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Incommensurate magnetism mediated by Weyl fermions in NdAlSi

Emergent relativistic quasiparticles in Weyl semimetals are the source of exotic electronic properties such as surface Fermi arcs, the anomalous Hall effect, and negative magnetoresistance, all observed in real materials. Whereas these phenomena highlight the effect of Weyl fermions on the electronic transport properties, less is known about what collective phenomena they may support. Here, we report a new Weyl semimetal, NdAlSi that offers an example. Using neutron diffraction, we report a long-wavelength magnetic order in NdAlSi whose periodicity is linked to the nesting vector between two topologically non-trivial Fermi pockets, which we characterize using density functional theory and quantum oscillation measurements. Our work provides a rare example of Weyl fermions driving collective magnetism.


Introduction
Weyl semimetals are topologically nontrivial phases of matter that sustain low energy excitations in the form of massless fermionic quasiparticles known as the Weyl fermions 1,2 . It is necessary to break either inversion or time-reversal symmetry to establish a Weyl semimetal [3][4][5][6][7][8][9] . A rare occasion arises, however, if a material breaks both symmetries and offers an opportunity to study the interplay between magnetism and Weyl fermions 10-13 . Here we present an extensive experimental and theoretical study of a new Weyl semimetal, NdAlSi, that breaks both symmetries. Our neutron diffraction experiment reveals the leading instability of the Weyl semimetal is to long-wavelength incommensurate order followed by a lower temperature transition to commensurate ferrimagnetism.
Using quantum oscillation measurements and density functional theory, we find that the incommensurate wavevector connects different branches of the Fermi surface that contain Weyl fermions.
This raises the interesting possibility of a magnetic order driven by relativistic electrons in NdAlSi.

Structural, Magnetic and Electronic Bulk Properties
Single crystals of NdAlSi were grown by a self-flux approach (see methods). The body-centered tetragonal unit cell and the first Brillouin zone (BZ) of NdAlSi are shown in Fig. 1A and B. The structure has two vertical mirror planes (m x and m y ) but lacks a horizontal mirror plane (m z ), and thus breaks inversion symmetry (Fig. 1B). Site mixing between Al and Si can, however, restore the m z mirror plane and change the space (point) group from non-centrosymmetic I4 1 md (C 4v ) to centrosymmetric I4 1 /amd (C 4h ). Contrary to x-ray scattering, the neutron scattering length of Al and Si are sufficiently different (b(Al) = 3.449 fm, b(Si) = 4.1491 fm) that site mixing is readily apparent in neutron diffraction. The refinement of our single crystal neutron diffraction pattern for NdAlSi in the I4 1 md space group is shown in Fig. 1C. A better fit (χ 2 = 5.70) is obtained in I4 1 md compared to refinement in the I4 1 /amd space group (χ 2 = 6.04), which yields a limit of 9% on Si-Al site mixing. The inset of Fig. 1C shows a strong second harmonic generation (SHG) signal (χ xxz = -115(3) pm/V, χ zxx = 94(2) pm/V and χ zzz = 564(5) pm/V) that originates from a bulk electric dipole and fits to the point group C 4v . Thus, we confirm the non-centrosymmetric space group I4 1 md as the correct structure for NdAlSi (Fig.S1).
The magnetic heat capacity (C mag p ) of NdAlSi is plotted in Fig. 1D, revealing a broad anomaly at approximately 18 K, as well as two peaks at T inc = 7.2(1) K and T com = 3.3(1) K. A total entropy of ∆S = 0.96(2)×R ln(10) is released between 2.35 K and 300 K (inset of Fig. 1D), which is the value expected for the ground-state spin-orbital manifold of Nd 3+ with L = 6, S = 3/2, and J = 9/2. Fitting C mag p (T ) indicates splitting of the 2J + 1 = 10-fold degenerate J-multiplet into a ground state doublet and four excited doublets at ∆ = 4(2) meV. The low temperature anomalies at T inc = 7.2 K and T com = 3.3 K mark two magnetic phase transitions.
The temperature dependence of the inverse out-of-plane magnetic susceptibility (1/χ c ) and its ratio with the in-plane susceptibility (χ c /χ a ) are reported in Fig. 2A. An isotropic Curie-Weiss fit with Θ CW = 10.2(8) K and µ ef f = 3.65(5) µ B describes the data well at high temperature, but fails at low temperature where the system develops an Ising-like easy axis anisotropy with χ c /χ a > 80 at T = 1.8 K. Comparing Fig. 1D and Fig. 2A shows that the signature of T inc is clearer in the heat capacity, whereas T com is most prominent in the magnetic susceptibility data. This is consistent with a putative antiferromagnetic (AFM) transition with k = 0 at T inc and a ferrimagnetic transition with k = 0 at T com . The magnetic transitions also affect the electrical resistivity (Fig. 2B), where the first small drop at T inc and the second larger drop at T com are correlated with the magnetic transitions.
As explained below, the resistivity minimum at approximately 6 K correlates with the appearance of an incommensurate order that is precursor to the ferrimagnetic state observed for T < T com .

Neutron Diffraction and Spin Structure
The magnetic order parameters were determined through single crystal neutron diffraction experiments that are summarized in Fig. 2C and 2D. Below T inc , we found antiferromagnetic Bragg peaks that are characterized by an incommensurate k inc = ( 2 3 + δ, 2 3 + δ, 0) ordering wavevector. We also observed magnetic Bragg peaks corresponding to the third harmonics of this wavevector, i.e. k = (3δ, 3δ, 0) whose intensity smoothly increases upon cooling from T inc to T com . The incommensurability δ(T ) shows a weak temperature dependence before dropping to zero for T < T com , where a sharp transition to commensurate ferrimagnetic order occurs (Fig. 2D).
The spin polarization of both the FM k = (000) and the AFM k com = ( 2 3 2 3 0) components of the T < T com spin structure were determined through refinement of magnetic neutron diffraction data (Fig. S2). The primitive unit cell of NdAlSi contains two Nd ions located at r 1 = (0, 0, 0) and r 2 = ( 1 2 , 0, 1 4 ) whose spin orientation and size can differ. For the k = (000) component of the spin structure, however, this difference would produce magnetic Bragg reflections at k = (110) positions that were not present in our data. Indeed, magnetic Bragg intensity was only detected at k = (000) positions such as Q = (101) (Fig. 3A). Furthermore, no magnetic Bragg intensity was detected at any of the Q = (00L) Bragg peaks (such as Q = (004) shown in the inset of Fig. 3A).
This precludes ordered in-plane components of the spins. Thus, we conclude that the k = (000) spin structure component is ferromagnetic with spins oriented along the c direction (see left inset of is any allowed nuclear Bragg peak. We found magnetic Bragg scattering at Q − positions that is two orders of magnitude greater than at Q + , which clearly points to the − state (Fig. S2). The Ising anisotropy for this order was established by examining theQ − ·ĉ dependence of the Bragg intensity, which easily distinguishes between isotropic, planar (XY), and Ising like order parameters ( Fig. 3B). The presence of weak Q + Bragg peaks suggests an in-plane moment of 0.14(5)µ b /Nd, but further work is needed to properly characterize this component.
The k FM = (000) and k com = ( 2 3 2 3 0) components of the NdAlSi spin structure are respectively shown in the inset of Fig. 3A and B. A refined moment of 1.1(2) µ B /Nd and 3.8(4) µ B /Nd were determined for the FM and AFM components respectively. While the phase relationship between these components is unconstrained by diffraction, a constant moment requirement fixes the phase.
The corresponding structure (Fig. 3C) can be described as a 1D down-up-up (d-u-u) chain in Fig. 3D, where each arrow represents a plane of spins perpendicular to k com (Fig. 3E).
The magnetization curve for H c features two plateaus (Fig. 3F). The first plateau for anisotropy, mean field theory shows that amplitude modulated incommensurate spin density wave ordering is generally unstable towards a lower T transition into a commensurate spin structure 17 .
NdAlSi can be understood within this theoretical framework: An amplitude modulated spin density wave with k inc = ( 2 3 + δ, 2 3 + δ, 0) first appears below T inc , its amplitude growing upon cooling until T com . As the amplitude grows, so does the third harmonic k = (3δ, 3δ, 0), which indicates that the sine wave is "squaring up" and the magnitude of the moment on each site is becoming more constant through the lattice. For an Ising-like incommensurate state, full magnetization on every site is expected at T = 0 and it requires abrupt soliton-like defects. At T com the commensurate d-u-u state with constant moment size per site becomes energetically favorable. Though its periodicity deviates from that favored by fermi-surface nesting, the commensurate square wave structure allows all Nd sites to achieve the saturation magnetization (Fig. 2C).   3F) and the 2.9(1) µ B /Nd sublattice magnetization of the commensurate low T state.
We used the Wannier90 21 package to accurately interpolate the first-principles band structure (see Methods). From here, we find a total of 56 Weyl nodes in this FM phase whose locations within the first BZ are shown in Fig. 5B,C. The colors of the Weyl nodes reflect their chiralities, with red and blue dots representing nodes with chiralities +1 and −1, respectively. Because the magnetic point group has eight symmetry elements, the Weyl points come in groups of eight nodes that are all degenerate in energy, four with chirality +1 and four with chirality −1. We label the nodes as W 1 (one group), W 2 (two groups), W 3 (two groups), W' 3 (one group), and W" 3 (one group), accounting for a total of 56 Weyl nodes. The type and momentum space location of these Weyl nodes are strongly impacted by the f -electron magnetism. For example, Weyl nodes in the FM phase appear close to the glide mirror plane lying along the Γ-X direction. These arise due to the crossing of spin-up and spin-down bands so they are not stabilized in the PM phase ( Fig.S3).
Furthermore, Weyl nodes along the S -Z direction change from type I to type II in going from the PM to the FM phase.
Having presented the quantum oscillations and DFT calculations for NdAlSi, we now combine these to obtain the Fermi surface in the FM state ( Fig. 5D and E). The Fermi level was determined by comparing the calculated β frequency from DFT and the experimental value of k β from quantum oscillations, as well as two other quantum oscillation frequencies that were resolved in our high-field experiment (Fig. S5). The best agreement was obtained with .7473 eV is the charge-neutrality point.

Incommensurate Magnetism and Nesting Wavevector
The where nesting conditions are satisfied. This indicates that the nesting between Fermi pockets lying along the (110) direction could be responsible for the magnetic order observed in NdAlSi. We note that this result is not the fruit of a fine tuning of E F since the nesting condition is satisfied over a broad range of E F , which extends well beyond the constraints of our quantum oscillations analysis.
We can then rationalize the low-temperature magnetic order starting from the FM state where the Zeeman term completely dominates and all the spins point along the c axis. Upon decreasing the field to H c , the magnetic susceptibility develops a peak at k inc due to RKKY interactions that favor the nesting wavevector. At still lower fields the corresponding modulated magnetic order emerges to gap the soft mode. This phenomenology is well established for various rare-earth metallic and intermetallic systems 18,19 .

Discussion and Conclusion
NdAlSi is special in that the nesting Fermi pockets contain W 3 Weyl nodes 41 meV above Weyl nodes promotes a spin-density-wave at q while every single node yields a q = 0 ferromagnetic channel through intra-node electron scattering. This latter interaction naturally explains the k FM = 0 spin component that ultimately develops in NdAlSi, as it is favored by every Weyl hole and electron pocket. This suggest that the AFM spin component must come from inter-node scattering processes. The multitude of Weyl nodes in NdAlSi creates many potential AFM ordering channels from which the k inc = 2 3 + δ, 2 3 + δ, 0 order would arise. To understand this, we note that nearly commensurate nesting wavevectors produce orders that support saturated sublattice magnetization and therefore are energetically favored. Moreover, k inc approximately connects not one but four symmetry-related pairs of W 3 Weyl nodes. Furthermore, the third harmonic 3k inc , which must be present in a sublattice-saturating square wave state and was observed at (3δ, 3δ, 0) (Fig. 2C), is near the Γ-point. Thus, in contrast to other Weyl-Weyl ordering channels, the third harmonic of k inc is favored by q → 0 intra-node scattering. Taken together, these observations provide a coherent provisional explanation of the observed magnetic order.
In conclusion, we have discovered uniaxial incommensurate amplitude modulated magnetic ordering in NdAlSi -an inversion symmetry breaking semimetal with topologically protected Weyl nodes. Upon cooling, the magnetic wavevector k inc = ( 2 3 +δ, 2 3 +δ, 0) approaches the commensurate point (δ → 0) while the third harmonic grows and the incommensurate structure acquires square wave character. This culminates in a first order phase transition to low T commensurate ferrimagnetism with characteristic wavevectors k = 0 and k com = (

Structural Characterization of NdAlSi
We have characterized the structure of NdAlSi using powder x-ray diffraction, single crystal neutron diffraction, and Energy-Dispersive X-ray spectroscopy (EDX). The results are summarized in Table. S1. Both neutron diffraction and EDX measurements detected Si and Nd vacancies.
The noncentrosymmetric I4 1 md (#109) and centrosymmetric I4 1 /amd (#141) space groups are indistinguishable with X-ray diffraction, and their contrast in neutron diffraction is small. We thus used second harmonic generation (SHG) measurements to distinguish between these space group.
SHG is sensitive to a center of inversion in the unit cell. SHG is negligible in a centrosymmetric space group. The large SHG signal reported in the main text points to the noncentrosymmetric space group I4 1 md (#109) and is inconsistent with the centrosymmetric spacegroup I4 1 /amd (#141).
The SHG data were fit to functions appropriate to four different experimental configurations: 1) incoming polarization rotating, output polarizer fixed with polarization parallel to the crystalline In the electric dipole approximation, the mathematical forms of these various responses for the [101] crystal face in the I4 1 md space group (C 4 v point group) are given by I eee (φ) = 1 32 cos 2 (φ) [(−2χ eee xxz − χ eee zxx + χ eee zzz ) cos(2φ) + 6χ eee xxz + 3χ eee zxx + χ eee zzz ] 2 (1) The data were fit to expressions [1][2][3][4] accounting for a rotation of the sample axes relative to the laboratory x-axis to produce excellent fits to the data, as seen in Fig. S1. The SHG susceptibilities extracted from those fits are χ eee xxz = − 115 ± 3pm/V , χ eee zxx = 94 ± 2pm/V , and χ eee zzz = 564 ± 5pm/V . The competing space group assignment I4 1 /amd (point group D 4h ) is centrosymmetric and thus should not produce as strong of a SHG response as we have shown here.

Neutron scattering of NdAlSi
We refined the spin polarization of the k = (000) magnetic structure of NdAlSi by acquiring rocking We also determined the spin polarization of the AFM components of the spin structure in both the commensurate (δ = 0) and incommensurate phases (δ = 0). To do so, rocking scans at 46 symmetrically non-equivalent Bragg positions were collected within the manifold of Bragg peaks Q + = G ± ( 1 3 + δ, 1 3 + δ, 0) and Q − = G ± ( 2 3 + δ, 2 3 + δ, 0) for both T = 1.6 K and 6 K. Here G refers to the manifold of nuclear Bragg peaks. The symmetry analysis of the δ = 0 phase reveals six possible basis vectors divided into two different irreps (Γ 1 and Γ 2 ) 2 , for which their real parts are shown in Fig. S2B. The two Nd ions located at r 1 =(0,0,0) and r 2 =(1/2,0,1/4) in the chemical unit cell have their spins anti-parallel to each other for spin structures described by ψ 1 + ψ 2 , ψ 4ψ 5 , or ψ 3 . These anti-parallel spin structures leads to strong Q − peaks and no intensity at Q + peaks.
On the other hand, spin structures described by ψ 1ψ 2 , ψ 4 + ψ 5 , or ψ 6 have Nd spins at r 1 and r 2 that are parallel. This situation leads to strong Q + peaks and no intensity at Q − peaks. As seen in Fig. S2C, we observed intensities at Q − positions that are two order of magnitude greater than at Q + so the spin structure is predominantly of the anti-parallel variety. Fig. 3B of main paper shows it is described by the Ising ψ 3 basis vector. The final refinement is shown in Fig. S2D using a ψ 3 basis vector with a moment of µ AF M = 1.9(2)µ B . A 4(3)°shift away from a perfect Ising anisotropy is indicated by weak, but finite Q + Bragg peaks. However, additional work is needed to properly quantify this possible spin canting. Finally, the relative intensities of the k = ( 2 3 + δ, 2 3 + δ, 0) Bragg peaks within the incommensurate phase are indistinguishable from those of the Ising-like commensurate phase so no spin reorientation was observed above T inc .
For the AFM component, the spatial variation of the Nd moments is expressed as: This expression includes both the k = ( The band structure calculation for the FM phase of NdAlSi are presented in the main manuscript within the PBE+U +SOC approximation (Fig. 5). As compared to the nonmagnetic case, the Weyl nodes W 1 near the k z = 0 plane are unaffected by the magnetization along [001], but the nodes lying off the k z plane along the S − Z direction undergo further splitting around the m x and m y mirror planes, which we defined as two subtypes: W 2 nodes close to the m x and m y mirror planes, and W' 2 nodes away from them. Furthermore, splitting of the W 3 nodes along the m xy and m xȳ mirror planes is observed, which gives rise to new pairs of Weyl nodes categorized into three types: W 3 nodes close to the k z = 0 plane, W' 3 off the k z = 0 plane, and W" 3 close to the k z = 0 plane but away from the location of the W 3 nodes. To visualize their dispersions, we plot the projected band structure in the vicinity of some representative Weyl nodes in Fig. S4. Figure S4A shows the projected band structure plotted along the k y direction with Fermi level at the energy position of Weyl nodes of type W 1 with positive chirality, located 98.6 meV above the charge-neutrality Fermi level. The corresponding two-dimensional Fermi surface projected in the k x − k y plane is shown in Fig. S4B. Similarly, the band structure for a Fermi level aligned with Weyl node W 2 is projected along k y in Fig. S4C, and the same is done for W' 2 in Fig. S4E. The Fermi surfaces projected in the k x − k y plane at these energy positions are shown in Fig. S4D and F respectively. The cyan-colored circles indicate the Weyl node positions for which the band structures are plotted. The band structure in Fig. S4E, around the W' 2 nodes, appear to be of type II Finally, the PBE+U +SOC band structure for the commensurate AFM d-u-u phase of NdAlSi is shown in Fig. S3F for which the presence of 54 Weyl nodes is also predicted. The electronic state appears to have a semimetallic character whose details are extremely sensitive to the structural parameters of NdAlSi, which we know from our neutron diffraction experiment is further complicated by the presence of Si and Nd vacancies on the order of few % (Table. S1). Thus, a robust characterization of the Weyl nodes in the d-u-u phase requires further analysis that is beyond the scope of the present manuscript.

Quantum Oscillations, Determination of E F , and nesting vector in NdAlSi
In the main text, we have determined the Fermi surface of NdAlSi in its ferromagnetic (FM) phase using the quantum oscillations (QOs) observed in our high-field resistivity measurements. Fig. S5A shows the resistivity of NdAlSi measured up to 35 T, where QOs are apparent at high fields. A smooth background was fitted to the data for fields above the metamagnetic (MM) transition, which is marked by a jump in resistivity. To isolate the QOs, we subtracted a smooth background for each temperature. The resulting resistivity data are shown in Fig. S5B. These data show that the ∆R = 0 axis passes through the center of the oscillations, and a monotonic decrease of the oscillatory amplitudes at all fields. Both observations justify our choice of the smooth background.
Next, we performed a fast Fourier transformation (FFT) of the data in Fig. S5B. The resulting FFT spectrum is plotted in Fig. S5C. Based on this spectrum, three distinct frequencies δ, β, and γ are identified, together with the second harmonic of the β frequency (2β). Note that the β frequency shifts downward as the temperature increases, which agrees with the data presented in the main text (Fig. 4C). We also determined the effective masses for the three Fermi pockets associated with the QOs peak at δ, β, and γ. To do this, we fit the standard Lifshitz-Kosevich (LK) formula 4, 5 to their QOs amplitudes as a function of temperature (Fig. S5D). The frequencies and effective masses extracted from Fig. S5C and D are listed in Table S2.
With the experimental QOs frequencies analyzed, we now turn to the theoretical QO frequencies given by DFT calculations, and determine the E F by matching the experiment with the theory.
Each E F corresponds to a specific Fermi surface, which generates a set of QO frequencies. We start from the E F determined by DFT (E DF T F = 6.7473 eV , the neutrality point), adjust the E F below and above E DF T F , compute QO frequencies for different E F 's until the calculated QO frequencies match with those we observed in the high-field experiment. For E F being 30 meV and 33 meV above E DF T F , the QO frequencies comparable to δ, β and γ are listed in Table S2. As shown in     β, and γ pockets derived from resistivity measured up to 35 T, are listed with the results from the DFT calculations. δ is identified to be an electron pocket, while β and γ are hole pockets. The "+XX meV" in the column "Sources" means the shift in E F from the DFT-determined E F for that particular calculation. As a result of such shift, the frequencies change between the calculations "DFT, +30 meV" and "DFT, +33 meV". The error in E F is determined to be 3 meV based on the change in E F from matching F γ to F δ .