Nature of the spin resonance mode in CeCoIn5

Spin-fluctuation-mediated unconventional superconductivity can emerge at the border of magnetism, featuring a superconducting order parameter that changes sign in momentum space. Detection of such a sign-change is experimentally challenging, since most probes are not phase-sensitive. The observation of a spin resonance mode (SRM) from inelastic neutron scattering is often seen as strong phase-sensitive evidence for a sign-changing superconducting order parameter, by assuming the SRM is a spin-excitonic bound state. Here we show that for the heavy fermion superconductor CeCoIn5, its SRM defies expectations for a spin-excitonic bound state, and is not a manifestation of sign-changing superconductivity. Instead, the SRM in CeCoIn5 likely arises from a reduction of damping to a magnon-like mode in the superconducting state, due to its proximity to magnetic quantum criticality. Our findings emphasize the need for more stringent tests of whether SRMs are spin-excitonic, when using their presence to evidence sign-changing superconductivity.

fermion superconductor CeCoIn 5 , its SRM defies expectations for a spin-excitonic bound state, and is not a manifestation of sign-changing superconductivity. Instead, the SRM in CeCoIn 5 likely arises from a reduction of damping to a magnon-like mode in the superconducting state, due to its proximity to magnetic quantum criticality. Our findings emphasize the need for more stringent tests of whether SRMs are spin-excitonic, when using their presence to evidence sign-changing superconductivity.
Understanding the physics of unconventional superconductors, which include cuprate, ironbased, 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 interactions [1][2][3] . The proximity to magnetically ordered states in these materials suggests spin fluctuations as a common thread that can pair electrons in unconventional superconductors [3][4][5] . Unlike phonon-mediated conventional superconductors with superconducting order parameters Δ(k) that depend weakly on momentum k, spinfluctuation-mediated superconductivity requires a Δ(k) that changes sign in momentum space 4 . 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 experiments 6,7 , 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 angleresolved photoemission, can probe the magnitude of the superconducting order parameter and its momentum dependence 7 , 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 parameter 4,[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 E r and an intensity that tracks the superconducting order parameter 8 .
While such behaviors of the SRM are consistent with the spin-exciton scenario, alternative explanations have also been proposed 8,[12][13][14][15][16] . Moreover, phenomenologically similar enhanced scattering in the superconducting state have been observed in systems without sign-changing superconductivity, including phonons 17,18 and hydrogen tunneling excitations 19 in conventional superconductors and the resonant magnetic exciton mode in semiconducting rare-earth borides 20,21 , 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 CeCu 2 Si 2 that demonstrated it exhibits nodeless superconductivity [22][23][24][25] , despite the observation of a SRM which suggests nodal d-wave superconductivity in the spin-exciton scenario 26,27 .
In this work, we use inelastic neutron scattering to systematically study the SRM in the prototypical heavy fermion superconductor CeCoIn 5 (T c = 2.3 K) 28,29 , which exhibits signchanging d x 2 − y 2-wave superconductivity similar to the cuprates [30][31][32] . Contrary to expectations for a spin-excitonic SRM with a prominent downward dispersion, our results show that the SRM in CeCoIn 5 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 CeCoIn 5 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.

Dispersion of the SRM in CeCoIn 5 at zero-field.
In the spin-exciton scenario, the SRM is a bound state residing below the PHCO with E(Q) < min(|Δ(k)| + |Δ(k + Q)|), resulting from a sign-change in the superconducting order parameter 4,8 . For cuprates with a d x 2 − y 2-wave superconducting order parameter, the SRM peaks at the antiferromagnetic wavevector Q AF = (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 Q AF towards Q n , which connects the nodal points of the superconductivity order parameter, the PHCO is progressively pushed towards zero. The reduction of the PHCO away from Q AF requires a spin-excitonic SRM to exhibit a downward dispersion away from Q AF , 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-exciton 4,5 . Here, unlike the cuprates, the PHCO depends weakly on momentum Q, allowing spin-excitonic SRMs to exhibit upward dispersions, as observed in electron-doped 39 and hole-doped compounds 40 .
In the prototypical heavy fermion superconductor CeCoIn 5 , like the cuprates, the superconducting order parameter is d x 2 − y 2-wave 30-32 and the SRM peaks around Q AF in momentum and E r ≈ 0.6 meV in energy 11 ; thus also like the cuprates, the PHCO is gradually suppressed moving from Q AF towards Q n ( 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 E ≳ E r , contrary to expectations in the spin-exciton picture 41,42 . These upward-dispersing features and the strong L-dependence of the SRM in CeCoIn 5 suggest it is a magnon-like mode, rather than a spin-exciton 16,41 . While the SRM in CeCoIn 5 is dominated by an upward-dispersing branch for E ≳ E r , whether a downward-dispersing branch expected in the spin-exciton scenario also exists for E < E r , remains unclear.
To elucidate whether the SRM in CeCoIn 5 has any downward-dispersing features, we carried out detailed inelastic neutron scattering measurements of the SRM in CeCoIn 5 using PANDA, along the (H, H, 0.5) direction for E ≲ E r ≈ 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 (H, H, 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

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 energy 46 . In CeCoIn 5 , the application of an in-plane magnetic field splits the SRM into a doublet, rather than a triplet 47,48 , likely due to the presence of magnetic anisotropy 41,49 . 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 CeCoIn 5 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 CeCoIn 5 Fig. 4 reveal dramatic changes to the SRM away from Q AF 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 Q AF 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 Q AF for E ≲ E r 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 Q AF 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 (H, H, 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 CeCoIn 5 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-(H, H, 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 CeCoIn 5 (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 over-damped 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 CeCoIn 5 results from the removal of damping to a pre-existing magnetic mode in the superconducting state 16,41 , rather than being a spin-exciton.

Discussion
Our results demonstrate the SRM in CeCoIn 5 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 CeCoIn 5 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 superconductors 50,51 , there is strong experimental evidence for d x 2 − y 2-wave superconductivity in CeCoIn 5 with a robust nodal d x 2 − y 2-wave superconducting order parameter [30][31][32]41,52 . Therefore, our findings indicate the SRM in CeCoIn 5 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 CeCoIn 5 . 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 CeCoIn 5 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 observed 38,[53][54][55][56][57] . However, these upward-dispersing excitations are different from what we have observed in CeCoIn 5 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 cuprates [35][36][37] , these SRMs were shown to be consistent with spin-excitons residing in a different region of momentum space 35,58 , where the PHCO energy is large (region with |Q| < |Q n | in Fig. 1b). In CeCoIn 5 , based on the electronic structure extracted from scanning tunneling microscopy measurements 31,43 , 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 CeCoIn 5 . While the observation of a spin-excitonic SRM indicates sign-changing superconductivity, the absence of a SRM (as seen experimentally in sufficiently underdoped cuprates 56 ) or the observed SRM not being spin-excitonic (as in CeCoIn 5 ) 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 CeCoIn 5 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 < E r is suggested to evidence that the SRM is a precursor 42,47,59 to the field-induced spin-density-wave phase (Q-phase) that orders at Q = (0.5 ± δ Q , 0.5 ± δ Q , 0.5) 60,61 . For E ≲ 0.45 meV, our data in Fig. 2 shows that δ ≈ 0.034, significantly smaller than δ Q = 0.05. While CeCoIn 5 is magnetically disordered, it can be tuned towards commensurate magnetic order at Q AF through Cddoping 62 , Rh-doping 63 , or Hg-doping 64 , as well as incommensurate magnetic order at Q = (0.5 ± δ Q , 0.5 ± δ Q , 0.5) through Nd-doping 65 or applying magnetic field 60 . The proximity of CeCoIn 5 to two types of magnetic orders indicates that fluctuations associated with both may be present in CeCoIn 5 , also suggested by two types of fluctuations unveiled by halfpolarized neutron scattering experiments 48 . In such a scenario, the overlap of fluctuations at Q = (0.5 ± δ Q , 0.5 ± δ Q , 0.5) and Q AF results in the observed δ < δ Q for E ≲ E r (see Supplementary Fig. 1 and Supplementary Note 2 for details). Such a coexistence of two types of magnetic fluctuations has also been observed in FeTe 66,67 ; in both CeCoIn 5 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 field 46 , when an easy-plane magnetic anisotropy perpendicular to the field direction (110) is taken into consideration, it is possible to account for the doublet splitting in CeCoIn 5 41,49 . However, the SRM has been demonstrated to have an Ising character at zerofield, with the easy-axis along (001) 42 ; therefore, to account for the doublet splitting of the SRM in CeCoIn 5 , it is necessary to consider modifications to the form of magnetic anisotropy under applied magnetic field, as demonstrated for magnetically ordered CeRhIn 5 68 . At zero-field, CeRhIn 5 exhibits an easy-ab-plane spin anisotropy, an applied magnetic field along (110) 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 CeCoIn 5 , at zero-field it exhibits an easy-axis spin anisotropy along (001), and if an applied magnetic field along (110) also eases the spin anisotropy along (110) as in CeRhIn 5 , 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 CeCoIn 5 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 upwarddispersing 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 CeCoIn 5 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.

Sample preparation and neutron scattering experimental setups.
Single crystals of CeCoIn 5 were prepared by the indium self-flux method 69 . Hundreds of CeCoIn 5 single crystals with a total mass ~1 g were co-aligned in the [H, H, L] scattering plane on aluminum plates using a hydrogen-free glue. Magnetic field is applied perpendicular to the scattering plane, along the (110) direction.
Neutron scattering experiments were carried out on the PANDA cold three-axes spectrometer at the Heinz Maier-Leibnitz Zentrum 70 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 constantenergy 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. 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, 6-8). The zero-field data shown in Fig. 4 and Supplementary Fig. 2 are reproduced from ref. 41 , to compare with data under applied field.

Supplementary Material
Refer to Web version on PubMed Central for supplementary material.  Blues squares are data at T = 0.45 K, well below T c = 2.3 K. Red circles are data at T = 2.5 K, just above T c . Solid blue lines are fits to two Gaussian peaks centered at (0.5 ± δ, 0.5 ± δ, 0.5) for data in the superconducting state, except for E = 0.375 meV, which is fit to a single Gaussian peak. Solid red lines are fits to a single Gaussian peak for data in the normal state.