## Abstract

Magnetic materials with noncoplanar magnetic structures can show unusual physical properties driven by nontrivial topology. Topologically-active states are often multi-**q** structures, which are challenging to stabilize in models and to identify in materials. Here, we use inelastic neutron-scattering experiments to show that the insulating double perovskites Ba_{2}YRuO_{6} and Ba_{2}LuRuO_{6} host a noncoplanar 3-**q** structure on the face-centered cubic lattice. Quantitative analysis of our neutron-scattering data reveals that these 3-**q** states are stabilized by biquadratic interactions. Our study identifies double perovskites as a highly promising class of materials to realize topological magnetism, elucidates the stabilization mechanism of the 3-**q** state in these materials, and establishes neutron spectroscopy on powder samples as a valuable technique to distinguish multi-**q** from single-**q** states, facilitating the discovery of topologically-nontrivial magnetic materials.

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## Introduction

Most magnetic materials order with simple magnetic structures in which spins are collinear or coplanar. Noncoplanar magnetic structures are relatively rare, but are of great current interest, because they can exhibit topological character and exotic physical properties^{1,2}. For example, the finite scalar spin chirality of noncoplanar spin textures can generate a topological magneto-optical effect^{3} and anomalous quantum Hall effect^{4,5}, even in the absence of spin-orbit coupling. Topologically-nontrivial spin textures are typically multi-**q** structures, which superpose magnetic modulations with symmetry-related wavevectors **q**^{2}. Multi-**q** spin textures with long-wavelength modulations, such as skyrmion and hedgehog crystals, are well-studied as hosts of topology-driven phenomena^{6,7,8}. In this context, multi-**q** antiferromagnets are increasingly important^{9}, because they offer higher densities of topological objects with the potential to generate stronger physical responses^{10}.

To probe the relationships between spin structure, interactions, topology, and physical response, it is crucial to identify real materials that host noncoplanar spin textures. This has proved a challenging task, for three main reasons. First, it is necessary to identify noncoplanar spin textures that are robust to subleading effects such as magnetic anisotropies, spin-lattice coupling^{11,12}, fluctuations^{13,14,15,16}, and anisotropic interactions^{17}, which usually favor collinear states. Second, most noncoplanar states are found in metals, such as USb^{18,19} and *γ*-Mn alloys^{20,21,22,23,24,25}, and are often stable only under an applied magnetic field^{6,26}. On the one hand, itinerant electrons can support the generation of physical responses; on the other hand, modeling the magnetic interactions of metals presents fundamental challenges^{27,28,29,30,31,32}, such that insulators are often more suitable as model materials. Third, neutron-diffraction measurements play a central role in solving magnetic structures, but suffer from a “multi-**q** problem”: Such measurements are generally unable to distinguish 1-**q** from multi-**q** structures^{33}, due to orientational averaging for powders or domain averaging for single crystals. Therefore, multi-**q** spin textures are challenging to stabilize in models, and to identify in real materials.

Here, we identify the Mott-insulating double perovskites Ba_{2}YRuO_{6} and Ba_{2}LuRuO_{6}^{34,35,36,37} as prototypical examples of noncoplanar 3-**q** magnetism on the face-centered cubic (FCC) lattice in zero magnetic field. We obtain evidence for 3-**q** magnetism from a spin-wave analysis of neutron spectroscopy data. By optimizing the magnetic structure and interactions simultaneously against our data, we show that the 3-**q** structure is stabilized by biquadratic interactions within an antiferromagnetic Heisenberg-Kitaev model. Our study experimentally establishes that noncoplanar multi-**q** states are stabilized in frustrated FCC antiferromagnets, identifies cubic double perovskites as model materials to realize this behavior, and identifies guiding principles to facilitate design of materials with noncoplanar magnetic states.

## Results

### Theory and materials selection

Our study is motivated by theoretical results for the FCC antiferromagnet^{13,38,39,40,41}. The nearest-neighbor Heisenberg-Kitaev spin Hamiltonian on the FCC lattice can be written as

where **S**_{i} is a Ru^{5+} spin with quantum number *S* = 3/2, *J* and *K* denote the Heisenberg and Kitaev interactions, respectively^{39}, and \(\gamma \in \left\{x,y,z\right\}\) is perpendicular to the cubic face containing the bond between neighbors \(\langle i,j\rangle\). For antiferromagnetic *J* > 0 only, the model is frustrated, and orderings with **q** ∈ [1, *q*, 0] are degenerate^{13,40,41}. The degenerate manifold includes **q** = [1, 0, 0] (“Type I”) ordering, which is favored by fluctuations^{13,14,42} and is observed in Ba_{2}YRuO_{6} and Ba_{2}LuRuO_{6}^{34}. Henceforth, we therefore restrict our discussion to **q** = [1, 0, 0] ordering. For a collinear structure, spins may be either parallel or perpendicular to **q**; the former is stabilized by *K* < 0 and the latter by *K* > 0^{39,40,41}.

Figure 1a shows the collinear (1-**q**) and noncollinear (multi-**q**) structures associated with Type I antiferromagnetism. An unusual property of the FCC lattice is that 1-**q**, 2-**q**, and 3-**q** structures are energetically degenerate for *all* bilinear interactions that stabilize Type I ordering^{40,41}. Moreover, uniaxial anisotropy (~ \({S}_{z}^{2}\)) and antisymmetric exchange terms are forbidden by \(Fm\bar{3}m\) symmetries, and quartic anisotropy (~ \({S}_{x}^{4}+{S}_{y}^{4}+{S}_{z}^{4}\)) is forbidden for *S* = 3/2 operators. Consequently, interactions that would usually favor collinear magnetic structures are inactive in the *S* = 3/2 FCC antiferromagnet. This remarkable property potentially allows noncollinear structures to appear.

To identify candidate systems for 3-**q** spin textures among the diverse magnetic ground states of double perovskites^{43,44,45,46,47,48,49,50,51}, we consider two criteria: Type I antiferromagnetic ordering, and strictly cubic symmetry below the magnetic ordering temperature, *T*_{N}. The second criterion is key because 3-**q** structures have cubic symmetry, while 1-**q** and 2-**q** structures have tetragonal or orthorhombic symmetry that could drive a crystallographic distortion *via* spin-lattice coupling [Fig. 1a]. We investigate Ba_{2}YRuO_{6} and Ba_{2}LuRuO_{6} because they are chemically well-ordered and show no evidence for low-temperature deviations from cubic symmetry^{34,36}. Moreover, recent first-principles calculations predict that their magnetic structures might not be collinear^{52}, in apparent contradiction with interpretations of previous experiments^{34}.

### Experiments

We prepared ~ 8 g polycrystalline samples of Ba_{2}YRuO_{6} and Ba_{2}LuRuO_{6} by solid-state reaction (see Methods). The magnetic ordering temperature *T*_{N} ≈ 37 K is the same for both samples, and is suppressed compared to the Weiss temperature *θ* ~ −500 K, indicating strong magnetic frustration^{36}. Rietveld refinements to X-ray diffraction data confirm cubic symmetry above and below *T*_{N} (see Methods and Supplementary Fig. 1). The level of Y/Ru site mixing was undetectable (<3%) and Lu/Ru site mixing was 6(2)%, with minor impurity phases of ≈ 1 wt.% and ≈ 4 wt.% for M = Y and Lu, respectively (see Supplementary Table 1). We performed inelastic neutron-scattering measurements on the SEQUOIA instrument at ORNL^{53} using incident neutron energies *E*_{i} = 62 and 11.8 meV, yielding elastic energy resolutions *δ*_{ins} ≈ 1.68 and 0.27 meV, respectively.

### Rietveld refinements to elastic magnetic scattering

Figure 1b shows magnetic Rietveld refinements to our elastic neutron-scattering data measured at *T* ≈ 5 K. Applying the **q** = [1, 0, 0] propagation vector to \(Fm\bar{3}m\) crystal symmetry generates two magnetic irreducible representations (irreps), notated \(m{{{{\rm{X}}}}}_{3}^{+}\) and \(m{{{{\rm{X}}}}}_{5}^{+}\)^{54,55,56,57}. These irreps can be distinguished by their magnetic Bragg profiles. The \(m{{{{\rm{X}}}}}_{5}^{+}\) irrep agrees well with our elastic-scattering data for both materials; Rietveld refinements yield ordered magnetic moment lengths of 2.56(2) and 2.43(2) *μ*_{B} per Ru for Ba_{2}YRuO_{6} and Ba_{2}LuRuO_{6}, respectively. Since the Ru^{5+} magnetic form factor has not been accurately determined, we tested several 4*d* magnetic form factors^{58}; while this choice does not qualitatively affect our results, the form factor for Zr^{+} (isoelectronic with Ru^{5+}) yields optimal agreement with our data and is used throughout. The Zr^{+} form factor also yielded better agreement with our data than the interpolated Ru^{5+} form factor used in ref. ^{59}. In contrast to the \(m{{{{\rm{X}}}}}_{5}^{+}\) irrep, the \(m{{{{\rm{X}}}}}_{3}^{+}\) irrep strongly disagrees with our data, as it yields zero intensity for the strong (100) magnetic Bragg peak. This can be understood intuitively for a collinear 1-**q** structure, because neutrons are only sensitive to spin components perpendicular to the scattering wavevector, and the \(m{{{{\rm{X}}}}}_{3}^{+}\) irrep has **S**∥**q** while the \(m{{{{\rm{X}}}}}_{5}^{+}\) irrep has **S** ⊥ **q** [Fig. 1a]. A previous neutron-diffraction study of Ba_{2}YRuO_{6} and Ba_{2}LuRuO_{6} considered only collinear 1-**q** structures^{34}, but could not rule out multi-**q** structures, due to the multi-**q** problem.

### Overview of inelastic magnetic scattering

To overcome the multi-**q** problem, we consider the energy dependence of neutron-scattering data^{60}. To explain this choice of approach, Fig. 2 shows calculated spin-wave spectra for the 1-**q**, 2-**q** and 3-**q** structures with the \(m{{{{\rm{X}}}}}_{5}^{+}\) irrep. Here, only antiferromagnetic Heisenberg interactions are included. Qualitative differences between the calculated powder-averaged spectra for 1-**q**, 2-**q** and 3-**q** structures are apparent; e.g., the 3-**q** calculation shows a cutoff of magnetic signal for energies above ~ 0.4*J**S*, whereas the magnetic signal for the the 1-**q** calculation extends to ~ 0.7*J**S*. These calculations motivate the use of powder-averaged inelastic neutron scattering data to distinguish between these candidate ground states, and will inform our discussion of the experimental spectra.

Figure 3a shows our experimental inelastic data measured with *E*_{i} = 62 meV at *T* ≈ 5 K. A structured inelastic signal appears at *T* < *T*_{N} for small scattering wavevectors, *Q* ≲ 2 Å^{−1}, which we identify as magnon scattering. The magnon scattering is very similar for M = Y and Lu, except the features for M = Lu appear slightly broader, which may reflect the slightly greater degree of M/Ru site mixing in this compound (see Supplementary Fig. 1 and Supplementary Table 1). The top of the magnetic band overlaps with an intense phonon signal for *Q* ≳ 2 Å^{−1}. Figure 3b shows the scattering intensity integrated over 4.0 ≤ *Q* ≤ 4.5 Å^{−1}, from which we extract the average energy *E*_{ph} and width *σ*_{ph} of this phonon band via Gaussian fits for each material. The energy overlap of magnon and phonon modes suggests that spin-lattice coupling may be significant, which we consider below.

### Optimization of magnetic structure and Hamiltonian

Our starting point for modeling the magnetic scattering is the nearest-neighbor Heisenberg-Kitaev model, Eq. (1). Antiferromagnetic *J* > 0 has been shown to be important in Ba_{2}YRuO_{6}^{37}, and we include *K* > 0 because it is needed to stabilize magnetic ordering with the \(m{{{{\rm{X}}}}}_{5}^{+}\) irrep, as observed experimentally^{34}. We consider three additional interactions. First, the symmetric off-diagonal interaction \({H}_{\Gamma }=\Gamma {\sum }_{{\left\langle i,j\right\rangle }_{\gamma }}({S}_{i}^{\alpha }{S}_{j}^{\beta }+{S}_{i}^{\beta }{S}_{j}^{\alpha })\) is the only additional bilinear nearest-neighbor interaction allowed by symmetry^{39}. Second, the Heisenberg next-nearest neighbor interaction \({H}_{2}={J}_{2}{\sum }_{\left\langle \left\langle i,j\right\rangle \right\rangle }{{{{\bf{S}}}}}_{i}\cdot {{{{\bf{S}}}}}_{j}\) has been invoked for Ba_{2}YRuO_{6}^{37}; we require *J*_{2 }≤ 0 to stabilize Type I ordering. Third, the nearest-neighbor biquadratic coupling \({H}_{{{{\rm{bq}}}}}={J}_{{{{\rm{bq}}}}}{\sum }_{\left\langle i,j\right\rangle }{({{{{\bf{S}}}}}_{i}\cdot {{{{\bf{S}}}}}_{j})}^{2}\) has been invoked in density-functional-theory calculations for 4*d* double perovskites due to their increased electron hopping relative to 3*d* analogs^{52}. While the 4-spin exchange enters into the Hamiltonian at the same order as biquadratic exchange, these terms can be combined for the FCC lattice^{52}, so we do not consider them separately. For *J*_{bq} = 0, the classical energy of 1-**q**, 2-**q**, and 3-**q** structures is equal for all *K*, Γ, and *J*_{2} that stabilize Type I ordering. Nonzero *J*_{bq} removes this degeneracy and stabilizes 1-**q** ordering for *J*_{bq} < 0 and 3-**q** ordering for *J*_{bq} > 0 [Fig. 4a]. Importantly, since single-ion anisotropies are forbidden for *S* = 3/2 in a cubic environment, biquadratic exchange is the only physically-plausible mechanism that can remove the degeneracy of 1-**q** and 3-**q** structures.

We performed extensive fits to our inelastic neutron-scattering data to optimize the magnetic interactions for each candidate magnetic structure. For each structure associated with the \(m{{{{\rm{X}}}}}_{5}^{+}\) irrep (1-**q**, 2-**q**, or 3-**q**), we optimized three spin Hamiltonian parameters (*J*, *K*, and either Γ, *J*_{2}, or *J*_{bq}) against the broadband inelastic data shown in Fig. 5a and the energy dependence near the (100) magnetic Bragg position shown in Fig. 5b. The broadband inelastic data provides an overview of the key features of the spectrum, while the low-energy data provides information about the magnon gap, which is an important constraint as we discuss further below. The powder-averaged magnon spectrum was calculated within the established approach of linear spin-wave theory^{61,62} using the SpinW program^{63}, updated to include *J*_{bq}^{64}. We included a renormalization factor that takes into account higher-order corrections in the 1/*S* expansion, which is necessary to extract a correct value of *J*_{bq} since the unrenormalized theory would yield a value of *J*_{bq} that is 2.25 times too small^{65}. The parameter values were optimized to minimize the sum of squared residuals using nonlinear least-squares refinement; our general approach is similar to^{66,67,68}. For each refined model, several refinements were performed with different initial parameter values, to check for alternative *χ*^{2} minima, and derivative-based and derivative-free optimizers were used (see Methods). An overall intensity scale factor was optimized in our refinements; we also optimized a linear-in-*E* intensity offset to account for phonon scattering. To account for magnon-phonon coupling, we calculated the energy-dependent broadening of the magnon spectrum as \(\delta (E)={\delta }_{{{{\rm{ins}}}}}(E)+A{e}^{-{(E-{E}_{{{{\rm{ph}}}}})}^{2}/2{\delta }_{{{{\rm{ph}}}}}^{2}}\), where *δ*(*E*) is the overall Gaussian energy width, *δ*_{ins}(*E*) is the instrumental resolution, and *A* is a refined parameter that phenomenologically accounts for magnon broadening due to coupling with phonons at *E* ~ *E*_{ph}.

Figure 5a compares our broadband inelastic data (*E*_{i} = 62 meV) with the best fit for each of the 1-**q**, 2-**q**, and 3-**q** structures. The data show two V-shaped features centered near the (100) and (120) magnetic Bragg peaks at ≈ 0.76 and ≈ 1.70 Å^{−1}, respectively, with a sharp cutoff of magnetic signal for energies above ~ 14 meV. For both materials, these characteristics are best reproduced by the 3-**q** structure, while the 1-**q** structure disagrees with our experimental data. These observations are confirmed by the goodness-of-fit metric *R*_{wp} [Fig. 4b]. For both materials and for every interaction model we considered, the 3-**q** structure yields better agreement with our data than the 1-**q** or 2-**q** structure. Notably, the goodness-of-fit is more sensitive to the structure than the precise magnetic interactions; indeed, the qualitative differences between 1-**q** and 3-**q** spectra are more apparent for Heisenberg exchange only [Fig. 2]. The global best fit is for the 3-**q** structure and *J*, *K*, and *J*_{bq} interactions with the refined values given in Table 1. Refined values of *A* indicate magnon broadening due to magnon-phonon coupling is larger for Ba_{2}LuRuO_{6}. Importantly, for both materials, the biquadratic term is significant, with *J*_{bq}/*J* ~ 0.06. Hence, our key results are that only the 3-**q** spin texture agrees well with our neutron data, and this state is stabilized by biquadratic interactions. Similar results for both materials suggest that this state is insensitive to a low level of chemical disorder, as is observed in Ba_{2}LuRuO_{6} only (see Supplementary Fig. 1 and Supplementary Table 1).

### Origin of magnon gap

Our model provides insight into the mechanism of gap opening^{35}. Low-energy inelastic neutron scattering measurements (*E*_{i} = 11.8 meV) shown in Fig. 5b reveal gaps of 2.59(6) meV and 2.49(11) meV for M = Y and Lu, respectively, estimated by fits to a sigmoid function, \(I(E)\propto 1/[1+{e}^{-{\sigma }_{g}(E-{E}_{g})}]\), where *E*_{g} and *σ*_{g} are the fitted gap energy and width, respectively. Figure 5b also compares our low-energy inelastic data with the 3-**q** magnon spectrum for the optimal *J*-*K*-*J*_{bq} model [Table 1]. This calculation reproduces the observed gap. In contrast, the *J*-*K*-*J*_{2} model that yields the next-best *R*_{wp} does not yield a gap for any of the candidate magnetic structures, and can therefore be discounted (see Supplementary Fig. 2). Since single-ion anisotropies are forbidden here, the mechanism of gap opening is subtle. A fully isotropic model (*J* > 0 only) possesses gapless Goldstone modes and accidental zero-energy modes^{42,69}. Both types of excitation must become gapped to explain the observed gap in neutron spectra, and whether this occurs depends on both structure and interactions. For a 1-**q** structure with **S**∥**q** (stable for *J*_{bq} < 0 and *K* < 0), the spectrum is fully gapped^{70}; however, this magnetic structure was ruled out by elastic neutron data [Fig. 1]. The 1-**q** structure with **S**⊥**q** (stable for *J*_{bq} < 0 and *K* > 0) is consistent with elastic neutron data, but its spectrum remains gapless for nonzero *K* because of the continuous rotational symmetry in the spin plane. By contrast, if the structure is 3-**q**, a gap is present at the linear spin-wave level if both *J*_{bq} > 0 and *K* > 0, since these interactions together favor \(\left\langle 111\right\rangle\) spin alignment. We conclude that it is difficult to explain the elastic neutron data and the observed gap with a 1-**q** structure. However, these observations can be explained by a 3-**q** model with *J*_{bq} > 0 and *K* > 0, which also yields optimal agreement with our high-energy inelastic neutron-scattering data.

Continuous symmetries and their associated gapless Goldstone modes are maintained when considering quantum effects beyond linear spin-wave theory, which may have a similar effect to negative *J*_{bq} by favoring collinear structures^{71} and gapping the accidental zero-energy modes^{42,70}. We note that the energy scale of quantum order-by-disorder is relatively small, on the order of 10^{−3}*J*^{42}, and can be easily surpassed by the expected biquadratic exchange interaction in these materials. Ref. ^{52} suggests that the combined *J*_{bq} and 4-spin interaction is on the order of 0.1*J* for Ba_{2}YRuO_{6}, in reasonable agreement with our own fits. Importantly, this *J*_{bq} value results in an energy difference between 1-**q** and 3-**q** orderings that is much larger than any energy difference induced by order by disorder.

## Discussion

We have identified Ba_{2}YRuO_{6} and Ba_{2}LuRuO_{6} as model materials that host noncoplanar 3-**q** structures stabilized by biquadratic interactions in zero applied field. Macroscopic topological physical responses may be generated synthesizing thin films of these materials with [111] strain^{72}. Our experimental results strikingly confirm recent first-principles predictions^{52}. The positive sign of *J*_{bq} suggests that the effect of inter-site electron hopping outweighs spin-lattice coupling, since the latter would give a negative contribution to *J*_{bq}^{11,12}. Crucially, we quantify the interactions that stabilize the noncoplanar state, in contrast to proposed 3-**q** structures in NiS_{2}^{73,74,75}, MnTe_{2}^{76}, and UO_{2}^{77,78,79,80}, where the relevant interactions are not yet well understood. Our work provides guiding principles to facilitate the identification of multi-**q** spin textures. First, double perovskites offer a rich materials space in which 1-**q** and multi-**q** structures may be nearly degenerate on the FCC lattice. In candidate materials, the crystal symmetry should be higher than a 1-**q** model would imply. Second, magnets that are not deep inside the Mott-insulating regime are expected to have larger *J*_{bq} and, consequently, more robust 3-**q** orderings. This criterion hints that cubic Ba_{2}YOsO_{6}^{81,82} may also host a 3-**q** state, due to its extended Os 5*d* orbitals, potentially offering a route to investigate the effect of increased electron hopping. For small *J*_{bq}, we anticipate a thermally-induced transition from 3-**q** to 1-**q** ordering, since thermal fluctuations favor collinear states. Third, quartic single-ion anisotropy may play a role in FCC magnets with *S* > 3/2; in particular, easy-〈111〉 axis anisotropy should favor 3-**q** ordering. The relevance of 3-**q** ordering extends beyond the FCC lattice: two themes are the presence of 3-fold or 6-fold symmetries, such as in triangular, honeycomb, and kagome lattices, and the presence of higher-order interactions than Heisenberg exchange, such as biquadratic^{83,84} and ring exchange terms^{85}. A methodological insight that may be relevant for several lattices^{60} is that established spin-wave methods can distinguish 1-**q** and multi-**q** structures by optimizing structure and interactions simultaneously. This result highlights that neutron-scattering experiments on bulk polycrystalline insulators are complementary to approaches such as *γ*-ray emission spectroscopy^{21} and scanning-tunneling spectroscopies on conducting or thin-film materials^{86}.

## Methods

### Sample preparation and characterization

Polycrystalline samples were prepared by conventional solid state reactions^{35}. Rare-earth oxides and RuO_{2} powder were first dried at 900 °C overnight. A stoichiometric mixture of BaCO_{3}, M_{2}O_{3} (M = Ba or Lu), and RuO_{2} was thoroughly mixed, pelletized, and fired at 1315 °C for a week with intermittent regrinding. Magnetic properties were measured with a Quantum Design Magnetic Property Measurement System in the temperature range 2 < *T* < 300 K. The temperature dependence of magnetization suggests both compounds order antiferromagnetically below ~ 37 K, consistent with refs. ^{34,35,36}.

### Neutron scattering data

Inelastic neutron-scattering data (*E*_{i} = 60 and 11.8 meV) were corrected for detector efficency using a vanadium standard, for absorption, and for background scattering by subtraction of empty-container measurements. The data were placed in absolute intensity units (mb meV^{−1} Ru^{−1}) by normalization to the nuclear Bragg profile.

Rietveld refinements were performed using the Fullprof program^{87,88}. The peak-shape was modeled as a Gaussian with *H*^{2} = \({(U{\tan }^{2}\theta +W)}^{2}\), where *H* is the full-width at half-maximum of the peak, and *U* and *W* are refined parameters^{89}. We also refined the cell parameter, oxygen position parameter, magnetic moment length, intensity scale factor, zero offset, and fourth-order polynomial background terms, and a scale factor for the Y_{2}O_{3} or Lu_{2}O_{3} impurity phase. Atomic displacement parameters were neglected due to the low sample temperature and limited *Q*-coverage of the data.

### X-ray diffraction data

X-ray diffraction data were collected at 20 K and 300 K on portions of the same samples measured by neutrons, using a Panalytical XPert Pro diffractometer and Cu K*α* radiation (*λ* = 1.540598 Å, Ge(111) monochromator). The Bragg-Brentano geometry was used. The low-temperature collection used an Oxford Phenix cryostat.

Rietveld refinements were performed using the Fullprof program^{87,88}. The peak-shape was modeled using a pseudo-Voigt function, with refined *U*, *V*, *W*, *X* and *η*_{0} parameters, and four refined asymmetry parameters. Background was modeled using 6th-order Chebychev polynomials. Sample displacement and micro-absorption corrections were refined. Reflections from the Al sample holder were fitted using LeBail profile matching.

We obtained excellent agreement between refined and experimental profiles within the published cubic \(Fm\bar{3}m\) model^{34} at 20 K and 300 K (see Supplementary Fig. 1). No peak splitting or selective broadening was observed. This result confirms that Ba_{2}LuRuO_{6} and Ba_{2}YRuO_{6} are cubic above and below *T*_{N}, in agreement with previous studies^{34,36}.

The level of chemical disorder and impurity phases were quantified using Rietveld refinements. Refined values of all structural parameters are given in Supplementary Table 1, including the level of M/Ru site mixing, an overall atomic-displacement (*B*_{iso}) factor, and the oxygen position parameter *x*. Two minor impurity phases were identified: M_{2}O_{3} (0.34(3) wt.% for M=Y, 0.93(9) wt.% for M=Lu) and Ba_{3}MRu_{2}O_{9} (0.71(10) wt.% for M=Y, 2.6(4) wt.% for M=Lu). The value of *B*_{iso} is larger for Ba_{2}LuRuO_{6} than for Ba_{2}YRuO_{6}, in agreement with ref. ^{34}. In Ba_{2}YRuO_{6}, the level of Y/Ru site mixing refines to zero with an uncertainty of a few percent, consistent with previous high-sensitivity ^{89}Y magic-angle-spinning NMR measurements^{36}. In Ba_{2}LuRuO_{6}, we refined a small but nonzero amount of Lu/Ru site mixing of 6(2)%. As discussed in ref. ^{36}, the difference of *x* from \(x=\frac{1}{4}\) is an indicator of the level of site ordering. As such, the values of *x* reported in Supplementary Table 1 are consistent with slightly increased site mixing for M=Lu compared with M=Y, which may also be reflected in the larger *B*_{iso} for M=Lu.

### Spin-wave model fitting

Refinements were performed against two inelastic neutron-scattering data sets simultaneously: a slice with 0.3 ≤ *Q* ≤ 2.5 Å^{−1} and 2.5 ≤ *E* ≤ 20 meV (*E*_{i} = 60 meV data), and a cut at *Q* = 0.7450 ± 0.0175 Å^{−1} with 0.75 ≤ *E* ≤ 6.25 meV (*E*_{i} = 11.8 meV data). All data points were weighted by their uncertainties as 1/*σ*^{2}. We minimized the function

where subscript “expt” and “calc” indicate measured and calculated magnon spectra, respectively, *σ* is an experimental uncertainty, *s* is a refined overall scale factor, and *b* is a refined linear-in-*E* background term; it is assumed that *s* and *b* are equal for both data sets. To account for the instrumental resolution, calculations were convoluted with an energy-dependent Gaussian energy broadening (elastic FWHM = 1.68 and 0.27 meV for *E*_{i} = 62 and 11.8 meV, respectively) and a constant Gaussian *Q*-broadening (FWHM = 0.08 and 0.054 Å^{−1} for *E*_{i} = 62 and 11.8 meV, respectively). The parameter values were optimized using the Minuit nonlinear least-squares program using its derivative-based MIGRAD and derivative-free SIMPLEX algorithms^{90,91}. For each refined model, at least four separate refinements were performed with different initial parameter values, to check for alternative *χ*^{2} minima. Notably, minima with similar overall *χ*^{2} were found for the 3-**q** structure with *J*, *K* and *J*_{bq} interactions. These minima corresponded to smaller values of *K*, which has a noticeable effect on the magnon spectra only at low energies. These minima yield a magnon gap that is significantly smaller than the observed gap, and were therefore neglected.

## Data availability

Experimental data and simulation results supporting this study are available from the corresponding authors on reasonable request.

## Code availability

Custom codes used in this study are available from the corresponding authors on reasonable request.

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## Acknowledgements

This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. This research used resources at the Spallation Neutron Source, a DOE Office of Science User Facility operated by the Oak Ridge National Laboratory.

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J.A.M.P. analyzed the data and wrote the paper with input from all co-authors. J.Y. synthesized the samples and performed bulk characterization measurements. M.A.M. performed X-ray diffraction measurements. M.B.S., A.D.C., S.-H.D., S.G. and J.A.M.P. performed neutron-scattering experiments. M.J.C. and J.A.M.P. developed spin-wave fitting code. H.Z., D.D., K.B. and C.D.B. calculated spin-wave renormalization. A.D.C., C.D.B. and J.A.M.P. designed the study.

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Paddison, J.A.M., Zhang, H., Yan, J. *et al.* Cubic double perovskites host noncoplanar spin textures.
*npj Quantum Mater.* **9**, 48 (2024). https://doi.org/10.1038/s41535-024-00650-6

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DOI: https://doi.org/10.1038/s41535-024-00650-6