Abstract
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, electronicnematic 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 stronglycorrelated antiferromagnet CeIn_{3}, that a multiorbital periodic Anderson model embedded with input from ab initio bandstructure calculations can be reduced to a simple KondoHeisenberg model, which captures the magnetic interactions quantitatively. We validate this tractable Hamiltonian via highresolution neutron spectroscopy that reproduces accurately the magnetic soft modes in CeIn_{3}, which are believed to mediate unconventional superconductivity. Our study paves the way for a quantitative understanding of metallic quantum states such as unconventional superconductivity.
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Introduction
The Kondo lattice model has been key in qualitatively demonstrating how a myriad of correlated quantum matter states emerge^{1,2,3,4,5,6,7,8} from the interplay of local and itinerant electrons^{8,9,10}. Beyond the strongly correlated electron materials for which this archetypal model was conceived, it applies to a growing list of novel quantum systems with potential for applications including the electronic transport through quantum dots^{11}, voltagetunable magnetic moments in graphene^{12}, magnetism in twistedbilayer graphene^{13} and in twodimensional organometallic materials^{14}, the electronic structure in layered narrowelectronicband materials^{15}, electronic resonances of Kagome metals^{16}, metallic spin liquid states^{17,18,19} that may even be of chiral character^{20}, skyrmions in centrosymmetric magnets^{21,22} and fully tunable electronic quasiparticles in semiconductor moiré materials^{23}. Further, Kondo lattice models have been used to study flatband materials^{24} and predict novel topological states such as topological superconductivity^{25} and quantum spin liquid states^{26}, including the highly soughtafter 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 prototypical 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 welllocalized 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 longrange magnetic exchange interaction between the moments known as the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction^{29,30,31} (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 Fermiliquid 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^{8}. To understand this emergence near QPTs in a quantitative manner requires a materialsspecific 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 nonmagnetic side of the QPT, a pioneering timeofflight (TOF) neutron spectroscopy study^{33}, complemented by a subsequent resonant inelastic Xray (RIXS) scattering measurement^{34}, on a selected material with a relatively large Kondo interaction (≈60 meV) has only recently achieved quantitative agreement with dynamical mean field theory (DMFT). 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. Materialsspecific theoretical investigations of emergent phenomena in felectron materials^{35,36} are often limited by the difficulty of validating lowenergy effective models derived from complex highenergy 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 pivotal requirement is stateoftheart momentumtransfer resolution. Notably, the longrange nature of RKKY exchange in real space (Fig. 1b) universally results in extremely sharp magnetic excitations in momentum space, as we elucidate below.
Here we illustrate an approach to reduce a multiorbital periodic Anderson model (MOPAM) imbued with materialsspecific hopping parameters to a minimal KondoHeisenberg lattice model that can form the basis for understanding emergent states of matter in various Kondo lattice materials (Fig. 2a). Using controlled fourthorder perturbation theory and the high combined momentumtransfer and energy resolution of the latestgeneration TOF spectrometers, we resolve the full magnetic interaction over the entire range of relevant length scales. We thereby demonstrate that we can determine accurately the emergent energy scales four orders of magnitude smaller than the bare parameters of the initial MOPAM. In doing so, we critically evaluate the assumptions made to date about heavyfermion materials.
Results
Due to a simple cubic structure that facilitates numerical calculations and its characteristic Doniach phase diagram as a function of pressure p (Fig. 1a), CeIn_{3} is ideally suited to revisit the role of magnetic interactions on a Kondo lattice. As the MOPAM was originally conceived to explain the localization of shielded electron shells^{37} and, in turn, offers a realistic theoretical treatment of coupled charge and spin degrees of freedoms in correlated metals, we begin with a 25orbital PAM for CeIn_{3},
where \({c}_{{{{{{{{\bf{k}}}}}}}},s}^{{{{\dagger}}} }\) (c_{k,s}) is the creation (annihilation) operator of band electrons with wavevector k and band index s, which includes the spin index. \({f}_{i,m\sigma }^{{{{\dagger}}} }\) (f_{i,mσ}) denotes the creation (annihilation) operator of an felectron state with l_{z} = m (−3 ≤ m ≤ 3) and spin σ on lattice site i. The matrix elements \({h}_{{m}^{{\prime} }{\sigma }^{{\prime} };m\sigma }^{{{{{{{{\rm{SI}}}}}}}}}\) of the singleion felectron Hamiltonian include the crystalfield coefficients and intraatomic spinorbit coupling (see Methods for details). For CeIn_{3} and related materials, it is well established that the f^{2} state is energetically considerably less favorable than the f^{0} excited state^{38,39}. Thus, we assume that the onsite repulsive interaction between felectrons is infinitely large, implying that f^{2} configurations are excluded from the Hilbert space: \({f}_{i,m\sigma }^{{{{\dagger}}} }\,{f}_{i,{m}^{{\prime} }{\sigma }^{{\prime} }}^{{{{\dagger}}} }=0\). The dispersion of the conduction electrons (ϵ_{k,s}) and the hybridization between f and conduction electron states (V_{k,mσs}) are obtained from a tightbinding 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 Inp, 3 Ins, 5 Ced and 1 Ces) to account fully for the electronic structure of the conduction bands near the Fermi level E_{F}, and to obtain well localized forbitals. The spinorbit 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^{40}. Hence, the energy of the Γ_{7} state (\({\epsilon }_{{{{\Gamma }}}_{7}}^{f}\)) is the one remaining free parameter in our model. We further demonstrate in the Supplementary Information that our ab initio band structure calculation is in agreement with the electronic structure determined via angleresolved photo emission spectroscopy (ARPES) experiments^{41}.
As the hybridization between the conduction electrons and the felectrons is small, we can use degenerate perturbation theory (see Supplementary Information for details) to derive an effective KondoHeisenberg Hamiltonian,
where \({\tilde{f}}_{i,\sigma }^{{{{\dagger}}} }\) creates an felectron in the lowest energy Γ_{7} doublet state (σ = {↑, ↓}) of site \(i,{S}_{{{{{{{{\bf{q}}}}}}}}}^{\nu }\) is the Fourier transform of the effective spin1/2 operator \({S}_{i}^{\nu }\) (see Methods). The Kondo coupling is \({J}_{i,\sigma {\sigma }^{{\prime} }}^{{{{{{{{\bf{k}}}}}}}}s,{{{{{{{{\bf{k}}}}}}}}}^{{\prime} }{s}^{{\prime} }}=\frac{1}{N}{e}^{i({{{{{{{\bf{k}}}}}}}}{{{{{{{{\bf{k}}}}}}}}}^{{\prime} })\cdot {{{{{{{{\bf{r}}}}}}}}}_{i}}{\tilde{V}}_{{{{{{{{\bf{k}}}}}}}},\sigma s}{\tilde{V}}_{{{{{{{{{\bf{k}}}}}}}}}^{{\prime} },{\sigma }^{{\prime} }{s}^{{\prime} }}^{*}/({E}_{F}{\varepsilon }_{{{{\Gamma }}}_{7}}^{f})\) where \({\tilde{V}}_{{{{{{{{\bf{k}}}}}}}},\sigma s}\) is the hybridization projected to the Γ_{7} doublet. \({{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{KH}}}}}}}}}\) contains effectively only the two conductionelectron bands that are close to the Fermi energy (the sum \({\sum }^{{\prime} }\) is restricted to band states within a cutoff Λ = 0.5 eV: ∣ϵ_{k,s} − E_{F}∣ ≤ Λ and \( {\epsilon }_{{{{{{{{{\bf{k}}}}}}}}}^{{\prime} },{s}^{{\prime} }}{E}_{F} \le {{\Lambda }}\); cf. top of Fig. 2b), as opposed to the 18 conduction bands in the MOPAM. The remaining bands have been integrated out by including all fourthorder particlehole and particleparticle processes that give rise to two types of magnetic interactions between the fmoments in \({I}_{{{{{{{{\bf{q}}}}}}}}}^{\mu \nu }\) (last term of Eq. (2)). Notably, in addition to shortranged superexchange involving particlehole and particleparticle processes with excited states outside the cutoff Λ (\({I}_{{{{{{{{\bf{q}}}}}}}}}^{{{{{{{{\rm{SE}}}}}}}}}\)), longranged interactions arise from fourthorder particleparticle processes with excited states inside the cutoff (\({\hat{I}}_{{{{{{{{\bf{q}}}}}}}}}^{({{{{{{{\rm{pp}}}}}}}})}\)) (cf. Fig. 2). This derivation provides the microscopic justification for including a Heisenberg term in model studies of the Kondo lattice (e.g.^{42,43}). As we shall see below, it also plays an important role in understanding the magnetically ordered state of CeIn_{3}. Importantly, this minimal KondoHeisenberg model still contains the materialsspecific information through the dispersion relation of the conduction electrons and the Heisenberg exchange coupling.
To validate this new minimal KondoHeisenberg 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)^{44,45}. The resulting effective spin Hamiltonian is
The effective exchange interaction \({\tilde{I}}_{{{{{{{{\bf{q}}}}}}}}}\) is then a sum of the RKKY interaction \({I}_{{{{{{{{\bf{q}}}}}}}}}^{{{{{{{{\rm{RKKY}}}}}}}}}\), the superexchange \({I}_{{{{{{{{\bf{q}}}}}}}}}^{{{{{{{{\rm{SE}}}}}}}}}\) and the particleparticle contribution \({\hat{I}}_{{{{{{{{\bf{q}}}}}}}}}^{\,({{{{{{{\rm{pp}}}}}}}})}\). It turns out to be practically isotropic, \({\tilde{I}}_{{{{{{{{\bf{q}}}}}}}}}^{\mu \nu }\approx \,{\delta }_{\mu \nu }{\tilde{I}}_{{{{{{{{\bf{q}}}}}}}}}\), as a consequence of the weak influence of the spinorbit 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. The inset on the upper right corner of Fig. 2b shows the position of these high symmetry points in the Brillouin zone that define this path. 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 \({{{{{{{{\bf{q}}}}}}}}}_{{{{{{{{\rm{AFM}}}}}}}}}=\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\)^{46} 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 \({I}_{{{{{{{{\bf{q}}}}}}}}}^{{{{{{{{\rm{SE}}}}}}}}}\) (purple line) exhibits a global minimum at the R point. Finally, the particleparticle contribution near E_{F} (\({\hat{I}}_{{{{{{{{\bf{q}}}}}}}}}^{({{{{{{{\rm{pp}}}}}}}})}\), 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 interaction \({\tilde{I}}_{{{{{{{{\bf{q}}}}}}}}}\) (bold cyan line) at the ordering wavevector R arises from the combination of two effects: a compensation between lowenergy particleparticle and particlehole contributions around the Γ point and the shortrange antiferromagnetic superexchange interaction generated by the highenergy processes in both channels.
To go beyond the magnetic ground state and to test the theoretically calculated exchange interaction \({\tilde{I}}_{{{{{{{{\bf{q}}}}}}}}}\) quantitatively against experiment, we have carried out highresolution 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 \({E}_{{{{{{{{\bf{q}}}}}}}}}=S\sqrt{\left({\tilde{I}}_{{{{{{{{{\bf{q}}}}}}}}}_{{{{{{{{\rm{AFM}}}}}}}}}}{\tilde{I}}_{{{{{{{{\bf{q}}}}}}}}}\right)\left({\tilde{I}}_{{{{{{{{{\bf{q}}}}}}}}}_{{{{{{{{\rm{AFM}}}}}}}}}}{\tilde{I}}_{{{{{{{{{\bf{q}}}}}}}}}_{{{{{{{{\rm{AFM}}}}}}}}}+{{{{{{{\bf{q}}}}}}}}}\right)}\) 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 longranged 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^{40}. In contrast, the magnon dispersion determined from our neutron spectroscopy data as well as the observed magnetic intensity are in quantitative agreement with the calculated dispersion and intensity, respectively (see Fig. 3). Here the neutron intensities in Fig. 3 are expressed as the dynamic magnetic susceptibility \({\tilde{\chi }}^{{\prime\prime} }({{{{{{{\bf{Q}}}}}}}},\, E)\) and were converted to absolute units of \({\mu }_{{{{{{{{\rm{B}}}}}}}}}^{2}{{{{{{{{\rm{meV}}}}}}}}}^{1}\) for comparison to theory. In particular, the energy and momentum transfer cuts through the magnon spectrum shown in Figs. 3c and d, respectively, demonstrate the excellent agreement between experiment and theory. 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 highsymmetry directions and confirm that the magnon gap is either absent or substantially smaller than the experimental resolution (ΔE = 106 μeV) as we show in detail in the Supplementary Information. A single fit parameter, \({\epsilon }_{{{{\Gamma }}}_{7}}^{f}=12.009\,{{{{{{{\rm{eV}}}}}}}}\), 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. In addition to being quantitatively accurate, our calculations are robust against small changes of the chemical potential of the order of 10 meV (resolution of our band structure calculation), which modify the slope and bandwidth of the magnon dispersion by less than 1
Finally, we consider the role of shortrange Heisenberg superexchange. Although the longrange RKKY interaction is widely credited with mediating magnetic order in Kondo lattice materials^{8}, shortrange superexchange interactions are commonly employed to fit the observed magnon dispersion^{47,48,49,50,51,52} and have been also used for CeIn_{3}^{40}. 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 \({W}_{\exp }=2.75(3)\,{{{{{{{\rm{meV}}}}}}}}\), as determined from the energy cuts in Fig. 3b. The experimental magnon velocity, \({v}_{\exp }=38.6(8)\,{{{{{{{\rm{meV/r.l.u.}}}}}}}}\), was inferred from linear fits to the measured dispersion near to the R point along the different highsymmetry directions and averaged over all directions (black dashed lines in Fig. 4 and Supplementary Information). The parameter \({\eta }_{\exp }=2.23(6)\) derived from our experiments compares favorably with η_{MO−PAM} = 2.27(5) computed from our model. In contrast, fits of the magnon dispersion with a single exchange constant J_{1} result in \({\eta }_{{J}_{1}}=0.58\), 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 higherorder exchange terms does not allow the large observed magnon velocity to be modeled accurately (see Supplementary Information), as was noted in previous studies^{47,51}.
Discussion
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 lowenergy 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 felectron materials, which highlight the broad relevance of our results. Either steep magnon dispersions were observed directly^{47,51,53} or this key feature is concealed due to inadequate resolution, resulting in reports of potentially spurious magnon gaps^{48,52,54} similar to CeIn_{3}^{40}. We note that the ability to resolve these steep lowenergy spin excitations also ushers in highresolution TOF spectroscopy as a complementary technique to access the electronic band structure of magnetically ordered heavyfermion materials.
A further remarkable insight revealed by our calculations is that, in addition to the RKKY interaction, contributions from the particleparticle 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 heavyfermion materials. First, it is consistent with neutron scattering studies of the magnon spectrum of various heavy fermionmaterials, where the dispersion towards the zone boundary is well explained using shortrange interactions^{47,48,49,49,50,51,52}. Second, the presence of shortrange interactions highlights why a large collection of heavy fermion materials exhibit commensurate AFM order^{46,48,55,56,57}, even though RKKY interactions arising from generic Fermi surfaces typically favor incommensurate order. Finally, the apparent lack of 4fbased metallic ferromagnets is now understood quite simply by the presence of particleparticle interactions in metals, which generically disfavor the ferromagnetic ground state.
In summary, the combination of our MOPAM approach with TOF spectroscopy establishes a straightforward recipe to obtain a quantitative, yet relatively simple, effective KondoHeisenberg 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 shortrange interaction promotes a localmoment magnetic state relative to the bare Kondo temperature, which may fundamentally alter the nature of the magnetic QPT observed under pressure^{58}. Adding the Γ_{8} fstate to our calculation should equally allow us to reproduce the magnetic anisotropy emerging as a function of magnetic field^{59}. Considering the everincreasing 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^{8}. Further, our discovery that both short and longrange 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 longranged interactions will turn out to be a generic feature of metallic systems whose starting point is the periodic Anderson model. Finally, our study paves the way for ab initio modeling of quantum systems described by Kondo lattices and therefore suggests an avenue beyond a phenomenological description of the ground states of their strongly correlated electron systems.
Methods
Multiorbital periodic Anderson model
The input parameters for the multiorbital periodic Anderson model (MOPAM) presented in Eq. (1) specific to CeIn_{3} were obtained by deriving a tightbinding 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 4forbitals. The underlying DFT band structure calculations were performed using the QUANTUM ESPRESSO package^{60} and fully relativistic projector augmentedwave (PAW) pseudopotentials with the PerdewBurkeErnzerhof (PBE) exchangecorrelation functional, which are available in PSlibrary^{61}. A realistic tightbinding Hamiltonian with 50 Wannier functions (25 orbitals times 2 for spin) was constructed using the Wannier90 package^{62}. Details for all steps are provided in the Supplementary Material.
The lowtemperature magnetic properties of CeIn_{3} at zero field are dominated by the lowenergy Ce 4f Γ_{7} doublet that results from diagonalizing the singlesite felectron Hamiltonian [second term of Eq. (1)],
which includes the crystal field coefficients \({B}_{m,{m}^{{\prime} }}\) and the intraatomic spinorbit coupling λ (\({\zeta }_{{m}^{{\prime} }{\sigma }^{{\prime} };m\sigma }={\delta }_{{m}^{{\prime} },m}{\delta }_{{\sigma }^{{\prime} },\sigma }m\sigma /2+{\delta }_{{m}^{{\prime} },m+\sigma }{\delta }_{{\sigma }^{{\prime} },\sigma }\sqrt{12m(m+\sigma )}/2\)). The resulting Γ_{7} doublet reads
By taking the limit of infinite intraatomic ff Coulomb repulsion which eliminates the f^{2} configurations, we project \({{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{MO}}}}}}}}{{{{{{{\rm{PAM}}}}}}}}}\) into the lowenergy subspace generated by the Γ_{7} doublet and obtain the periodic Anderson model
where the constrained operators \({\tilde{f}}_{i,\sigma }^{{{{\dagger}}} }\) (\({\tilde{f}}_{i,\sigma }\)) create (annihilate) an felectron in the Γ_{7} doublet with σ = {↑, ↓} and \({\tilde{f}}_{i,\sigma }^{{{{\dagger}}} }{\tilde{f}}_{i,{\sigma }^{{\prime} }}^{{{{\dagger}}} }=0,\, {\epsilon }_{{{{\Gamma }}}_{7}}^{f}\) is the energy of the Γ_{7} states, and \({\tilde{V}}_{{{{{{{{\bf{k}}}}}}}},\sigma s}\) is the hybridization between the Γ_{7} doublet and the conduction electron states (1 ≤ s ≤ 36).
By treating the small hybridization \({\tilde{V}}_{{{{{{{{\bf{k}}}}}}}},\sigma s}\) as a perturbation, the periodic Anderson model can be reduced to the effective Kondo–Heisenberg Hamiltonian shown in Eq. (2). Furthermore, for strongly localized felectrons, the Kondo lattice model can be further reduced to the RKKY Hamiltonian via second order degenerate perturbation theory in the Kondo interaction (see Supplementary Information for details). The final effective spin Hamiltonian \({{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{spin}}}}}}}}}\) is presented in Eq. (3), where the pseudo spin\(\frac{1}{2}\) operator is defined as \({{{{{{{{\bf{S}}}}}}}}}_{i}\equiv \frac{1}{2}{\tilde{f}}_{i,\alpha }^{{{{\dagger}}} }{{{{{{{{\boldsymbol{\sigma }}}}}}}}}_{\alpha \beta }{\tilde{\,f}}_{i,\beta }\) (σ^{ν} are the Pauli matrices with ν = x, y, z), and its Fourier transform is defined as \({S}_{{{{{{{{\bf{q}}}}}}}}}^{\nu }=\frac{1}{N}{\sum }_{i}{e}^{i{{{{{{{\bf{q}}}}}}}}\cdot {{{{{{{{\bf{r}}}}}}}}}_{i}}{S}_{i}^{\nu }\). N is the total number of Ce atoms on the lattice (we take N → ∞ in the following calculation).
Sample preparation
To overcome the high absorption of cold neutrons by indium, platelike single crystalline samples of CeIn_{3} were grown by the indium selfflux method and polished to a thickness of around 0.7 mm. To maximize the total scattering intensity 24 pieces were carefully coaligned on an aluminum sampleholder using a hydrogenfree adhesive (CYTOP) (see Sec. II.A of the Supplementary Information).
Neutron spectroscopy
Inelastic neutron scattering was carried out at the cold neutron chopper spectrometer CNCS at ORNL^{63}. For the measurements, the crystal array was oriented in such a way that the crystallographic \(\left[1\overline{1}0\right]\) 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 \({{{{{{{{\bf{b}}}}}}}}}_{\nu }=\frac{2\pi }{a}{\hat{{{{{{{{\bf{a}}}}}}}}}}_{\nu }\) (ν = 1, 2, 3) represent primitive translation vectors of the reciprocal cubic lattice (a = 4.689 Å). Throughout the manuscript and the SI, when stating components of momentum transfers the reciprocal lattice unit (\(1\cdot {{{{{{{\rm{r.l.u.}}}}}}}}:=\frac{2\pi }{a}\)) is omitted. To a given momentum transfer Q (uppercase 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} (labeled by a lowercase letter). Δq denotes the modulus of Δq. Reciprocalspace distances are also given in units \(1\cdot {{{{{{{\rm{r.l.u.}}}}}}}}:=\frac{2\pi }{a}\).
Timeofflight neutron spectroscopy was performed with two different incident neutron energies. Highresolution experiments with incident neutron energy E_{i,1} = 3.315 meV (ΔE = 106 μeV) permitted the study of the steep magnon dispersion in the vicinity of the reciprocal space position \({{{{{{{{\bf{Q}}}}}}}}}_{0}=\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\), which represents the R point. A “highenergy” 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^{64}, 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 ( ± 16 degrees) coverage, this allows one to obtain four dimensional data sets covering all three momentum transfer directions and energy transfer. Detailed information is provided in Sec. II of the Supplementary Information.
For Figs. 3, 4 the neutron intensity recorded in our experiments has been expressed as the imaginary part of the dynamic magnetic susceptibility \({\tilde{\chi }}^{{\prime\prime} }({{{{{{{\bf{Q}}}}}}}},\, E)\) in absolute units of \({\mu }_{{{{{{{{\rm{B}}}}}}}}}^{2}{{{{{{{{\rm{meV}}}}}}}}}^{1}\) to enable direct comparison with the calculations of the dynamic magnetic susceptibility via the MultiOrbital Periodic Anderson Model (see above). For this purpose, the recorded neutron intensities have been corrected for neutron absorption, put on an absolute scale via comparison to the wellknown incoherent scattering of the sample, and finally background subtracted via data sets obtained below T_{N}(for details of this procedure, we refer to the Supplementary Information). Further, the resulting corrected and normalized magnetic intensity was then corrected by the Bose factor accounting for the thermal population of magnon states, and subsequently divided by the square of the magnetic formfactor for Ce^{3+} as well as the ratio \(\left{{{{{{{{\bf{K}}}}}}}}}_{f}\right/\left{{{{{{{{\bf{K}}}}}}}}}_{i}\right\), whereby K_{i} and K_{f} denote the wavevectors of incident and scattered neutrons, respectively. For further information, we refer to the Supplementary Information. In the following, we additionally detail the integration ranges for the data shown in Figs. 3, 4. The data in Fig. 3a on the paths ΓR, RΓ, ΓX, XM, and MΓ were taken from the reciprocal space lines between the Qpositions \(\left(1,1,\,1\right)\) and \(\left(\frac{1}{2},\frac{1}{2},\,\frac{1}{2}\right)\), between \(\left(\frac{1}{2},\frac{1}{2},\, \frac{1}{2}\right)\) and \(\left(0,\, 0,\, 0\right)\), between \(\left(1,1,0\right)\) and \(\left(1,\, 1,\, \frac{1}{2}\right)\), between \(\left(1,\,\frac{1}{2},\, 1\right)\) and \(\left(\frac{1}{2},\frac{1}{2},\, 1\right)\), and between \(\left(\frac{1}{2},\, \frac{1}{2},\, 0\right)\) and \(\left(1 \,,\, 1,\,0\right)\), respectively. The intensity was integrated within a distance of ± 0.17 r.l.u. along the two Qdirections that are perpendicular to the path ΓXM and a distance of ± 0.09 r.l.u. along the two Qdirections that are perpendicular to the paths RΓ and MΓ. For the energy cuts shown Fig. 3b the intensity was integrated within a distance of ± 0.17 r.l.u. along three different reciprocal space directions. 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 tripleaxis neutron spectroscopy is often credited with the best combined momentumenergytransfer resolution, it fails to properly identify the dispersion even when used with the best resolution due to socalled CurratAxe spurions^{65}. To demonstrate this we have carried out additional measurements on the multiplexing triple axis spectrometer CAMEA at Paul Scherrer Institute^{66} 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 analyzed during the current study have been deposited on the ONCat platform of Oak Ridge National Laboratory (ORNL) under the following digital object identifier: 10.14461/oncat.data.64d61647fd6850c0afce4da3/1994514 (https://doi.org/10.14461/oncat.data.64d61647fd6850c0afce4da3/1994514). Currently, the DOIminting functionality in ONCat is still in beta and behind a firewall and you need to be on ORNL’s network to access it. This can be achieved going to https://analysis.sns.gov/ and registering an account, signing and starting a session. As soon as the DOIminting functionality is publically available the data will directly available via the link above. The processed neutron spectroscopy data shown in the manuscript are available at the Zenodo database under accession code digital object identifier: 10.5281/zenodo.10146787 (https://zenodo.org/records/10146787)^{67}.
Code availability
All code for the absorption correction of the neutron spectroscopy data is available in the Zenodo database under accession code digital object identifier: 10.5281/zenodo.10147126 (https://zenodo.org/records/10147126)^{68}.
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Acknowledgements
M.J. would like to thank Georg Ehlers for useful discussions concerning the setup of the CNCS experiment and acknowledges fruitful discussions with Bruce Normand. Work at Los Alamos National Laboratory was performed under the U.S. DOE, Office of Science, BES project “Quantum Fluctuations in Narrow Band Systems”. This research used resources at the Spallation Neutron Source, a DOE Office of Science User Facility operated by the Oak Ridge National Laboratory. WS is supported through funding from the European Union’s Horizon 2020 research and innovation programme under the Marie SklodowskaCurie grant agreement No 884104 (PSIFELLOWIII3i). Z.W. was supported by funding from the Lincoln Chair of Excellence in Physics. During the writing of this paper, Z.W. was supported by the U.S. Department of Energy through the University of Minnesota Center for Quantum Materials, under Award No. DESC0016371. FR thanks the hospitality of the University of Tokyo. Y.N. and R.A. acknowledge funding through GrantinAids for Scientific Research (JSPS KAKENHI, Japan) [Grant No. 20K14423 and 21H01041, and 19H05825, respectively] and “Program for Promoting Researches on the Supercomputer Fugaku” (Project ID:hp210163) from MEXT, Japan.
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C.D.B., F.R., and M.J. conceived the study. Y.N., R.A., and F.R. carried out the electronic band structure calculations. Z.W., E.A.G., and C.D.B. derived the multiorbital periodic Anderson model and Kondo–Heisenberg model. D.M.F. and M.J. designed the TOF experiments. W.S., D.G.M., and M.J. designed the CAMEA experiments. D.M.F., W.S., A.P., J.L., S.F., J.V., D.G.M., C.N., and M.J. carried out the experiments. J.V. comissioned and set up the lowtemperature sample environment for CAMEA. N.H.S. and E.D.B. synthesized and characterized the samples. D.M.F. assembled the singlecrystal mosaic. W.S. and M.J. analyzed the data. W.S., Z.W., C.D.B., F.R., and M.J. interpreted the results. W.S., Z.W., C.D.B., F.R., and M.J. wrote the paper with input from all the authors.
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Simeth, W., Wang, Z., Ghioldi, E.A. et al. A microscopic Kondo lattice model for the heavy fermion antiferromagnet CeIn_{3}. Nat Commun 14, 8239 (2023). https://doi.org/10.1038/s4146702343947z
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DOI: https://doi.org/10.1038/s4146702343947z
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