A microscopic Kondo lattice model for the heavy fermion antiferromagnet CeIn3

Electrons at the border of localization generate exotic states of matter across all classes of strongly correlated electron materials and many other quantum materials with emergent functionality. Heavy electron metals are a model example, in which magnetic interactions arise from the opposing limits of localized and itinerant electrons. This remarkable duality is intimately related to the emergence of a plethora of novel quantum matter states such as unconventional superconductivity, electronic-nematic states, hidden order and most recently topological states of matter such as topological Kondo insulators and Kondo semimetals and putative chiral superconductors. The outstanding challenge is that the archetypal Kondo lattice model that captures the underlying electronic dichotomy is notoriously difficult to solve for real materials. Here we show, using the prototypical strongly-correlated antiferromagnet CeIn3, that a multi-orbital periodic Anderson model embedded with input from ab initio bandstructure calculations can be reduced to a simple Kondo-Heisenberg model, which captures the magnetic interactions quantitatively. We validate this tractable Hamiltonian via high-resolution neutron spectroscopy that reproduces accurately the magnetic soft modes in CeIn3, which are believed to mediate unconventional superconductivity. Our study paves the way for a quantitative understanding of metallic quantum states such as unconventional superconductivity.

and quantum spin liquid states [26], including the highly sought-after fractional quasiparticles [27]. Despite this continued relevance, quantitative predictions for real materials based on Kondo lattice models remain a formidable computational hurdle.
Metals containing cerium are firmly established model systems for the interplay between itinerant and localized electronic degrees of freedom and are ideal candidates to make progress on this issue. A phenotypical case is CeIn 3 , whose phase diagram as a function of temperature T and hydrostatic pressure p is shown in Fig. 1a [28]. The formation of well-localized magnetic moments occurs due to Ce 4f orbitals that are buried close to the nuclei. A weak hybridization with conduction electron bands leads to a long-range magnetic exchange interaction between the moments known as the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction [29-31] (see Fig. 1b). By increasing the strength of the hybridization, one can screen the magnetic moment through the Kondo effect, leading to a strongly renormalized electronic density of states near the Fermi energy. This heavy Fermi-liquid state borders the magnetically ordered state, from which it is separated by a magnetic quantum phase transition (QPT) that can be accessed via an external control parameter (here: pressure). Interestingly, novel quantum states (here: superconductivity [32]) emerge generically in the vicinity of the QPT suggesting that they are mediated by the associated magnetic quantum critical fluctuations [3]. To understand this emergence near QPTs in a quantitative manner requires a materials-specific microscopic model that incorporates relevant interactions to account for the magnetically ordered state and the resulting fluctuations.
Both for experiment and theory, the challenge in understanding real materials is the extreme energy resolution (∼meV) required to capture the inherently small energy scales that emerge in the renormalized electronic state (see Fig. 2a). On the non-magnetic side of the QPT, a pioneering time-of-flight (TOF) neutron spectroscopy study on a selected material with a relatively large Kondo interaction (≈ 60 meV) has only recently achieved quantitative agreement with dynamical mean field theory (DMFT) [33]. In contrast, for the magnetic (or superconducting) state of interest here, the relevant energy scale is typically of the order of meV, making this a formidable issue. Materials-specific theoretical investigations of emergent phenomena in f -electron materials [34,35] are often limited by the difficulty of validating low-energy effective models derived from complex high-energy input. Consequently, studies of such emergent states of matter are generally restricted to oversimplified models of real materials. Experimentally, in addition to the demand for energy resolution, the piv-electrons ( k,s ) and the hybridization between f and conduction electron states (V k,mσs ) are obtained from a tight-binding fit to the ab initio bandstructure calculation (see Methods and Supplementary Information for details). We find it important to include 18 conduction electron orbitals per spin (9 In-p, 3 In-s, 5 Ce-d and 1 Ce-s) to account fully for the electronic structure of the conduction bands near the Fermi level E F , and to obtain well localized f -orbitals. The spin-orbit coupling and crystal fields of CeIn 3 allow that, out of the 14 f states, only the Γ 7 ground state doublet needs to be included, which is separated by 12 meV from the Γ 8 quartet excited state [39]. Hence, the energy of the Γ 7 state ( f Γ 7 ) is the one remaining free parameter in our model.
wheref † i,σ creates an f -electron in the lowest energy Γ 7 doublet state (σ = {↑, ↓}) of site i, S ν q is the Fourier transform of the effective spin-1/2 operator S ν i (see Methods). The Kondo coupling is J ks,k s i,σσ = 1 N e i(k−k )·r iṼ k,σsṼ * k ,σ s /(E F − ε f Γ 7 ) whereṼ k,σs is the hybridization projected to the Γ 7 doublet. H KH contains effectively only the two conduction-electron bands that are close to the Fermi energy (the sum is restricted to band states within a cut-off Λ = 0.5 eV: | k,s − E F | ≤ Λ and | k ,s − E F | ≤ Λ; cf. top of Fig. 2b), as opposed to the 18 conduction bands in the MO-PAM. The remaining bands have been integrated out by including all fourth-order particle-hole and particle-particle processes that give rise to two types of magnetic interactions between the f -moments in I µν q (last term of Eq. 2). Notably, in addition to short-ranged superexchange involving particle-hole and particle-particle processes with excited states outside the cut-off Λ (I SE q ), long-ranged interactions arise from fourth order particle-particle processes with excited states inside the cut-off (Î (pp) q ) (cf. Fig. 2).
This derivation provides the microscopic justification for including a Heisenberg term in model studies of the Kondo lattice (e.g. [40,41]). As we shall see below, it also plays an important role in understanding the magnetically ordered state of CeIn 3 . Importantly, this minimal Kondo-Heisenberg model still contains the materials-specific information through the dispersion relation of the conduction electrons and the Heisenberg exchange coupling.
To validate this new minimal Kondo-Heisenberg model for CeIn 3 against experiments, and to illustrate the importance of the different contributions to magnetic interactions identified above, we further derive the RKKY Hamiltonian from the Kondo lattice term (first two terms of Eq. (2)) via an additional Schrieffer-Wolf transformation (see Supplementary   Information) [42,43]. The resulting effective spin Hamiltonian is The effective exchange interactionĨ q is then a sum of the RKKY interaction I RKKY q , the superexchange I SE q and the particle-particle contributionÎ (pp) q . It turns out to be practically isotropic,Ĩ µν q ≈ δ µνĨq , as a consequence of the weak influence of the spin-orbit interaction on the f − c hybridization amplitudes and the suppression of the f 2 magnetic virtual states.
In Fig. 2b, we show all resulting contributions along the path RΓXMΓ. For comparison with experiment, we note that the antiferromagnetic order of CeIn 3 below a Néel temperature T N = 10 K is characterized by a magnetic propagation vector q AFM = 1 2 , 1 2 , 1 2 [44] corresponding to the R point (cf. Figs. 1 and 2). Crucially, as shown in Fig. 2b, the RKKY interaction (blue line) typically thought to mediate magnetic order in Kondo lattice materials, has a global minimum near Γ = (0, 0, 0) and only a local minimum at R, demonstrating that it is not responsible for the onset of the observed order. Instead, the superexchange I SE q (purple line) exhibits a global minimum at the R point. Finally, the particle-particle contribution near E F (Î (pp) q , pink line) is mostly flat, but is characterized by a pronounced, cusplike maximum at Γ. It is then essential to note that the global minimum of the net interactionĨ q (bold cyan line) at the ordering wavevector R arises from the combination of two effects: a compensation between low-energy particle-particle and particle-hole contributions around the Γ point and the short-range antiferromagnetic superexchange interaction generated by the high-energy processes in both channels.
To go beyond the magnetic ground state and to test the theoretically calculated exchange interactionĨ q quantitatively against experiment, we have carried out high-resolution neutron spectroscopy to measure the dispersion of the magnons in CeIn 3 , which is determined by the magnetic interactions derived above. Notably, the dispersion is given by Ĩ q AFM −Ĩ q AFM +q with S = 1/2 and is shown as the solid blue line in Fig. 3a, which presents an overview of our results at T = 1.8 K. The key signature predicted by our model is a extremely dispersive magnon around the R point arising due to the steep local minimum of the long-ranged RKKY contribution; this was not identified in a previous study with modest resolution by Knafo et al., who instead reported a large magnon gap of more than 1 meV [39]. In contrast, the magnon dispersion determined from our neutron spectroscopy data is in quantitative agreement with the calculated dispersion (see Fig. 3a). Figure 4 showcases the steep magnon dispersion in a small region around the R point, which could only be uncovered by the most modern spectrometers (see Methods). Here panels a1, b1, and c1 show slices through the R point along the three cubic high-symmetry directions and confirm that the magnon gap is either absent or substantially smaller than the experimental resolution (∆E = 106 µeV) as we show in detail in the Supplementary Information.
A single fit parameter, f Γ 7 = 12.195 eV, reproduces the experimental dispersion both with regard to the bandwidth and the magnon velocity. This showcases that our microscopic model, which starts with an ab initio bandstructure calculation with energy scales of 10 eV, is able to predict magnetic interactions on the order of meV.
Finally, we consider the role of short-range Heisenberg superexchange. Although the longrange RKKY interaction is widely credited with mediating magnetic order in Kondo lattice materials [3], short-range superexchange interactions are commonly employed to fit the observed magnon dispersion [45][46][47][48][49][50] and have been also used for CeIn 3 [39]. As expected shortrange superexchange allows us to model the magnon dispersion at high energy and the zone boundary well (cf. Fig. 3). Besides providing a microscopic justification for including shortrange superexchange interactions, our model also demonstrates that they are not sufficient to explain the magnons near the magnetic zone center. We quantify this statement via the dimensionless parameter η that describes ratio of the magnon velocity v to the magnon bandwidth W . The experimental magnon bandwidth is W exp = 2. 69(2) meV, as determined from the energy cuts in Fig .3b. The experimental magnon velocity, v exp = 38.0(7) meV/r.l.u., was inferred from linear fits to the measured dispersion near to the R point along the different high-symmetry directions and averaged over all directions (black dashed lines in Fig. 4 and Supplementary Information). The parameter η exp = 2.25(4) derived from our experiments compares favorably with η MO-PAM = 1.92(6) computed from our model. In contrast, fits of the magnon dispersion with a single exchange constant J 1 result in η J 1 = 0.58, which stems from a substantial underestimation of the magnon velocity near the R point (cf. light green line in Fig. 4). We note that even adding a large number of additional higher-order exchange terms does not allow the large observed magnon velocity to be modelled accurately (see Supplementary Information), as was noted in previous studies [45,49].
Our study demonstrates that it is now possible to derive a minimal microscopic Kondo lattice model for a real material starting from ab initio electronic bandstructure calculations starting at energy scales of eV. This enables us to reproduce quantitatively the magnetic order and all relevant magnetic interactions in the prototypical Kondo lattice CeIn 3 at energy scales 10,000 times smaller. Thus, it provides a tractable path for the accurate calculation of the low-energy spin excitations that arise due to strong electronic correlations, and are believed to mediate the emergence of novel quantum phases (cf. Fig. 1). Due to the longrange nature of the RKKY interaction, these magnetic soft modes are remarkably steep.
This is also borne out by neutron scattering studies on further f -electron materials, which highlight the broad relevance of our results. Either steep magnon dispersions were observed directly [45, 49,51] or this key feature is concealed due to inadequate resolution, resulting in reports of potentially spurious magnon gaps [46, 50, 52] similar to CeIn 3 [39]. We note that the ability to resolve these steep low-energy spin excitations also ushers in high-resolution TOF spectroscopy as a novel technique to verify the electronic band structure of magnetically ordered heavy-fermion materials.
A further remarkable insight revealed by our calculations is that, in addition to the RKKY interaction, contributions from the particle-particle channel are equally crucial to quantitative description of the magnetic order and the magnon spectrum. In turn, our approach resolves several puzzles concerning magnetic order in heavy-fermion materials.
First, it is consistent with neutron scattering studies of the magnon spectrum of various heavy fermion-materials, where the dispersion towards the zone boundary is well explained using short-range interactions [45][46][47][48][49][50]. Second, the presence of short-range interactions highlights why a large collection of heavy fermion materials exhibit commensurate AFM order [44,46,[53][54][55], even though RKKY interactions arising from generic Fermi surfaces typically favor incommensurate order. Finally, the apparent lack of 4f -based metallic ferromagnets is now understood quite simply by the presence of particle-particle interactions in metals, which generically disfavor the ferromagnetic ground state.
In summary, the combination of our MO-PAM approach with TOF spectroscopy establishes a straightforward recipe to obtain a quantitative, yet relatively simple, effective Kondo-Heisenberg Hamiltonian, which for CeIn 3 includes only 2+1 orbitals (two conduction electron bands and one f level). In turn, studying this effective model as a function of pressure has the promising prospect of establishing the emergence of unconventional superconductivity in CeIn 3 quantitatively. Notably, the short-range interaction promotes a local-moment magnetic state relative to the bare Kondo temperature, which may fundamentally alter the nature of the magnetic QPT observed under pressure [56]. Adding the Γ 8 f -state to our calculation should equally allow us to reproduce the magnetic anisotropy emerging as a function of magnetic field [57]. Considering the ever-increasing computational power available, our approach is equally in reach for more complex materials with lower symmetry and more orbitals, which will allow us to unlock the microscopic understanding of a large number of quantum matter states with functional properties [3]. Further, our discovery that both short-and long-range interactions are key to understanding magnetic order in heavy fermion materials offers a straightforward explanation of their rich magnetic phase diagrams, where changing the balance of interactions by applying external tuning parameters allows us to select distinct magnetic ground states. Similarly, we anticipate that the combination of short-and long-ranged interactions will turn out to be a generic feature of metallic systems whose starting point is the periodic Anderson model. Finally, beyond a quantitative understanding of the ground states of strongly correlated electron systems, our study paves the way for ab initio modeling of all quantum systems described by Kondo lattices.

Multi-Orbital Periodic Anderson Model
The input parameters for the multi-orbital periodic Anderson model (MO-PAM) presented in Eq. (1) specific to CeIn 3 were obtained by deriving a tight-binding model based on density functional theory (DFT) that accurately captures the electronic structure of the conduction bands near the Fermi level E F , and yields the hybridization between these bands and the 4f -orbitals. The underlying DFT band structure calculations were performed using the Quantum ESPRESSO package [58] and fully relativistic projector augmented-wave (PAW) pseudopotentials with the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional, which are available in PSlibrary [59]. A realistic tight-binding Hamiltonian with 50 Wannier functions (25 orbitals times 2 for spin) was constructed using the Wannier90 package [60]. Details for all steps are provided in the which includes the crystal field coefficients B m,m and the intra-atomic spin-orbit coupling λ (ζ m σ ;mσ = δ m ,m δ σ ,σ mσ/2 + δ m ,m+σ δ σ ,−σ 12 − m(m + σ)/2). The resulting Γ 7 doublet reads By taking the limit of infinite intra-atomic f -f Coulomb repulsion which eliminates the f 2 configurations, we project H MO-PAM into the low-energy subspace generated by the Γ 7 doublet and obtain the periodic Anderson model where the constrained operatorsf † i,σ (f i,σ ) create (annihilate) an f -electron in the Γ 7 doublet with σ = {↑, ↓} andf † i,σf † i,σ = 0, f Γ 7 is the energy of the Γ 7 states, andṼ k,σs is the hybridization between the Γ 7 doublet and the conduction electron states (1 ≤ s ≤ 36).
By treating the small hybridizationṼ k,σs as a perturbation, the periodic Anderson model Neutron Spectroscopy Inelastic neutron scattering was carried out at the cold neutron chopper spectrometer CNCS at ORNL [61]. For the measurements, the crystal array was oriented in such a way that the crystallographic 110 axis was vertical. Momentum transfers of neutrons are given in the reference frame of the sample by means of Q = Hb 1 +Kb 2 +Lb 3 , where H, K, and L denote the Miller indices and b ν = 2π aâ ν (ν = 1, 2, 3) represent primitive translation vectors of the reciprocal cubic lattice (a = 4.689Å). Throughout the manuscript and the SI, when stating components of momentum transfers the reciprocal lattice unit (1 · r.l.u. := 2π a ) is omitted. To a given momentum transfer Q (upper-case letter), the reduced momentum transfer that equals the equivalent reciprocal space position in the cell 0 ≤ q ν < 1, is given by q = q 1 b 1 + q 2 b 2 + q 3 b 3 (labelled by a lower-case letter). ∆q denotes the modulus of ∆q. Reciprocal-space distances are also given in units 1 · r.l.u. := 2π a . Time-of-flight neutron spectroscopy was performed with two different incident neutron energies. High-resolution experiments with incident neutron energy E i,1 = 3.315 meV (∆E = 106 µeV) permitted the study of the steep magnon dispersion in the vicinity of the reciprocal space position Q 0 = − 1 2 , − 1 2 , 1 2 , which represents the R point. A "high-energy" setting with incident neutron energy E i,2 = 12 meV was performed to determine the magnon dispersion across the entire Brillouin zone. Data were collected in terms of a socalled Horace scan [62], where the crystal is rotated around the vertical axis. In combination with the CNCS detector, which has large horizontal (from -50 to 140 degrees) and vertical between the Q-positions (−1, −1, 1) and − 1 2 , − 1 2 , 1 2 , between − 1 2 , − 1 2 , 1 2 and (0, 0, 0), between (1, 1, 0) and 1, 1, 1 2 , between −1, − 1 2 , 1 and − 1 2 , − 1 2 , 1 , and between 1 2 , 1 2 , 0 and (1, 1, 0), respectively. The intensity was integrated within a distance of ±0.17 r.l.u. along the two Q-directions that are perpendicular to the path γ =RΓXMΓ. For the energy cuts shown Fig. 3b the intensity was integrated within a distance of ±0.17 r.l.u. along γ. For the reduced momentum transfer cuts and slices shown in Fig. 4, the intensity was integrated within a distance of 0.08 r.l.u. along the two reciprocal space directions that are perpendicular to the reduced momentum transfer q. For the cuts in the lower panels, intensity was integrated over energies ±0.1 meV.
We note that to resolve the sharp magnon dispersion high momentum resolution is required in addition to extreme energy resolution. Although triple-axis neutron spectroscopy is often credited with the best combined momentum-energy-transfer resolution, it fails to properly identify the dispersion even when used with the best resolution due to so-called Currat-Axe spurions [63]. To demonstrate this we have carried out additional measurements on the multiplexing triple axis spectrometer CAMEA at Paul Scherrer Institute [64] using the identical sample and incident energies E i = 4.15 meV, 4.8 meV, and 5.5 meV. Details of these measurements and the spurious features close to the zone center are shown in the Supplementary Information. This highlights the importance of modern TOF spectroscopy for the investigation of strongly correlated metals.
Data Availability The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Author information
The authors declare no competing financial interests. Correspondence and requests for materials should be addressed to M. Janoschek (marc.janoschek@psi.ch). . It is also oscillatory in nature as we illustrate for a spherical Fermi-surface with radius k F (right panel), which results in a period λ ≈ π/k F in real space (bottom panel). However, as we demonstrate in Fig. 2, magnetic order on a Kondo lattice is not only mediated by RKKY interactions but additionally requires short-range superexchange and fourth-order particle-particle interactions. The Kondo interaction contains the spin exchange interaction with the conduction electrons (blue dashed line), while the Heisenberg interaction is a combination of short-range interactions arising from completely filled or completely unoccupied conduction bands (purple p-like orbital between the local moments) and a long range interaction due to particle-particle processes not captured through the Kondo exchange (right process in the MO-PAM sketch, and pink wavy line in the KHM sketch). This Hamiltonian encapsulates the physics of heavy-fermion materials such as CeIn 3 . Finally, the low-energy conduction electrons can be integrated out to yield an effective spin Hamiltonian between local f -moments, which contains both the long-range RKKY interaction from the Kondo exchange (blue wavy line), and the other contributions explained above. b Results derived for CeIn 3 using this approach are shown. The upper panel shows the electronic bandstructure that serves as input for the MO-PAM. The cut-off Λ = 0.5 eV allows short-range superexchange and long-ranged interactions to be seperated (see text for details). The lower panel shows the various contributions to the net magnetic interactionĨ q in CeIn 3 shifted by the indicated energies for visibility. The color of each contribution denotes the corresponding process of the same color in a. (S1b) By taking the limit of infinite intra-atomic f -f Coulomb repulsion which eliminates the f 2 configurations, we project H MO-PAM into the low-energy subspace generated by the Γ 7 doublet and obtain the following periodic Anderson model where the constrained operatorsf † i,σ (f i,σ ) create (annihilate) an f -electron in the Γ 7 doublet with σ = {↑, ↓} andf † i,σf † i,σ = 0, f Γ 7 is the energy of the Γ 7 states, andṼ k,σs is the hybridization between the Γ 7 doublet and the conduction electron states (1 ≤ s ≤ 36). s is a combined spin and orbital index for the conduction electrons.
By treating the small hybridizationṼ k,σs as a perturbation, the periodic Anderson model can be reduced to an effective Kondo lattice model, which appears to second order in the hybridization, plus an effective exchange interaction between the Γ 7 doublets that is generated to fourth order in the hybridization, which is necessary to properly account for the magnetic interactions as detailed below: where the local Hilbert space is constrained by the condition σf † i,σf i,σ = 1, which implies that the f -electrons are localized in their orbitals and their effective spin σ is the only remaining f -degree of freedom. It is important to note that we have only retained the conduction electron states whose distance to the Fermi level is smaller than a cut-off Λ for both the hopping and the Kondo coupling terms (indicated in the constrained sums). The Kondo interaction is then given by where we have approximated the energy of the virtual processes k,s − f Γ 7 by E F − f Γ 7 . The Heisenberg Hamiltonian (H Heis ) includes all magnetic interactions between the Γ 7 states generated through a fourth order degenerate perturbation theory of the periodic Anderson model Eq. (S2), which are not captured in the KLM of Eq. (S4). This includes all particleparticle (pp) processes at all energies (Fig. S3), which are not included in the Kondo model [42], and particle-hole (ph) processes that involve at least one virtual excited state outside S3. Diagrams of the virtual processes f 1 → f 0 → f 1 in the particle-hole a and particleparticle b channels. Dashed lines indicate the energy cut-off that restricts the excited states that contribute to the RKKY interactions (ph within the cut-off Λ), the superexchange (ph and pp involving at least one virtual state outside the cut-off), and a long-range interactionK (pp) (pp within the cut-off). where and H spin can also be derived directly from the periodic Anderson model using fourth order perturbation theory. In doing so, H spin includes all the ph and pp fourth order processes in the hybridizationṼ k,σs , which are depicted in Fig. S3. The contributions from the ph channel, shown in Fig. S3a, produce an effective interaction where B stands for the first Brillouin zone. While the contributions from the pp channel, shown in Fig. S3b, produce an effective coupling The Fourier transform of these effective interactions is: (S12) In order to separate the contributions to H RKKY and H Heis , we must split both interactions into two terms, Consequently, the sum of these three terms are defined as our Heisenberg contribution, and must be added to our low energy effective model Eq. (S3) if we aim to accurately account for magnetic interactions.
To derive the single-magnon dispersion, it is convenient to express the effective spin Hamiltonian in terms of the the pseudo spin-1 2 operators S i ≡ 1 2f † i,α σ αβfi,β : The effective spin-spin interaction turns out to be practically isotropic because of the absence of f 2 virtual states and the presence of the cubic symmetry: withĨ q = I RKKY q + I q . Indeed, our numerical evaluation confirms thatĨ q ≡Ĩ xx q =Ĩ yy q =Ĩ zz q . Furthermore, the terms {Ĩ xy q ,Ĩ yz q ,Ĩ zx q } that correspond to the cubic anisotropies are found to be much smaller than 0.01 meV.  Fig. S4(a) illustrates the similarity, and hence, the validity in the approximation used to derive the Kondo lattice model. Fig. S4

The Heisenberg term is
. The effective superexchange interaction I SE q ≡Ǐ (ph) q +Ǐ (pp) q involves at least one virtual excited state outside the energy interval defined by the cut-off Λ.
In the calculation, the position of the Fermi level, E F = 12.588 eV, is fixed by the number of non-f electrons per unit cell. The only free parameter of the theory is the diagonal energy f Γ 7 of the Γ 7 doublet. By fitting the bandwidth of the measured single-magnon dispersion we obtain f Γ 7 ≈ 12.26 eV for the effective spin-spin interaction derived from the PAM (usinĝ I (ph) q ) and f Γ 7 ≈ 12.195 eV for the effective spin-spin interaction derived from the KLM (using I RKKY q ), which is the case presented in the main text.

II. EXPERIMENTAL METHODS
In the following, we elaborate on the experimental methods. After a description of the sample preparation, details of the neutron-spectroscopy experiments are presented.
A. Sample preparation Figure S5 shows a photograph of the CeIn 3 sample-mosaic on the aluminum holder. The mosaic spread with respect to the (110) rotation axis of the single-crystal pieces around the ideal orientation, which is indicated schematically on the right, is within ±1.4 deg rotation angle. Shown in b is an examplary Laue picture that indicates the high quality of one of the single-crystal pieces.
FIG. S5. a Photograph of the CeIn 3 single-crystal mosaic. The crystallographic directions along that the crystals were coaligned are indicated on the right. b X-ray Laue diffraction pattern of one of the used CeIn 3 single-crystal samples illustrating the high quality. The data were recorded with the X-ray beam parallel to the (001) crystallographic direction.

B. Neutron spectroscopy experiments
To determine the dispersion of magnetic excitations in CeIn 3 at T = 1.8 K, inelastic neutron scattering (INS) experiments were performed using the time-of-flight technique at the cold neutron chopper spectrometer (CNCS) at Oak Ridge National Laboratory (ORNL) [61].
The orientation of the CeIn 3 sample mosaic for the experiment was such that the crystallographic direction (110) was vertical and (110) as well as (001) were in the horizontal scattering plane of CNCS (see Fig. S4).
The CNCS spectrometer was operated in the following two complementary instrumental settings: • In the first setting, which we refer to as high-energy setting, the incident neutron energy was E i,1 = 12 meV (wave vector K i,1 = 2.406Å −1 ) and the choppers of CNCS were operated in the high-flux mode with 180 Hz double-disk rotation frequency.
• In the second setting, which we refer to as high-resolution setting, the incident neutron energy was E i,2 = 3.315 meV (wave vector K i,2 = 1.265Å −1 ) and the choppers were operated in the high-flux mode with 300 Hz double-disk rotation frequency.
The first setting provides an overview of the magnetic excitations in a large area of reciprocal space and up to high energy transfers, whereas the second setting permits high resolution measurements of the magnon dispersion in the vicinity of the momentum transfer Q 0 = − 1 2 , − 1 2 , 1 2 , which corresponds to probing excitations at the magnetic propagation vector q AFM = 1 2 , 1 2 , 1 2 [44] located at the R point. For the mapping of the excitation spectrum, data were recorded at 1.8 K (foreground) and at 20 K (background), i.e., in the magnetically ordered state and well above the ordering temperature. To determine the imaginary part of the dynamic susceptibility, χ (Q, E), the background was subtracted from the foreground, and the resulting magnetic intensity was corrected by the Bose-factor at 1.8 K and divided by the square of the magnetic form factor of Ce 3+ as well as by |K f | / |K i |, whereby K i and K f denote the wavevectors of incident and scattered neutrons, respectively. Neglecting the Debye-Waller factor as well as the polarization factor in the neutron scattering cross section, which depends on the microscopic direction of magnetic moments, the corrected data are equal to the imaginary part of the dynamic susceptibility apart from a constant prefactor (cf. Ref. [72]).
For the data sets in the high-energy and high-resolution setting, respectively, the angular coverage of sample rotation amounted to 33 deg and to 50 deg with a step size of 1 deg.
Data reduction was done with the software-package HORACE [62].  −1, 1). b In the high-resolution setting the accessible range in reciprocal space is restricted to the close vicinity around the magnetic Bragg peak position (−0.5, −0.5, 0.5). Both panels show background-subtracted neutron scattering intensity that was integrated over the energy range ∆E = ±0.1 meV.

III. MAGNON DISPERSION INFERRED FROM EXPERIMENTAL DATA
In the following, we present a detailed account of the magnetic excitations in CeIn 3 as derived from our INS experiments. At first, we show neutron spectroscopy data that are complementary to the data in the main text. Subsequently, the excitations at the R point are studied in detail and the presence of a putative gap, previously reported by Knafo et al.
[39], is discussed. Finally, the trajectory of the magnetic dispersion throughout the Brillouin zone is inferred from our data.

B. Magnetic excitations at the R point
The INS data recorded in the high-resolution setting reveal a relatively steep slope of the magnon dispersion emerging from the magnetic Bragg peak at q AFM = 1 2 , 1 2 , 1 2 , as presented in Fig. 4 of the main text. As elaborated on in the following, the data indicate further, that a gap at the R point is either absent or smaller than the energy resolution of CNCS. In particular, the data pose an upper bound on the size of a putative gap at the R point.
The instrumental energy resolution may be inferred from the incoherent scattering intensity at a Q-position where elastic magnetic Bragg scattering is absent. The energy-dependent foreground intensity at Q 1 = − 1 2 , − 1 2 , 1 2 − 0.1 , which is obtained by integration of timeof-flight intensity over a cuboid Q-volume centered at Q 1 and extended over a distance of ±0.018 r.l.u. along the directions (110), (110), and (001) displays a Gaussian profile around the elastic line given by: The width of the profile, which is characterised by σ = 45 µeV and a full width at half maximum FWHM = 2 √ 2 ln 2σ = 106 µeV, essentially represents the resolution of CNCS in the high-resolution setting at zero energy transfer. The value is close to resolution of CNCS reported in literature (cf. Ref. [61,75]) and considerably smaller than 196 µeV, which is the FWHM for incoherent Vanadium scattering estimated by the Violini -algorithm by means of the software Takin (see Ref. [76]).
The R point was investigated at the momentum transfer Q 0 = − 1 2 , − 1 2 , 1 2 . Figure S8a presents background-subtracted INS data as a function of energy transfer E at the position Q 0 , where the elastic magnetic Bragg peak is located, as well as at the shifted position Q 1 , where magnetic intensity is absent. The magnetic Bragg peak at Q 0 as a function of energy, E, displays a Gaussian profile with the same width as the instrumental resolution. The magnon dispersion intersects due to its steep slope with the integration dome of Q 0 up to energies around 1 meV (see Fig. S8b).
The onset of substantial scattering intensity at energy transfers right above the elastic Bragg peak indicates that the dispersion is gapless within the energy resolution of CNCS.

C. Trajectory of Magnon Dispersion
The trajectory in reciprocal space of the magnon dispersion was inferred from INS data from the high-energy and high-resolution setting, as explained in the following.
Sufficiently far away from the points R and Γ and in major parts of the Brillouin zone, the  Close to the R point, where the magnetic ordering vector is located, the gapless magnetic excitation dispersion displays a steep slope and the trajectory is visible in cuts at constant energy transfers through the high-resolution INS data set. Figs. S10, S11, and S12 present constant-energy cuts for momentum transfers around Q 0 = − 1 2 , − 1 2 , 1 2 . For the cuts at finite energies, i.e., E > 0, the data were corrected by the Bose-factor and the resulting  symmetric around the center q AFM , i.e., they appear at values +q and −q, but possibly with different spectral weights.
At the lowest energies, the separation of intersection points is smaller than the FWHM of the experimental momentum-transfer-resolution (purple shading) and the cuts are well fitted by a single-Gaussian profile (orange shading) denoted 1G. For higher energies the
Magnon velocities at the R point, i.e., q AFM = 1 2 , 1 2 , 1 2 , were determined by linear fits to high-resolution INS data with momentum around Q 0 = − 1 2 , − 1 2 , 1 2 and compared with computations. At lowest energies, where the distance of the two intersection points in constant-energy cuts as a function of ∆q = |q − q AFM | (see Figs. S10, S11, and S12) is smaller than the FWHM of the instrumental resolution, an estimate for the magnon velocity is obtained from the resolution. For the cubic space diagonal RΓ, the splitting is smaller than the experimental resolution for energies E ≤ 0.6 meV, which imposes a conservative lower bound on the magnon velocity: where we used 1 r.l.u. ≈ 1.340Å −1 for CeIn 3 at ambient pressure.
We assume a gapless magnon dispersion and determine the magnon velocities by linear weighted least-squares fits to the data points inferred from constant-energy cuts at energies E = 0.8 meV, 1 meV, .., 1.6 meV. We performed four fits, where we considered the N = 2, diagonal, the same energies E = 0.8 meV, 1 meV, .., 1.6 meV were considered and fits were carried out for the N = 2, 3, ..., 5 data points with lowest energies. In turn, for the cubic edge, constant-energy cuts at E = 0.6 meV, 0.8 meV, .., 1.6 meV were considered and fits were carried out for the N = 2, 3, ..., 6 data points with lowest energies.
The magnon velocities obtained from the fits as well as the mean values for the three reciprocal space directions are presented in Tab. S1. The velocity averaged over the three directions RΓ, RX, and RM is given by v ∆q = (38.0 ± 0.7) meV/ r.l.u.. The size of the bandwidth is well reproduced by fits with Heisenberg models. The bandwidths of the J 1 fit, J 1 -J 2 fit, J 1 -J 2 -J 3 fit, and J 1 -J 2 -J 3 -J 4 fit on the path RΓXMΓ are given by 3.00 meV, 3.24 meV, 2.95 meV, and 3.11 meV, respectively. Here J ν denotes the Heisenberg exchange connecting the νth nearest neighbor pair.

VI. RATIO OF MAGNON VELOCITY AND BANDWIDTH
The steep dispersion at the R point manifests itself in the extremely large ratio of magnon velocity to bandwidth, which is typically expressed in dimensionless units. The ratio as inferred from experimental data amounts to: Calculations that are based on MO-PAM result in a similarly large value given by: η MO-PAM = v ∆q W MO-PAM · 1 · r.l.u. 2π = 1.92 ± 0.06.
In contrast, finite series expansions using exchange constants up to a few nearestneighbors typically feature substantially smaller values. The corresponding fits are shown in section VII. The J 1 fit, J 1 -J 2 fit, J 1 -J 2 -J 3 fit, and J 1 -J 2 -J 3 -J 4 fit lead to η = 1 √ 3 ≈ 0.58, η = 0.56, η = 0.88, and η = 0.88 (for the ratio of magnon velocity at R in direction RΓ to bandwidth on the path RΓXMΓ), respectively.

VII. FIT OF EXPERIMENTAL DATA BY HEISENBERG MODELS
The magnon dispersion on the path RΓXMΓ, which is presented in Fig. 3 of the main text, was fitted with low-order nearest neighbour Heisenberg models. To reproduce the bandwidth of measured magnons, the models were fitted by means of least-squares statistics to the 14 data-points that were recorded in the high-energy setting.

VIII. CURRAT-AXE SPURIONS AND STEEP MAGNON DISPERSIONS
We now illustrate why triple-axis spectroscopy is not well-suited to investigate the steep dispersion that we observe in CeIn 3 . This is due to spurious scattering that is caused by neutrons scattered incoherently at the monochromator or analyzer crystals of a triple-axis spectrometer. When the spectrometer is set to a non-zero energy transfer, i.e., |K i | = |K f |, but the angle between K i and K f as well as the orientation of the sample correspond to a geometry that allows for Bragg scattering from the sample, the incoherently scattered neutrons from the monochromator that have a wave vector |K i,inc | = |K f | lead to the observation of accidental Bragg scattering. These scattering events, which may also alternatively FIG. S13. Fit of high-energy experimental data by Heisenberg models. The neutron scattering data obtained via the high-energy setting (a) and the high-resolution setting (a and b) are compared with fits to a J 1 model, J 1 -J 2 model, J 1 -J 2 -J 3 model, and J 1 -J 2 -J 3 -J 4 , respectively, as explained in the text.
arise from neutrons incoherently scattered at the analyzer, result in spurious intensity that appears as a dispersive feature close to a magnetic zone center with a linear dispersion. In the literature, this is well-known as Currat-Axe spurion [63].
To show this, we have repeated our experiment on CeIn 3 previously carried out on CNCS at ORNL, using the multiplexing triple-axis spectrometer CAMEA [64]. For each of the three incident neutron energies, spectroscopy data were recorded with two different positions of the multiplexing detector, namely 2θ = −45 deg and −41 deg, and with different angles of vertical sample rotation covering an angular range of 60 deg with a step size of 0.5 deg.
The branches of Currat-Axe spurions originating from the monochromator as well as from the analyzer were calculated by means of the software MJOLNIR [79] and are shown in Fig. S14. Note that these lines disperse in close vicinity of the real magnon dispersion, that is, however, substantially weaker. In the constant-energy cut shown in b the Currat-Axe spurions from incoherent analyzer scattering result in a peak at around L = 0.47 r.l.u.
(Gaussian at smaller L). The signal in the center (Gaussian at larger L indicated by a black arrow) may be superposition of magnon scattering as well as Currat-Axe spurions arising from incoherent scattering at the monochromator.
FIG. S14. Triple axis spectroscopy and Currat-Axe spurions Shown in a is neutron spectroscopy intensity around momentum transfers (0.5, 0.5, 0.5), which represents the R-point. Background recorded at 20 K were subtracted from the foreground data recorded at 2 K. INS intensity was integrated within a reciprocal space distance of ±0.09 r.l.u. along the axis (110). The location of Currat-Axe spurions from the analyzer crystal and from the monochromator are indicated by solid and interrupted lines, respectively. The spurions were calculated for incident neutron energies 4.2 meV (cyan color), 4.8 meV (yellow color), and 5.5 meV (black), respectively. The blue dotted line shows the trajectory of the magnon dispersion resulting from MO-PAM calculations. Shown in b is an exemplary Q-cut at constant energy 0.65 meV. INS intensity was integrated over a reciprocal space distance ±0.035 r.l.u. parallel to the axis (110) and over energies between 0.5 meV and 0.8 meV. The binning for the (001) direction amounts to 0.02 r.l.u.. The data were fitted with a superposition of two Gaussian profiles. The peak at smaller L arises from spurious scattering from the analyzer, whereas the peak in the center (black arrow) may be a superposition of spurious scattering from the monochromator and of magnons.

IX. PREVIOUS EXPERIMENTAL STUDIES OF MAGNETIC EXCIATIONS IN CEIN 3
The magnetic excitations in CeIn 3 were previously characterised by Knafo et al. [39] using triple-axis spectroscopy. The data, which are presented in Fig. S15, are qualitatively in agreement with our study in major parts of the Brillouin zone. In particular, the recorded dispersion features the same bandwidth that we identified in our study. In the vicinity of Γ and in the vicinity of R, the energy resolution of IN22 was not sufficient to resolve the extremely steep excitation dispersion.