is very hard to quantify by REXS experiments. On the other hand, single crystal neutron diffraction can answer Referee 1’s question, although naturally abundant Gd is a strong neutron absorber and its neutron scattering length shows a very sharp energy dependence – thus making neutron scattering on Gd-based compounds difficult [Physica B 552, 58-61 (2019)].\nWe performed elastic neutron scattering experiments on GdAlSi at the D9 hot neutron diffractometer of ILL, operating in four-circle mode (E = 327meV). In the experiment, a i\nJ 7/2\n25\nsingle crystal of GdAlSi with dimensions of 2.4×2.4×0.3mm3 was fixed to an aluminum pin attached to the goniometer, ensuring that the c axis was oriented vertically. The sample was theninstalledinadisplexdevicecompatiblewiththefour-circleconfigurationofD9,enabling access to a large region of reciprocal space. Reciprocal space analysis was performed using the Int3D software [Crystals 11, 897 (2021)]. A small area detector measuring 6 × 6cm (32×32pixels)facilitatedreciprocalspacesurveying, optimizationofnuclearpeakpositions, andthesearchforincommensuratemagneticreflections. Themeasurementinvolvedω-scans inbisectinggeometry(transmissionmode). Finally,integrationofω-scansandLorentzfactor correction was conducted using the RACER program [Acta Crystallogr. A 30, 580–584 (1974)].\nWe first analyze nuclear reflections with integer (H, K, L) indices. The observed nuclear structure factor |F | is compared to model calculations, |F | in the polar space obs calc group I4 md (Supplementary Fig. R12a). By carefully considering equally distributed polar 1 structural domains in the sample, the crystal structure was refined with Gd = (0, 0, 0), Al = (0, 0, 0.5946), and Si = (0, 0, 0.4217) in lattice coordinates, all occupying the Wyckoff 4a site. These internal coordinates are similar to the sister compound NdAlSi [Nat. Mater. 20, 1650–1656 (2021)]. The reliability factor R, defined as R = Σ|F −F |/Σ|F |, is obs calc obs 0.086, indicating good agreement between model and fit.\nLet us consider the incommensurate magnetic reflections at Q = (±q, ±q, 0) with q = 0.673. As shown in Supplementary Fig. R12b, the magnetic structure factor is well reproduced when carefully considering a six-fold cell expansion and two types of Q-direction domains. The elliptical distortion of the cycloid is small, with m (Q)/m (Q) = 0.95, ∥ c consistent with a perfectly harmonic cycloidal structure. The refinement yields a similarly low R-value as the lattice refinement, further supporting the robustness of the magnetic model. The amplitude of the magnetic moment was extracted by comparing the linear slopes of nuclear and magnetic refinements: 6.2µ /Gd, close to the theoretical value of B 7µ /Gd for the 4f magnetic shell of Gd.\nB\nImportantly, our neutron scattering experiments independently establish the harmonic\n26\ncycloidal ground-state magnetic structure of GdAlSi, in agreement with—but not reliant on—the REXS measurements in the Main Text.\nWe added Supplementary Note S6 for the neutron diffraction data and analysis.\nReferee Comment1.11 — 9. Can the authors comment on the energy and momentum resolution in the REXS measurements, and how these influence their ability to distinguish between harmonic and potentially weak anharmonic or elliptical components of the cycloid?\nReply: We thank Referee 1 for a technical question. The harmonicity of our cycloid is obtained as m (Q)/m (Q) = 0.966±0.007 from the fitting in Fig. 2g of the Main Text ∥ c (please also refer to our reply to Comment1.8). Crucially, the estimation of anharmonicity is based on the two polarization components and their ratio,\nR = I /I . (R6)\nπ-σ′ π-π′\nAs we are taking the ratio, we argue that neither the energy nor the momentum resolution significantly affect our conclusion. Below, we provide additional technical details on the REXS measurements and also discuss the limitations of efficiency for our analyzer plate.\na. Energy dependence.\nIn resonant elastic X-ray scattering (REXS), tuning the incident photon energy away from the resonance maximum can reduce the magnetic scattering intensity and potentially influence the analysis of the data. To address this concern, we emphasise that the Gd L 2 edge is a single resonance peak in GdAlSi; this simple line shape is in contrast to the more complex resonance structure of the Gd M edges. All experiments presented in the Main Text and Supplementary Information were performed at the resonance maximum identified in Fig. R13. This figure shows the incident photon-energy dependence of the magnetic re-\nflection (2−q, q, 0), which exhibits a sharp peak at 7.936keV, corresponding to the Gd L 2 edge. Conducting all measurements at this resonant energy ensures that the magnetic scattering intensity is maximised and that any influence on the analysis from reduced scattering intensity is negligible.\nWe added the energy dependence of the magnetic reflection in Supplementary Fig. S14.\nb. Momentum dependence.\nOur analysis is based on the intensity ratio of the two polarization components, following Eq. (R6). This excludes possible effects due to the projection of the beam onto the sample (due to the active volume of the sample in the scattering process), which may arise when comparing different momentum-transfer points for discussing the anharmonicity.\nFurther, in the Main Text, the polarization analysis was carried out by Gaussian fitting of ω-scan profiles. In principle, magnetic reflections form three-dimensional peaks in reciprocal\nspace, and restricting the analysis to the ω direction could influence the analysis. However, our REXS measurements show that the magnetic reflections are sharp along all reciprocal- space directions (∼ 3×10−3˚A −1 ). Figure R14 presents line scan profiles of the (2−q, q, 0) reflectionforω, χ, and2θ-ω, correspondingtothreeindependentreciprocalspacedirections. The sharpness observed in all scans confirms that the analysis in the Main Text remains robust.\nWe added the line scan profiles to the SI as Supplementary Fig. S15.\nc. Efficiency of analyzer plate\nWe used a pyrolytic graphite (PG) plate to separate the polarization components I π-σ′ and I . In principle, a small leakage of the unwanted polarization component occurs in π-π′ this setup if the scattering angle 2θ from the analyzer plate is not 90◦. However, at PG\n29\nthe experimental energy used here, the scattering angle of the PG (0, 0, 6) reflection is 88.6◦, which results in only about 0.6% admixture of the respectively other polarization component. Moreover, possible effects arising from the finite size of the PG plate can be excluded: the plate used in our experiment was approximately 3cm×3cm, far larger than the incident beam size (∼ 1mm×1mm), ensuring complete beam coverage and negligible geometrical influence.\nWe added a discussion regarding the analyzer plate to Methods.\nReferee Comment1.12 — Are higher harmonics within the detection limits of the current setup?\nReply: We thank Referee 2 for a final technical comment. Higher-harmonic magnetic reflections are commonly regarded as signatures of distortions in the magnetic structure, with the chiral soliton lattice being a representative case [Phys. Rev. Lett. 108, 107202 (2012)]. As shown in the Main Text, GdAlSi has the propagation vector Q = (q, q, 0) with q = 0.673. The line scan in Fig. R15 shows that the higher-harmonic reflection (4, 2, 0)−2Q with Q = (q, q, 0) is absent at zero magnetic field, allowing us to exclude 2 2 the presence of higher harmonics that are up to two orders of magnitude weaker than the first harmonic. This experiment thus confirms that the ground-state cycloidal structure remains undistorted, within the resolution of the experiment.\nWe added Supplementary Note S7 to discuss the absence of higher harmonics.\nResponse to Referee 2\nReferee Comment2.1 — The authors have used several experimental techniques including the resonant-elastic x-ray scattering (REXS) together with model calculations to study the nature of magnetic order in the Weyl semimetal GdAlSi. Previous band-structure calculations predicted a collinear magnetic order arising due to momentum-dependent spin splitting associated with the so-called altermagnet instead of a conventional collinear magnetic order. The current work represents a significant progress in this direction as it establishes the nature of magnetic order in the weyl semimetallic GdAlSi, where it finds perfectly harmonic spin cycloid in the absence and multi-Q textures in the presence of magnetic field. The manuscript deserves publication provided that the authors can address the following issues:\nReply: We thank Referee 2 for carefully studying our manuscript and for stressing the ongoing international research efforts on this material family. We here do our best to address the suggestions and concerns raised by Referee 2, as detailed below.\nReferee Comment2.2 — (i) The Q vector, the author refers to, does not only appear as modulation vector for helimagnetic order but also seems to connect the Weyl points. The authors mention that this is accidental. This point needs further clarification because it may be possible that correlation between the different Weyl fermions is responsible for the magnetic order.\nReply: This is an important issue raised by both Referees: Do the Weyl points close to the Fermi energy have an impact on magnetic interactions in RAl(Si/Ge)? Providing more detailed calculations to respond to the question, we argue that the period of the magnetic order (i.e., the magnitude and direction of Q) is dominated by states far from E . In this sense – although Q seems to connect between regions with Weyl points in F the Brillouin zone [Nat. Mater. 20, 1650 (2021)] – we call this ’accidental’. On the other hand, our calculation and analysis cannot rule out that anisotropic exchange and Dzyaloshinskii-Moriya interactions, which are clearly important for our spin models, have a contribution from Weyl points. The relative contribution of Weyls to DMI or anisotropic exchange remains a question for future research and can be addressed, e.g., by detailed E F tuning using a series of doped single crystals.\nWe demonstrate these points in four stages, adding extensive additional band structure calculations.\na. Short-range exchange interactions in GdAlSi.\nmostly short-ranged and antiferromagnetic regardless of bond distance, cf. Fig. R16a. This suggests that superexchange due to states far from E dominates over ’true’ Ruderman- F Kittel-Kasuya-Yosida (RKKY) interactions, which should be long-ranged and oscillatory in form [Phys. Rev. B 109, L201108 (2024)]. The short-range interactions in Fig. R16 are also well consistent with our real-space spin Hamiltonian calculations (Supplementary Note S2), which show that magnetic order with propagation vector Q = (2/3,2/3,0) can be approximated by only two short-range, antiferromagnetic, and isotropic interactions, J 1 and J (cf. Fig. 1d of the Main Text).\nc\n33\nb. Electronic structure calculation reproduces observed propagation vector.\nImportantly, the magnetic interactions from the electronic structure calculation successfully reproduce the experimentally observed propagation vector Q : As shown in Fig. R16b, exp the incommensurate state is energetically favoured over the ferromagnetic state at Q′ = (0, 0, 0), with the maximum occurring at Q′ = (q′, q′, 0) with q′ = 0.29 ≈ 1/3. This is in excellent agreement with Q = (q, q, 0) where q = 0.67 ≈ 2/3.\nexp\nThe difference of 1/3 vs. 2/3 is merely a matter of the convention used for ab-initio calculations [cf. Phys. Rev. B 109, L201108 (2024) and its SI]: To account for two magnetic Gd sites in the (primitive paramagnetic) unit cell of GdAlSi, the Fourier-transform of the magnetic interactions are defined in matrix notation and the maximum eigenvalue J(Q′) is extracted from this matrix. Whether this Q′ corresponds to an ordering vector Q in the first Brillouin zone, e.g., (1/3,1/3,0), or in a higher zone, e.g., (2/3,2/3,0), exp can be examined by considering the real-space magnetic structures in Fig. 3 of Phys. Rev. B 109, L201108 (2024). In summary, the theoretical study clearly demonstrates that the maximum of J(Q′) at Q′ ≈ (1/3,1/3,0) corresponds to ordering with antiferromagnetic coupling between the two Gd sites in the primitive cell, consistent with experiment.\nPhys. Rev. B 109, L201108 (2024) also discusses how the short-range interactions in RAlSi (R =rare earth) can be understood in terms of superexchange via the p orbitals of nonmagnetic Si and Al atoms; this is fundamentally different from RKKY-type interactions involving only Weyl electrons. We note that recent theoretical studies on SmAlSi have also questioned the nesting scenario based on a Lindhard susceptibility analysis [Commun. Phys. 6, 134 (2023); arXiv:2508.11244 (2025)].\nc. Improved band structure calculations and Weyl points in GdAlSi\nIn the process of addressing this question, and aiming to obtain a more accurate description of electronic states near the Fermi level where Weyl nodes emerge, we introduced a Hubbard U into the electronic structure calculations. We then searched for Weyl points in the Bril-\nc\nlouin zone (BZ) by constructing a wannierised Hamiltonian. Fig. R17b shows the calculated electronic structure with U = 6eV and 24 Weyl points close to the k = 0 plane. The c Fermi surface of GdAlSi is consistent with a semimetallic electronic character. Note that a RKKY scenario generally requires a high density of states near the Fermi level, or at least, if Weyl points are to dominate, an absence of trivial electronic states at E . Nevertheless, F and consistent with many previous studies on this material class, our calculations clearly show that Q connects between Weyl points in the electronic structure.\nd. Possible Dzyaloshinskii-Moriya (DM) interactions and anisotropic exchange.\nThe isotropic interactions in Fig. R16 fix the period and direction of the magnetic ’wave’, but they do not fix the spin direction. This means that there remains a degree of freedom between cycloid or proper-screw (or a mixed character, as in certain multiferroics). The spin model calculations in the Main Text and SI show that DM interactions and anisotropic exchange play an important role in stabilizing the observed cycloid and multi-Q states. There- fore, the careful study of a possible impact of Weyl nodes on these anisotropic interactions\n35\nremains an important subject for future investigation. We assert that our experimental evidence for the spin cycloid and multi-Q state in GdAlSi, combined with numerical evidence for Weyl fermions that is consistent with the recent, peer-reviewed experimental literature [Phys. Rev. B 109, 035120 (2024)], warrants publication of our work even without a detailed calculation of (weak) DMI and anisotropic exchange density functional theory, and their explicit attribution to Weyl fermions.\nWe added this discussion in Supplementary Note S2 of the revised manuscript. We also updated Fig. 1e to show the band structure calculations with Hubbard U and Weyl points.\nWe further added the sentence below to the Main Text to more clearly communicate with our readers:\n“helimagnetism is not predominantly caused by the Weyl electrons and the matching of QtotheseparationofWeylpointsinmomentumspaceisaccidental,seealsoSupplementary Note S2.”\nNext, we revised Methods as follows:\n“the Gd atoms were treated with the generalised gradient approximation exchange-correlation functional with a Hubbard U (GGA+U). U was chosen to be 6eV.”\n“Effective magnetic interactions between the 4f magnetic atoms are computed using the infinitesimal rotation method.”\nReferee Comment2.3 — In that case, an important relevant question will be whether it would be possible for such correlation to introduce an ordering for magnetic moments without destroying or affecting the topologically protected Weyl points.\nReply: We appreciate this important comment on the impact of helimagnetic order on the Weyl points. In short, our additional electronic structure calculations suggest that some Weyl points are likely to remain intact in the helimagnetic state, while others are gapped.\nN\nFigureR18: Change of electronic structure by helimagnetic order. a, Fermisur- face of the field-aligned ferromagnetic state of GdAlSi in the k = 0 plane in the larger, c paramagnetic Brillouin zone (BZ, blue). The smaller, helimagnetic BZ is depicted in orange. The background colour corresponds to quasiparticle density of states (QDOS). Red circles represent Weyl nodes. b,c, Calculated Fermi surfaces of the ferromagnetic state folded into the helimagnetic BZ (b) and the cycloidal state (c) at k = 0. Orange c lines indicate the BZ boundary. Significant changes in the electronic structure, likely involving gap opening at Weyl nodes, are observed when comparing panels b and c.\nAs shown in Fig. R18a, some Weyl nodes in the ferromagnetic state of GdAlSi are located near the orange helimagnetic Brillouin zone (BZ) boundary. Figure R18b,c shows the ferromagnetic and cycloidal electronic structures, respectively, calculated in the folded helimagnetic BZ. When the periodic magnetic order expands the unit cell and thus folds the BZ, partial charge gaps open at the new BZ boundary. Significant modifications in the band structure are thus observed in the helimagnetic state. In GdAlSi, the change of the electronic structure due to the magnetic order is rather strong, and it is hard to track the fate of individual Weyl points. Moreover, we note that the search for Weyl nodes in the\n37\nhelimagnetic state remains numerically challenging. However, Weyl points close to the BZ boundary in Fig. R18a are very likely to be gapped, while others remain intact.\nModification of Weyl nodes by magnetism has previously been studied in the RAlSi/Ge (R=rareearth)compoundswithanetmagnetization, boththeoreticallyandexperimentally [Phys. Rev. B 97, 041104(R) (2018); Nat. Commun. 14, 7185 (2023)]. Such a net magnetization may also shift Weyls or generate additional Weyl nodes. Our study reveals the harmonic cycloidal magnetic order of GdAlSi, which will serve as a material platform for studying the modification of Weyl nodes by Q in presence of zero net magnetization. In the Main Text, we have already noted that:\n“Such states are realised by band-folding of Weyl fermions into a small Brillouin zone upon formation of periodically modulated order, and conditions for the hybridization of Weyl points have been derived and phrased in the language of special relativity.”\n“Indeed, controlling these Q-domains in magnetic Weyl semimetals, by magnetic field or current, provides a valley-selective way to open a (partial) charge gap around the Weyl cones, if they have an energy-like causal connection in the folded Brilloin zone.”\nIn response to this comment, we added Supplementary Fig. S7 and modified the Discussion & Conclusion to clearly describe our central motivation, as follows:\n“GdAlSi serves as a prototypical material for studying the modification of Weyl nodes by a perfect helimagnetic cylcoid, in contrast to the previously reported materials that exhibit a net magnetization (NdAlSi/Ge) ”\nReferee Comment2.4 — (ii) The authors perform a model calculation to describe the origin of the harmonic spin cycloid where they do not consider isotropic exchange interaction.\nReply: We apologise for a possibly confusing discussion in the previous version. In the following two responses, we would like to argue that the real-space and momentum space models are well consistent with each other, and make changes to the text to clarify this matter.\nFirst, we understand Referee 2’s comment to mean that the model calculation in Supple-\n38\nmentary Fig. S17c, Eq. (S3) and Fig. 1d does not include anisotropic exchange interactions (there seems to be a typo). This is correct, as these calculations are meant to show the origin of the ordering vector Q, which represents the period and direction of the helimagnetic ’wave’ without fixing the spin direction.\nThe previous version of the Main Text states that:\n“this natural choice of interactions (i) selects only propagation vectors of the type Q = (±q,±q,0), constrained by tetragonal symmetry; (ii) reproduces the experimentally relevant range of q for a reasonable J /J ratio; (iii) demonstrates the competition c 1 between incommensurate helimagnetism and commensurate up-up-down order in RAlSi, R =rare earth, as described in the Main Text and the Supplementary Note S8.”\nSubsequently, in Supplementary Note S8 and Supplementary Fig. S17d, we do add anisotropic exchange and Dzyaloshinskii-Moriya interactions to stabilise the cycloid in the real-space model. The Hamiltonian of the real-space model in Eq. (S10) is\n(cid:88)\nˆ\nH = H +H +H −µ SB· S (R7)\nex DM anis B\ni\nwith isotropic exchange interactions H , Dzyaloshinskii-Moriya interactios (DMI) H , and ex DM\nanisotropic exchange H . The last term is the Zeeman coupling.\nanis\nConsidering the previous discussion unclear, after reading Referee 2’s comment, we modified the Main Text:\n“The period and direction of the helimagnetic “wave” in GdAlSi is readily understood from the viewpoint of a minimal exchange Hamiltonian with isotropic nearest-neighbor interaction J < 0 and inter-layer coupling J < 0,”\n1 c\nReferee Comment2.5 — On the other hand, to describe multi-Q phase in the presence of magnetic field, they introduce anisotropic exchange interaction. I believe\n39\nauthors should justify the consideration of different models to describe different states in the same material system.\nReply: We appreciate this constructive comment. The previous version was confusing because, in the Main Text, we introduced two different spin Hamiltonians: the real-space Hamiltonian of Eq. (R7) and a Fourier-space Hamiltonian. The purpose of the real-space model is merely to give an intuitive understanding of the magnetic interactions, propagation vector, andspintextureinthezero-fieldgroundstate(discussedinSupplementaryNoteS8).\nIn contrast – and we would here like to stress this –, the Fourier-space Hamiltonian explains both the zero-field ground state and the field-induced multi-Q state, although this was maybe not clear enough in the previous version. Indeed, we prefer the Fourier-space model for the discussion of both magnetic phases in the Main Text, as it helps to avoid parameter fine-tuning due to competing interactions and robustly reproduces the magnetic orders in GdAlSi.\nTo clarify the purpose of both models, we revised the spin model calculations section of the Main Text as follows:\n“While this real-space spin Hamiltonian provides intuitive understanding, it suffers from an inherent frustration of model parameters. Therefore, to reproduce the helimagnetic multi-Q textures under a magnetic field, we fix the direction and amplitude of Q and introduce a Fourier-space model that includes both DMI and anisotropic exchange interactions. However, to reproduce the helimagnetic multi-Q textures under a magnetic field, we require both DMI and anisotropic exchange interactions. ”\nReferee Comment2.6 — (iii) The authors used linearly polarised x-rays in REXS to detect the orientation of magnetic moments in a particular ordering if I correctly understand. I think they should have briefly mentioned how the linearly polarised x-rays can couple to the magnetic moments in order to detect the underlying magnetic structure.\nf = C ε∗ ·ε +iC (ε∗ ×ε)·m(Q)+C ε†Oε, (R8)\nres 0 f i 1 f i 2 f i\nwhere C , ε, ε , and m(Q) are constants, the polarization vectors of the incident and i i f\nscattered X-rays, and the Fourier component of the magnetic moment, respectively. Among these terms, only the second term depends explicitly on m(Q), and thus encodes the magnetic information. Consequently, the resonantly enhanced scattering intensity I = f ·f∗ is given by Eq. (5) in Methods as\nres res\nI ∝ |(ε ×ε )·m(Q)|2. (R9)\ni f\nIn response to this comment, we added the discussion above to Methods.\nReferee Comment2.7 — (iv) Localized spin model with Dzyaloshinskii-Moriya interaction is used to describe the different magnetic ordering. However, has a microscopic model such as the Rasbha-Hubbard model been used earlier to study topological semimetallic state with such a complex magnetic ordering? (See e.g. in simpler cases of magnetic order or without magnetic order, Phys. Rev. B 107, 045123 (2023), Phys. Rev. B 110, 075134 (2024), Phys. Rev. B 111, 075166 (2025)).\nReply: We appreciate Referee 2’s insightful comment, which has motivated us to take anotherlookattherelevantliterature. Inourwork,weuseaneffectivespinHamiltonianwith\n41\nlocal magnetic moments, where the magnetic interactions can be derived from electronic structure calculations. We then discuss the magnetic texture from the viewpoint of the effective model, aiming to keep the discussion simple. As Referee 2 correctly states, the charge degree of freedom is thus ”integrated out”, contrary to more sophisticated models such as Rashba-Hubbard or Kondo models with spin-orbit interactions.\nIn the field of Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions from Weyl fermions, Kondo-type models with explicit spin-charge coupling have been considered: [Phys. Rev. B 92, 241103(R) (2015); Phys. Rev. B 92, 224435 (2015)]. Further, as Referee 2 indicates, Hubbard models, where the magnetic moment is itinerant and not localised, have been considered for Weyl systems.\nWe have added these references, but maintain that the local-moment discussion is suitable for discussing the static magnetic structure of GdAlSi with localised Gd-4f moments. However, future studies of spin dynamics or current-driven phenomena in GdAlSi should treat (dynamical) spin-charge coupling very carefully.\nIn response to this comment, we added the following sentence to the Main Text.\n“The present model describes the interactions between local magnetic moments, which are implicitly mediated by conduction electrons. However, models of itinerant magnetism may provide a complementary perspective on the spin cycloid and multi-Q order in GdAlSi.”\nConclusion of our response letter:\nWe again thank both Referees for their helpful comments and constructive suggestions. We made a sincere effort to answer all questions thoroughly, adding high-quality neutron scattering experiments as well as experimental data on energy dispersive X-ray spectroscopy, magnetization measurements under controlled magnetic history and resonant X-ray scattering. We further added theoretical discussions of exchange interactions and revised the\n42\nmanuscript to more clearly describe the logic of our paper. We believe that the revised manuscript, with extensive revisions, is suitable for publication in Nature Communications.\nThemanuscriptwasrevisedsubstantially,withnewdataandrelevantdiscussionsadded.\nThe revised sentences/paragraphs in the Main Text are marked in red for the conve- nience of the Referees.\n2nd paragraph of “Introduction”\n“Here, we report the undistorted, cycloidal helimagnetic ground state of GdAlSi with perfectly isotropic Gd3+ ions, a material in which Fermi arcs from Weyl nodes have been identified, and a possibly related anomalous Hall effect has been observed under a magnetic field”\n1st paragraph of “Electronic and magnetic state of GdAlSi”\nWe added References Ref.[23,25,26] at the end of the sentence starting “GdAlSi crys- tallises in...”\n3rd paragraph of “Electronic and magnetic state of GdAlSi”\n“The period and direction of the helimagnetic “wave” in GdAlSi is readily understood from the viewpoint of a minimal exchange Hamiltonian with isotropic nearest-neighbor interaction J < 0 and inter-layer coupling J < 0,”\n1 c\n4th paragraph of “Electronic and magnetic state of GdAlSi”\n“helimagnetism is not predominantly caused by the Weyl electrons and the match- ing of Q to the separation of Weyl points in momentum space is accidental, see also Supplementary Note S2”\n2nd paragraph of “Zero-field cycloid from resonant elastic X-ray scattering (REXS)”\n“this corresponds to a harmonic helimagnetic cycloidal structure at zero magnetic field,\n44\nwhich is further confirmed by quantitative analysis with m (Q)/m (Q) = 0.966(7) in ∥ c Fig. 2g.”\n1st paragraph of “Spin model calculations and field induced phase transition”\n“Whilethisreal-spacespinHamiltonianprovidesintuitiveunderstanding,itsuffersfrom an inherent frustration of model parameters. Therefore, to reproduce the helimagnetic multi-Q textures under a magnetic field, we fix the direction and amplitude of Q and introduce a Fourier-space model that includes both DMI and anisotropic exchange in- teractions. However, to reproduce the helimagnetic multi-Q textures under a magnetic field, we require both DMI and anisotropic exchange interactions.”\n2nd paragraph of “Spin model calculations and field induced phase transition”\n“m (Q ) with α = x,y,z and ν = 1–8 are Fourier components of the localised mag- α ν netic moment m(r ); ν = 5−8 are commensurate with the lattice, while ν = 1−4 are i incommensurate ν = 1,2 are incommensurate with the lattice, ν = 3,4 are commensu- rate, and ν = 5-8 represent higher harmonics formed by superposition of ν = 1-4”\n3rd paragraph of “Spin model calculations and field induced phase transition”\n“Inotherwords,thecalculationiswellconsistentwiththeexperimentalobservationsmagnetic structure in zero field ( in Fig. 3c) and in finite magnetic field ( and Supplementary Fig. S1).”\n3rd paragraph of “Spin model calculations and field induced phase transition”\n“In Supplementary Note S8, we further truncated the Fourier-space model of Eq. (3) afterafewnear-neighboursandobtainedrecovertheanintuitivereal-spaceHamiltonian in Fig. 1c. , to discuss the energetics of the helimagnetic ground state.”\n3rd paragraph of “Spin model calculations and field induced phase transition”\n“The present model describes effective magnetic interactions between local magnetic moments, which are implicitly mediated by conduction electrons. However, models of itinerant magnetism may provide a complementary perspective on the spin cycloid and\n45\nmulti-Q order in GdAlSi”\n3rd paragraph of “Spin model calculations and field induced phase transition”\nWe added References Ref.[34-36] at the end of the sentence starting “The present model describes...”\n1st paragraph of “Discussion & conclusion”\n“GdAlSi will serve as a prototypical material for studying the modification of Weyl nodes by a perfect helimagnetic cylcoid, in contrast to the previously reported materials that exhibit a net magnetization (NdAlSi/Ge).”\n2nd paragraph of “Discussion & conclusion”\n“Note that the screw-type DM interaction, which has been proposed for magnetic Weyl systems,isstrictlysymmetry-forbiddenbythepolarcrystalstructureofRAlSi,R =rare earth (Supplementary Note S4 and Supplementary Fig. S11).”\nFigure 1\nWe replaced Fig. 1d with the updated calculations on a finer parameter grid. We replaced Fig. 1e with the result from GGA+U and Weyl points.\nFigure 2\n“g, Undistortedhelimagneticcycloidstructurewithm (Q)/m (Q) = 0.966(7)obtained ∥ c by polarization analysis in REXS at various magnetic reflections.”\nReferences\nWe added Refs. [23,25,26, 34-36, 43-51]\nMethods\nWe revised “Resonant elastic X-ray scattering (REXS)”, “Fourier-space model cal- culations”, “Electronic structure calculations”, “Second harmonic generation (SHG)” sections.\nAuthor list and Author contributions\nSupplementary Information\nWe added Supplementary Figs. S5-16, S18 and Supplementary Note S2-7.\nWe would like to thank the two Referees for devoting their time to carefully reviewing our revised manuscript and response letter, and we are pleased that both Referees recommended our manuscript for publication in Nature Communications. Below, we provide a point-by-point response to the Referees’ final comments.\nResponse to Referee 1\nReferee Comment1.1 — The authors have thoroughly addressed all concerns raised by the referees, providing substantial new calculations, neutron data, and clearer dis- cussion of the topological–magnetic interplay. The revisions significantly strengthen the manuscript, and the responses are complete and technically convincing.\nReply: We sincerely thank Referee 1 for the positive and thoughtful evaluation, and we are pleased that the additional calculations, neutron data, and revised discussion are considered to significantly strengthen the manuscript.\nReferee Comment1.2 — A few minor improvements—such as slightly streamlining the Supplementary Information and tightening the distinction between the two spin models—could further enhance readability but do not affect the scientific validity.\nReply: We have reordered the Supplementary Information Notes for clarity. We also highlighted the terms real-space and Fourier-space in italics in Supplementary Note S5.\nReferee Comment1.3 — I recommend acceptance of the revised manuscript.\nReply: We thank Referee 1 for recommending acceptance of the revised manuscript.\nReferee Comment2.1 — Theauthorshavesatisfactorilyaddressedallthereviewer’s comments, and the revised manuscript is now suitable for publication.\nReply: WesincerelythankReferee2fortheircarefulevaluationandconstructivecomments, and for recommending our manuscript for publication in Nature Communications.\n2"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have thoroughly addressed all concerns raised by the referees, providing substantial new calculations, neutron data, and clearer discussion of the topological–magnetic interplay. The revisions significantly strengthen the manuscript, and the responses are complete and technically convincing. A few minor improvements—such as slightly streamlining the Supplementary Information and tightening the distinction between the two spin models—could further enhance readability but do not affect the scientific validity.\nI recommend acceptance of the revised manuscript.", "reviewer_2": "The authors have satisfactorily addressed all the reviewer’s comments, and the revised manuscript is now suitable for publication."}, "author_response": "We would like to thank the two Referees for devoting their time to carefully reviewing our revised manuscript and response letter, and we are pleased that both Referees recommended our manuscript for publication in Nature Communications. Below, we provide a point-by-point response to the Referees’ final comments.\nResponse to Referee 1\nReferee Comment1.1 — The authors have thoroughly addressed all concerns raised by the referees, providing substantial new calculations, neutron data, and clearer discussion of the topological–magnetic interplay. The revisions significantly strengthen the manuscript, and the responses are complete and technically convincing.\nReply: We sincerely thank Referee 1 for the positive and thoughtful evaluation, and we are pleased that the additional calculations, neutron data, and revised discussion are considered to significantly strengthen the manuscript.\nReferee Comment1.2 — A few minor improvements—such as slightly streamlining the Supplementary Information and tightening the distinction between the two spin models—could further enhance readability but do not affect the scientific validity.\nReply: We have reordered the Supplementary Information Notes for clarity. We also highlighted the terms real-space and Fourier-space in italics in Supplementary Note S5.\nReferee Comment1.3 — I recommend acceptance of the revised manuscript.\nReply: We thank Referee 1 for recommending acceptance of the revised manuscript.\nReferee Comment2.1 — Theauthorshavesatisfactorilyaddressedallthereviewer’s comments, and the revised manuscript is now suitable for publication.\nReply: WesincerelythankReferee2fortheircarefulevaluationandconstructivecomments, and for recommending our manuscript for publication in Nature Communications.\n2"}]

Decision letter