## Introduction

Understanding the physics of unconventional superconductors, which include cuprate, iron-based, and heavy fermion superconductors, remains a major challenge in condensed matter physics. Unlike conventional superconductors with phonons responsible for binding electrons into pairs, pairing in unconventional superconductors occurs due to electronic interactions1,2,3. The proximity to magnetically ordered states in these materials suggests spin fluctuations as a common thread that can pair electrons in unconventional superconductors3,4,5. Unlike phonon-mediated conventional superconductors with superconducting order parameters Δ(k) that depend weakly on momentum k, spin-fluctuation-mediated superconductivity requires a Δ(k) that changes sign in momentum space4. Therefore, the experimental determination of whether a sign-change occurs in Δ(k) is paramount for identifying and testing the spin-fluctuation-mediated pairing mechanism.

While sign-changing superconductivity in cuprate superconductors has been confirmed through phases-sensitive tunneling experiments6,7, such direct experimental evidence is lacking in most other systems where a sign-change has been proposed. Most experimental techniques, including penetration depth, specific heat, thermal conductivity, and angle-resolved photoemission, can probe the magnitude of the superconducting order parameter and its momentum dependence7, but are not phase-sensitive. The observation of a spin resonance mode (SRM) in inelastic neutron scattering is commonly regarded as strong phase-sensitive evidence for a sign-changing superconducting order parameter4,8,9,10,11, based on the assumption that the SRM is a spin-exciton appearing below the particle–hole continuum onset (PHCO), and at a momentum transfer Q that connects parts of the Fermi surface exhibiting a sign-change in the superconducting order parameter [Δ(k) = −Δ(k + Q)].

Experimentally, the SRM is typically identified through the appearance of additional magnetic scattering in the superconducting state relative to the normal state, peaking at a well-defined energy Er and an intensity that tracks the superconducting order parameter8. While such behaviors of the SRM are consistent with the spin-exciton scenario, alternative explanations have also been proposed8,12,13,14,15,16. Moreover, phenomenologically similar enhanced scattering in the superconducting state have been observed in systems without sign-changing superconductivity, including phonons17,18 and hydrogen tunneling excitations19 in conventional superconductors and the resonant magnetic exciton mode in semiconducting rare-earth borides20,21, indicating mechanisms other than sign-changing superconductivity that could account for the experimental signatures of the SRM. Therefore, it is important to test whether experimentally observed SRMs are indeed spin-excitonic in nature, given the presence of a SRM is often used to evidence sign-changing unconventional superconductivity. This is underscored by recent measurements on CeCu2Si2 that demonstrated it exhibits nodeless superconductivity22,23,24,25, despite the observation of a SRM which suggests nodal d-wave superconductivity in the spin-exciton scenario26,27.

In this work, we use inelastic neutron scattering to systematically study the SRM in the prototypical heavy fermion superconductor CeCoIn5 (Tc = 2.3 K)28,29, which exhibits sign-changing $${d}_{{x}^{2}-{y}^{2}}$$-wave superconductivity similar to the cuprates30,31,32. Contrary to expectations for a spin-excitonic SRM with a prominent downward dispersion, our results show that the SRM in CeCoIn5 disperses upward without downward-dispersing features. Under applied magnetic field, the SRM splits into two upward-dispersing branches, with the dispersive features becoming progressively smeared out due to an increase in damping. Taken together, our results suggest that the SRM in CeCoIn5 is not spin-excitonic, and therefore is not a manifestation of the $${d}_{{x}^{2}-{y}^{2}}$$-wave superconducting order parameter. Instead, it likely results from the removal of damping to a pre-existing magnetic mode in a more strongly coupled unconventional superconductor. Our findings underscore the importance of more stringent tests to verify the spin-excitonic nature of SRMs, when using their presence to evidence sign-changing unconventional superconductivity.

## Results

### Dispersion of the SRM in CeCoIn5 at zero-field

In the spin-exciton scenario, the SRM is a bound state residing below the PHCO with $$E({\bf{Q}}) \, < \, \min (| \Delta ({\bf{k}})| +| \Delta ({\bf{k}}+{\bf{Q}})| )$$, resulting from a sign-change in the superconducting order parameter4,8. For cuprates with a $${d}_{{x}^{2}-{y}^{2}}$$-wave superconducting order parameter, the SRM peaks at the antiferromagnetic wavevector QAF = (0.5, 0.5), which connects hot spots that are close to the antinodal points of the $${d}_{{x}^{2}-{y}^{2}}$$-wave superconducting order parameter (Fig. 1a). As the SRM disperses away from QAF towards Qn, which connects the nodal points of the superconductivity order parameter, the PHCO is progressively pushed towards zero. The reduction of the PHCO away from QAF requires a spin-excitonic SRM to exhibit a downward dispersion away from QAF, so that it stays below the PHCO. Inelastic neutron scattering measurements of the SRM in hole-doped cuprates demonstrated that it dominantly disperses downwards, consistent with expectations of the spin-exciton picture (Fig. 1b)33,34,35,36,37,38. For iron pnictide superconductors with isotropic s±-wave superconducting gaps, the SRM is also consistent with being a spin-exciton4,5. Here, unlike the cuprates, the PHCO depends weakly on momentum Q, allowing spin-excitonic SRMs to exhibit upward dispersions, as observed in electron-doped39 and hole-doped compounds40.

In the prototypical heavy fermion superconductor CeCoIn5, like the cuprates, the superconducting order parameter is $${d}_{{x}^{2}-{y}^{2}}$$-wave30,31,32 and the SRM peaks around QAF in momentum and Er ≈ 0.6 meV in energy11; thus also like the cuprates, the PHCO is gradually suppressed moving from QAF towards Qn (Fig. 1c and d), resulting in a downward dispersion of the SRM in the spin-exciton scenario (Fig. 1d, see Supplementary Note 1 for details)27. Experimentally, however, the SRM is found to be dominated by a robust upward dispersion for EEr, contrary to expectations in the spin-exciton picture41,42. These upward-dispersing features and the strong L-dependence of the SRM in CeCoIn5 suggest it is a magnon-like mode, rather than a spin-exciton16,41. While the SRM in CeCoIn5 is dominated by an upward-dispersing branch for EEr, whether a downward-dispersing branch expected in the spin-exciton scenario also exists for E < Er, remains unclear.

To elucidate whether the SRM in CeCoIn5 has any downward-dispersing features, we carried out detailed inelastic neutron scattering measurements of the SRM in CeCoIn5 using PANDA, along the (HH, 0.5) direction for EEr ≈ 0.6 meV, with results shown in Fig. 2. The magnetic scattering at E = 0.375 meV is weaker in the superconducting state compared to the normal state (Fig. 2a), demonstrating a partial gapping of the magnetic fluctuations at this energy upon entering the superconducting state. With increasing energy, scattering in the superconducting state becomes more intense compared to the normal state (Fig. 2b–f), and the SRM can be clearly identified by such enhanced magnetic scattering. Constant-energy scans along (HH, 0.5) for E ≥ 0.4 meV (Fig. 2b–f) clearly reveal two peaks at Q = (0.5 ± δ, 0.5 ± δ, 0.5), in good agreement with previous work (see Supplementary Fig. 1 and Supplementary Note 2 for details)42. While the magnetic scattering for E = 0.375 meV appears to be a single peak, its broad width compared to higher energies suggests the magnetic scattering at this energy also consists of two peaks. By fitting the results in Fig. 2b–f using two Gaussian peaks at Q = (0.5 ± δ, 0.5 ± δ, 0.5), we find δ does not change significantly for E ≤ 0.45 meV (Fig. 2b–d) and increases monotonically with increasing energy for E > 0.45 meV (Fig. 3a), ruling out any downward-dispersing features. Combined with similar measurements for EEr obtained using multi-axis crystal spectrometer (MACS) (see Supplementary Figs. 2 and3 and Supplementary Note 3 for details), we find the SRM in CeCoIn5 disperses only upward, inconsistent with calculations for the spin-exciton scenario, based on an electronic structure from scanning tunneling microscopy measurements (Figs. 1d and 3a, see Supplementary Note 1 for details)31,43. Instead, the dispersion of the SRM resemble spin waves in CeRhIn5 (Fig. 3b)44,45, suggesting it to be magnon-like (see Supplementary Fig. 4 and Supplementary Note 2 for additional comparisons).

### Splitting of the dispersive SRM under applied magnetic field

For a spin-excitonic SRM that is isotropic in spin space, the application of a magnetic field should split it into a triplet in energy46. In CeCoIn5, the application of an in-plane magnetic field splits the SRM into a doublet, rather than a triplet47,48, likely due to the presence of magnetic anisotropy41,49. The doublet splitting of the SRM under applied field, combined with the upward dispersion, raises the question of how the dispersive features of the SRM in CeCoIn5 evolve with applied field, and whether the absence of a downward-dispersing branch is robust under applied field.

To address these questions, we studied the SRM in CeCoIn5 using MACS, under an applied magnetic field perpendicular to the [HHL] scattering plane, with results shown in Figs. 4 and 5. Constant-energy scans along (HH, 0.5) and (0.5, 0.5, L) directions in Fig. 4 reveal dramatic changes to the SRM away from QAF under applied magnetic field. For E = 0.5 meV and E = 0.6 meV, the SRM broadens upon increasing the magnetic field from B = 0 T to B = 6 T (Fig. 4a–d). On the other hand, for E = 0.8 meV and E = 1.0 meV, two split peaks around QAF are clearly seen at B = 0 T, while increasing the magnetic field to B = 3 T significantly reduces the splitting and only a single peak can be resolved at B = 6 T (Fig. 4e–h). We note that while the SRM is peaked slightly away from QAF for EEr at zero-field, as demonstrated in Fig. 2, the resolution of our MACS measurements is insufficient to resolve such a small splitting, instead a single peak at QAF is observed (Fig. 4a and c).

These disparate behaviors at different energies can be understood to result from the doublet splitting of the upward-dispersing SRM, as schematically depicted in Fig. 4i. The broadening of the peaks along (HH, 0.5) at E = 0.5 and 0.6 meV under applied field is due to a downward shift of the lower branch of the SRM, and increased damping resulting from the PHCO also moving to lower energies. For higher energies E = 0.8 and 1.0 meV, the intensity of magnetic scattering is dominated by the upper branch, and because the upper branch moves to higher energies under applied field, a reduction in peak splitting is observed. Our results indicate the dispersive SRM in CeCoIn5 splits into two branches under an in-plane magnetic field, while maintaining its upward-dispersing character. This conclusion is also supported by the analysis of peak splittings for the data in Fig. 5 (see Supplementary Fig. 5 and Supplementary Note 4 for details).

In addition to splitting the SRM into two upward-dispersing branches, energy-(HH, 0.5) and energy-(0.5, 0.5, L) maps in Fig. 5 and Supplementary Fig. 6 (obtained from data shown in Supplementary Figs. 3, 7 and 8, see Supplementary Note 3 for details) suggest that applied magnetic field also results in significant damping to the SRM in CeCoIn5 (see Supplementary Fig. 9 and Supplementary Note 4 for additional evidence from constant-Q scans). While the dispersive features can be clearly observed in the B = 0 T data (Fig. 5a and b), with applied field the dispersive features become less prominent for B = 4 T (Fig. 5c and d) and for B = 6 T no dispersive features can be resolved (Supplementary Fig. 6d and h). These results suggest that with applied field, the SRM becomes progressively damped and its dispersive character smeared out, becoming similar to overdamped magnetic excitations in the normal state, as the applied field approaches the upper critical field (see Supplementary Fig. 10 and Supplementary Note 3 for details). The increase in damping is unexpected in the spin-exciton scenario. This is because the SRM and the PHCO are shifted in energy in unison by an applied magnetic field, the SRM should therefore remain undamped (see Supplementary Fig. 11 and Supplementary Note 1 for details). Instead, the observed damping of the SRM with increasing field suggests that the SRM and the PHCO move independently with increasing magnetic field, consistent with the suggestion that the SRM in CeCoIn5 results from the removal of damping to a pre-existing magnetic mode in the superconducting state16,41, rather than being a spin-exciton.

## Discussion

Our results demonstrate the SRM in CeCoIn5 disperses upward, without downward dispersing features, inconsistent with expectations for a spin-exciton in a $${d}_{{x}^{2}-{y}^{2}}$$-wave superconductor. This suggests that either the superconducting order parameter in CeCoIn5 is not $${d}_{{x}^{2}-{y}^{2}}$$-wave, or that the SRM is not spin-excitonic. While nodeless s± superconductivity has been proposed for Pu-based 115 heavy-fermion superconductors50,51, there is strong experimental evidence for $${d}_{{x}^{2}-{y}^{2}}$$-wave superconductivity in CeCoIn5 with a robust nodal $${d}_{{x}^{2}-{y}^{2}}$$-wave superconducting order parameter30,31,32,41,52. Therefore, our findings indicate the SRM in CeCoIn5 is not spin-excitonic in origin, and as such, it is not a manifestation of the sign-changing $${d}_{{x}^{2}-{y}^{2}}$$-wave superconducting order parameter in CeCoIn5. More broadly, our results highlight that while SRMs in different unconventional superconductors exhibit similar experimental signatures, they may have distinct origins. When a SRM is spin-excitonic in origin, it evidences sign-changing superconductivity and provides information about the system’s electronic structure. On the other hand, if the SRM has a different origin, it may not be appropriate to use the observation of a SRM for these purposes. We note that while a spin-excitonic contribution to the SRM with intensity weaker than our detection limit cannot be ruled out, this does not affect our conclusion that the detectable SRM in CeCoIn5 is not spin-excitonic.

In the cuprates, X-shaped or Y-shaped excitations with dominant upward dispersing branches, which may result from either localized or itinerant electrons, have been observed38,53,54,55,56,57. However, these upward-dispersing excitations are different from what we have observed in CeCoIn5 in that they are already present in the normal state, and exhibits little or no change upon entering the superconducting state, i.e. they are not SRMs. When a SRM is present, as seen through additional magnetic scattering uniquely associated with the superconducting state, it is always dominated by a downward dispersion, as shown in Fig. 1b. While a weaker upward-dispersing branch of the SRM has also been detected in some cuprates35,36,37, these SRMs were shown to be consistent with spin-excitons residing in a different region of momentum space35,58, where the PHCO energy is large (region with Q < Qn in Fig. 1b). In CeCoIn5, based on the electronic structure extracted from scanning tunneling microscopy measurements31,43, it can be seen that while a similar region of the PHCO is present (Fig. 3a), it does not account for our experimentally determined dispersion in CeCoIn5. While the observation of a spin-excitonic SRM indicates sign-changing superconductivity, the absence of a SRM (as seen experimentally in sufficiently underdoped cuprates56) or the observed SRM not being spin-excitonic (as in CeCoIn5) does not invalidate sign-changing superconductivity, but simply means that in these cases information on the superconducting order parameter do not directly manifest in magnetic excitations.

In addition to demonstrating the SRM in CeCoIn5 disperses upward at zero-field, our results in Fig. 2 also show that the SRM is appropriately described by two peaks at Q = (0.5 ± δ, 0.5 ± δ, 0.5) for all energies. The splitting of the SRM for E < Er is suggested to evidence that the SRM is a precursor42,47,59 to the field-induced spin-density-wave phase (Q-phase) that orders at Q = (0.5 ± δQ, 0.5 ± δQ, 0.5)60,61. For E 0.45 meV, our data in Fig. 2 shows that δ ≈ 0.034, significantly smaller than δQ = 0.05. While CeCoIn5 is magnetically disordered, it can be tuned towards commensurate magnetic order at QAF through Cd-doping62, Rh-doping63, or Hg-doping64, as well as incommensurate magnetic order at Q = (0.5 ± δQ, 0.5 ± δQ, 0.5) through Nd-doping65 or applying magnetic field60. The proximity of CeCoIn5 to two types of magnetic orders indicates that fluctuations associated with both may be present in CeCoIn5, also suggested by two types of fluctuations unveiled by half-polarized neutron scattering experiments48. In such a scenario, the overlap of fluctuations at Q = (0.5 ± δQ, 0.5 ± δQ, 0.5) and QAF results in the observed δ < δQ for EEr (see Supplementary Fig. 1 and Supplementary Note 2 for details). Such a coexistence of two types of magnetic fluctuations has also been observed in FeTe66,67; in both CeCoIn5 and FeTe, it results from the quasi-degeneracy of different magnetic states.

While a SRM that is isotropic in spin space is expected to split into triplets in energy under applied field46, when an easy-plane magnetic anisotropy perpendicular to the field direction $$(1\bar{1}0)$$ is taken into consideration, it is possible to account for the doublet splitting in CeCoIn541,49. However, the SRM has been demonstrated to have an Ising character at zero-field, with the easy-axis along (001)42; therefore, to account for the doublet splitting of the SRM in CeCoIn5, it is necessary to consider modifications to the form of magnetic anisotropy under applied magnetic field, as demonstrated for magnetically ordered CeRhIn568. At zero-field, CeRhIn5 exhibits an easy-ab-plane spin anisotropy, an applied magnetic field along $$(1\bar{1}0)$$ induces an anomalously large additional easy-axis anisotropy along (110), driving the system to exhibit an easy-axis spin anisotropy along (110) overall. In the case of CeCoIn5, at zero-field it exhibits an easy-axis spin anisotropy along (001), and if an applied magnetic field along $$(1\bar{1}0)$$ also eases the spin anisotropy along (110) as in CeRhIn5, the system may be driven to overall exhibit an easy-plane-like spin anisotropy, with the easy-plane spanned by (001) and (110) (perpendicular to the applied field).

In conclusion, our detailed inelastic neutron scattering measurements indicate the SRM in CeCoIn5 disperses upward without any downward dispersing features, indicating it is not spin-excitonic in origin. Under an applied magnetic field, the SRM splits into two upward-dispersing branches and progressively loses its dispersive characters with increasing field, suggesting the SRM in CeCoIn5 results from the removal of damping to a pre-existing magnetic mode in the superconducting state. As such, our results suggest the SRM in CeCoIn5 is not a result of the sign-change in its superconducting order parameter. Our findings demonstrate SRMs observed in unconventional superconductors can have origins other than spin-excitonic, in which case their presence may not provide information on the superconducting order parameter.

## Methods

### Sample preparation and neutron scattering experimental setups

Single crystals of CeCoIn5 were prepared by the indium self-flux method69. Hundreds of CeCoIn5 single crystals with a total mass  ~1 g were co-aligned in the [HHL] scattering plane on aluminum plates using a hydrogen-free glue. Magnetic field is applied perpendicular to the scattering plane, along the $$(1\bar{1}0)$$ direction.

Neutron scattering experiments were carried out on the PANDA cold three-axes spectrometer at the Heinz Maier-Leibnitz Zentrum70 and the MACS at the NIST Center for Neutron Research. The inelastic neutron scattering experiments on PANDA used fixed kf = 1.3 Å−1. A sapphire filter is used before the monochromator and a Be filter cooled to 40 K is used before the sample. The monochromator has horizontal and vertical variable focusing mechanics, vertical focusing of the analyzer is fixed (variable focusing is not needed because the detector is a vertically placed 1 inch 3He tube) and horizontal focusing is variable. In the focused mode, variable focusings are adjusted depending on the neutron wavelength based on empirically optimized values. The inelastic neutron scattering measurements at MACS used Be filters both before and after the sample with fixed Ef = 3.0 meV or Ef = 3.7 meV. Most of measurements on MACS were made using the 20 spectroscopic detectors simultaneously to efficiently obtain the magnetic scattering within the [HHL] scattering plane. Constant-Q scans at QAF shown in Supplementary Fig. 9 were carried out using MACS with a single detector. The analyzers are vertically focused, while the monochromator is doubly focused.

### Data analysis

Data shown in Fig. 2 and Supplementary Fig. 1 are obtained using PANDA. The constant-energy scans were fit to a single Gaussian peak or two Gaussian peaks equally displaced from the center; scans at different energy transfers are fit globally with the same peak center. Constant-Q scans in Supplementary Fig. 9 are measured on MACS using a single detector. All the rest of neutron scattering data are obtained using MACS by measuring maps of large portions of the [HHL] scattering plane, simultaneously using the 20 detectors available at MACS. The maps of [HHL]-plane are folded into a single quadrant to improve statistics. Cuts along (HH, 0.5) were obtained by binning data with 0.37 ≤ L ≤ 0.63 and a step size of 0.025; cuts along (0.5, 0.5, L) are obtained by binning data with (0.42, 0.42) ≤ (HH) ≤ (0.58, 0.58) and a step size of 0.05. Normal state magnetic excitations measured at T = 2.5 K have been subtracted in all the MACS data except Supplementary Fig. 10. The cuts along (HH, 0.5) are fit with a single Gaussian peak centered at Q = (0.5, 0.5, 0.5) or two Gaussian peaks at Q = (0.5 ± δ, 0.5 ± δ, 0.5). The cuts along (0.5, 0.5, L) are fit using a lattice sum of a single Lorentzian peak centered at Q = (0.5, 0.5, 0.5) or a lattice sum of two Lorentzian peaks at Q = (0.5, 0.5, 0.5 ± δ). B = 4 T data are collected using Ef = 3.0 meV, while measurements at other fields used Ef = 3.7 meV. Using MACS we collected high statistics data for selected energies (Fig. 4 and Supplementary Fig. 2) and lower statistics data with finer energy steps (Fig. 5 and Supplementary Figs. 3, 68). The zero-field data shown in Fig. 4 and Supplementary Fig. 2 are reproduced from ref. 41, to compare with data under applied field.