Optical and magnetic excitations in the underscreened quasiquartet Kondo lattice

The underscreened Kondo lattice consisting of a single twofold degenerate conduction band and a crystalline electric-ﬁeld (CEF) split 4 f -electron quasiquartet has nonconventional quasiparticle dispersions obtained from the constrained mean-ﬁeld theory. An additional genuinely heavy band is found in the main hybridization band gap of the upper and lower hybridzed bands whose heavy effective mass is controled by the CEF splitting. Its presence should profoundly inﬂuence the dynamical optical and magnetic response functions. In the former the onset of the optical conductivity is not the main hybridization energy but the much lower Kondo energy scale which appears in the direct transitions to the additional heavy band. The dynamical magnetic response is also strongly modiﬁed by the in-gap heavy band which can lead to unconventional resonant excitations that may be interpreted as coherent CEF-Kondo lattice magnetic exciton bands. Their instability at low temperature signiﬁes the onset of induced excitonic magnetism in the underscreened Kondo lattice.


I. INTRODUCTION
Fundamental electronic properties of correlated f-electron compounds can be qualitatively understood within the Anderson lattice or Kondo-lattice (KL) models [1][2][3][4][5].In the strongly correlated limit (forbidden double occupancies) with large f-electron repulsion U f f → ∞ the slave-boson meanfield treatment with correlations simulated by a charge constraint on fermion and boson fields provides the most direct access to a description of renormalized quasiparticle bands.The combined effect of conduction (s-) and f-electron hybridzation as well as the f-electron correlation leads to two fundamental properties.Firstly the bands appear in (degenerate) pairs with a hybridazation gap existing between them for general kpoints in the Brillouin zone (BZ).The size of the (indirect) effective gap is reduced to the order of the single ion Kondo temperature T * .Secondly, due to this small energy scale in the range of a few meV the quasiparticle bands close to the gap are very flat corresponding to large enhancement of the effective quasiparticle mass.
The latter explains the thermodynamic and also transport properties of heavy fermion metals at low temperatures.In these materials, mostly Ce-intermetallics like, e.g.CeAl 2 , CeB 6 , CeCoIn 5 and many others the chemical potential is located in the flat part of the lower quasiparticle band.Due to residual quasiparticle interaction heavy fermion metals are prone to instabilities resulting, as in the above compounds, in exotic low temperature magnetic, multipolar [6,7] and superconducting phase transitions [8].In rare cases like the much discussed SmB 6 or YbB 12 borides [9] the chemical potential resides inside the hybridization gap leading to a Kondo insulator or semiconductor state (in the former due to the mixed valence [10] of ≈ 2.5+, it should be better termed mixed valence or hybridization gap insulator).Likewise the low energy charge and spin response as represented by optical conductivity and inelastic neutron scattering (INS) can be qualitatively understood within the mean field slave boson approach * akbari@postech.ac.kr of the Kondo lattices [11][12][13][14] and also dynamical mean field technique [15].In particular the appearance of a collective spin exciton resonance observed in many f-electron materials (possibly superconducting or with hidden order) inside the hybridization gaps or those opened by symmetry breaking may be interpreted within this approach [16][17][18][19].
Generally for these purposes the simplified SU(N) Kondo lattice model is employed.It assumes that the degeneracy N of localized (4f-or 5f-) states is the same as that of conduction electron states.Without crystalline electric field (CEF) effect the former is (2J + 1) and this may be quite large (J is the f-electron total angular momentum).In practice the CEF splitting reduces the f-electron degeneracy to N=2, 4 (the latter only in cubic environment).However, if the splitting of CEF ground and first excited states is moderate both are involved in the Kondo screening leading to the heavy quasiparticle bands, thus possibly invoking a larger quasi-degeneracy.This poses a problem for the straightforward application of the SU(N) KL model.For general wave vector in the Brillouin zone conduction electron states are at most twofold Kramers degenerate when time reversal symmetry holds and their spin orbit coupling is neglected.Higher degeneracy can only appear at symmetry positions.Therefore for N > 2 the genuine KL model is rather artificial since degeneracies of f-and conduction states no longer matches.This problem can be treated for the impurity model [3,20] but in the lattice it is rather difficult to analyze properly.
Physically in most cases it is more reasonable to assume an 'underscreened' model with higher f-electron than conduction electron degeneracy, where the former may actually be of pseudo-type, i.e. with a small CEF splitting of the same order as the Kondo temperature.Such a model has recently been investigated in detail in view of its quasiparticle spectrum and how the latter deviates from the canonical two N-fold degenerate hybridized bands of the genuine SU(N) KL model (see Ref. 21 and earlier work in Refs.[22][23][24].Specifically a quasi-quartet KL model for f-states was studied which is e.g.relevant for the (reduced) Γ 6 − Γ 7 CEF-level scheme in YbRu 2 Ge 2 and similar tetragonal compounds [25][26][27], hybridizing with a simple twofold degenerate conduction arXiv:2003.01535v1[cond-mat.str-el]3 Mar 2020 band.As an important result it was obtained that in the underscreened case an additional heavy quasiparticle band appears within the main hybridization gap whose dispersion is controled by the interplay of CEF splitting and Kondo screening.This gives the model a much richer low energy band structure than the common SU(N) model with several more discernible hybridization gaps.It would be highly desirable to probe this unconventional KL quasiparticle spectrum with inelastic low energy probes such as optical conductivity, inelastic neutron scattering (INS) and STM-techniques, since ARPES does not have the resolution to probe such subtle features in heavy bands.Actually STM-QPI technique has given indications in two heavy fermion metals that the hybridized band structures are more complex than suggested by the common SU(N) KL model with upper and lower hybridized branches [28][29][30].
It is the main purpose of the present work to study in detail inelastic low energy response of the underscreened quasi-quartet KL model based on the results of the previous work [21] to predict the signatures of the additional heavy band of this model in optical conductivity σ(ω) and inelastic neutron scattering S(q, ω).We show that their signatures appear as additional shoulders and peaks in the frequency dependencies of these experimental quantities and for INS have a distinctly dispersive behaviour.We argue that our results on optical conductivity suggest a simple explanation for unconventional behaviour of this quantity found in the Kondo insulator YbB 12 [31].Furthermore we discuss the possibility of a hybrid CEF-Kondo magnetic exciton mode in the dynamical magnetic response in the heavy bands of the underscreened KL model.We also give a qualitative discussion for the appearance of induced excitonic magnetism due to dominating non-diagonal exchange in the quasi-quartet and how the corresponding instability criterion is influenced by the Kondo screening for CEF split f-electrons.

II. QUASIQUARTET KL MODEL
The quasiquartet model is illustrated in the inset of Fig. 1 showing the two CEF-split Kramers doublets (τ = 1, 2) which interact with the conduction electrons that are scattered both elastically (τ ↔ τ ) and inelastically (1 ↔ 2).The basic Hamiltonian of the lattice of quasiquartets interacting with the single (doubly Kramers degenerate) conduction band is then where k = −(D c /2)(cos k x + cos k y ) is the conduction band dispersion with band width 2D c , g J the f-electron gfactor corresponding to the CEF split total f-electron angular momentum (J) multiplet and I ex the exchange coupling strength.This Kondo lattice model is of the underscreened type because there are only N = 2 conduction states that interact with 2N = 4 localized quasi-quartet f-states.The heavy quasiparticle spectrum of this model was studied in detail in Ref. 21 using a fermionic representation of H K and treating it within a constrained mean field theory.Using the spinors ) where c † kσ and f † τ kσ create conduction and f-electrons this leads to a bilinear fermionic mean field Hamiltonian Here λ is the effective f-level and Vτ its effective hybridization with conduction electrons in the fermionic representation.These quantities are determined by f-occupation constraint n f = 1, conduction electron number n c and a selfconsistency relation [21].We define ∆ = 1  2 ∆ 0 so that the CEF split effective f-level energies are λ 01 = λ − ∆; The diagonalization of the mean field Hamiltionian leads to three quasiparticle bands ), ), ). ( where we used the auxiliary quantities with the definition . The essential problem is the selfconsistent determination of the effective f-level position λ with respect of Fermi energy µ and the effective hybridization Vτ of CEF split f-states [21].An example for the structure of quasiparticle bands in the symmetric case V1 = V2 is shown in Fig. 1.There are two hybridized bands E 1,2k of partly conduction and f-electron character that changes when crossing the Brillouin zone (BZ).They have alternating flat portions corresponding to heavy effective mass.On the other hand the central overall flat band E 3k has mostly f-electron character with only small c-electron admixture that causes the small band width given by which is nonzero only for finite CEF splitting ∆ 0 .Here, in the symmetric case is the low energy (Kondo) scale for the heavy bands that determines their mass and hybridization gaps.It is obtained from solving the selfconsistency equation for λ − µ.
J0; J z 12 = 0 where J0 = (gJ − 1)Iex is the bare Kondo exchange in Eq. ( 1).Here µ = −0.094.band mass.Furthermore the main indirect hybridization gap is given by . The whole heavy E 3k band lies within this gap.There are additional indirect and direct hybridization gaps to be identified as discussed in detail in Ref. 21.These features are nicely illustrated by the example of heavy band structure in Fig. 1(b) for the (particlehole) symmetric case.We emphasize that these bands should not be considered like ordinary non-interacting bands that can be rigidly filled up the chemical potential µ with an arbitrary position.This is not true due to the constraint n f = 1 enforced by the strong on-site correlations (U f f → ∞).Therefore when the chemical potential changes the effective level λ and hence the hybridization gap structure in Fig. 1 is tied and moving along with the chemical potential.This means the chemical potential always has to be in close vicinity to the hybridization gap structure in the DOS [21].

III. OPTICAL CONDUCTIVITY
The optical conductivity (real part) is the response function associated with the (q = 0) conduction electron current Then the conductivity (j x) is obtained as [3,32] with where v x k = ∇ kx k is the group velocity.Using the spectral representation where ρ c (k, ω) is the renormalized conduction electron spectral function given by Here Σ c (iω n ) denotes the conduction electron self energy due to hybridization with the two f-orbitals (τ = 1, 2).Its evaluation leads to a sum of delta functions at the quasiparticle energies (β = 1 − 3) weighted by c-electron residua: The explicit forms of the Zβk is given in Appendix A. Inserting Eq. ( 9) into Eq.( 8) and averaging out the velocity we obtain where ω 2 pl = 4πn c e 2 /m b is the plasma frequency with n c and, m b the conduction electron density and effective band mass, respectively.As for the magnetic susceptibilities (Sec.IV) we may evaluate this expression using the explicit conduction electron spectral function given above.Then, using Eqs.(8,12) and the residual weights given in Eq. (A3) we finally obtain for the optical conductivity Because the chemical potential is closely located below the upper edge of the lowest band due to the constraint n f = 1 [21] there wil be a Drude term from the corresponding intraband transitions.In the limit T → 0 f (E βk ) = Θ H (E βk ) the inter-band contributions (β = β) in Eq. ( 13) contain only pairs (β β) = (2, 3), and (2, 1) since only band β = 2 is occupied and β = 3, 1 are empty.Thus the optical conductivity will have several threshold frequencies given by the various hybridization gaps in Fig. 1, all of them corresponding to direct transitions (q=0).Because the quasipartcle (E βk ) and residual ( Zβk ) k -dependencies in Eqs.(3,11) stem entirely from the conduction electron dispersion we may convert the k-summation in Eq. ( 13) into an integral over the bare conduction electron DOS according to As models the square and tight-binding DOS have been used before [21] corresponding to a band width 2D c .In the former case the DOS may be taken outside the integral as the constant For the numerical calculations we use directly the general expression Eq. ( 13).The results for σ(ω) are presented in Fig. 2.There is a small Drude peak from the lowest band whose width is determined by the small imaginary part (γ = 0.001D c ) used in the integration.Most importantly two inelastic peaks in the frequency dependence are visible, corresponding to the two direct (q = 0) hybridization gaps identified in the quasiparticle spectrum [21] and visble in Fig. 1.In the terminology of Ref. 21 the lower one starting at ω l1 ∼ 0.17D c corresponds to . The former is determined by the low energy Kondo and CEF-splitting energy scales Dc and ∆, respectively while the upper one by the larger effective hybridisation scale 2 V since ∆ V .The lower peak is much less pronounced because it is associated with optical transitions from the occupied to the central heavy band whose Bloch functions have only small c-electron content [21].The presence of two energy scales and peaks in the optical conductivity due to the posssibility of two direct transition is a decisive difference to the conventional Kondo lattice model which exhibits only the 'high' energy scale of ∆ d h1 2 V in σ(ω).The Kondo scale T * is not associated with any direct feature in σ(ω) because in the fully screened SU(N) KL it only appears for the indirect transitions with BZ-boundary momentum transfer Q.There is indeed some evidence that the two energy scales of the more realistic underscreened model may have been observed experimentally (Sec.VIII).

IV. BARE MAGNETIC SUSCEPTIBILITIES
For the calculation of the dynamic magnetic response we first need to calculate the bare physical magnetic susceptibilities coming from particle hole excitations in the heavy bands due to the dynamics of magnetic moments g J µ B J. These are combinations of the pseudospin susceptibilities.In terms of the pseudospin operators for the two Kramers doublets (τ, τ = 1, 2) [26] the total angular momentum operators, constricted to the quasiquartet CEF system may be written as Here the coefficients c z τ τ and c τ τ are determined by the parameters of the CEF potential or their eigenstates [21,26] (see Appendix B and Table I for more details).
The in-plane (µ = x, y or ⊥) and out of plane (µ = z or ) suszeptibilities of the physical moment operators µ = g J µ B J are related to their reduced expressions according to Here µ = x, y(⊥) cartesian components are equivalent due to tetragonal symmetry.Using Eq. ( 16) the two independent components (µ =⊥, z) of the reduced susceptibility χl (q, iν l ) may then be expressed by the pseudospin susceptibilities according to The second term in the last equation ∼ c 2  12 is the inter-orbital or vanVleck contribution due to nondiagonal matrix elements between the two CEF doublets (inset in Fig. 1).In the bubble approximation [3] the bare pseudospin susceptibilities may then be expressed via the Green's functions of fermionic variables in the pseudospin representation of Eq. ( 15) according to where q = k + q.For further evaluation we use the spectral representation of Green's functions according to where the f-electron spectral densities are given by [21] Using Eq. ( 11) they are evaluated as with the residual weights (see also Appendix A) given by and the definition of Inserting Eq. ( 20) into Eq.( 19) and using the explicit form of spectral weights in Eqs.(21 and 10) and their residual form to carry out the frequency integrations we obtain:  and (c,d) the out of plane (⊥) magnetic response functions, at q = (0, 0) and q = (π, π), respectively.Peak features at ω l and ωu for both q = 0, Q stemming from large and small hybridization gaps in Fig. 1
Here the ksummation runs over the 2D BZ and the β, β = 1 − 3 summation over the three quasiparticle bands of Eq. (3) comprising in principle three intra-band and three inter-band transitions.However due to the constraint [21] n f = 1 the chemical potential lies in the lowest band E 2k and then only one intraband 2 ↔ 2 and two interband transitions 2 ↔ 1, 3 contribute to the susceptibilities.
The results for the bare susceptibilities for both moment directions and for BZ center and boundary wave vectors are shown in Fig. 3(a,b) for and Fig. 3(c,d) for ⊥ where we plotted both real and imaginary parts.For both moment directions the transitions across the two direct hybridsation gaps (Sec.III) show up clearly as sharp separate peaks in the spectrum (imaginary part, blue) associated with singular behaviour of the real part (red) at the zone center q = 0. On the other hand for the zone boundary wave vector Q = (π, π) a larger manyfold of indirect transitions is possible and the spectrum is more spread out in frequency, although certain individual idirect gap excitation energies are still discernible.This result leads one to consider identification of the two-peak structure not only in optical conductivity but also in inelastic neutron scattering that probes the magnetic response functions, albeit of the interacting system considered in the next section.

V. DIPOLAR RPA DYNAMIC SUSCEPTIBILITY AND SPECTRUM
From the two-impurity Kondo models it is known that it induces two competing effects: The on-site screening of moments that tend to form a singlet ground state and creation of effective inter-site (RKKY)-type couplings that prefer to align the moments to a magnetic ground state.In the periodic lattice the constrained fermionic mean-field treatment of the underscreened Kondo lattice model successfully captures the ingredients of the heavy quasiparticle states that form close to the Fermi level.However this approximation only involves a homogeneous global (site-independent) hybridization field and therefore does not lead to any effective intersite couplings.The latter would appear if fluctuations of this field and their exchange between sites would be included as a next step [33], but this could only capture long range interactions.To simulate such competition effects more flexibly even on the basis of the mean field quasiparticle picture it is customary to extend the model by adding an extra inter-site exchange term explicitly to Eq. ( 1) which may be thought to have been created by having already eliminated additional higher lying conduction band states by a Schrieffer-Wolff transformation.This procedure has been formally carried out before in the case of the fully screened conventional KL model [11].However the result has a rather singular behaviour in k-space and therefore one has to resort to a phenomenological form of intersite exchange.This leads to an extended Kondo-Heisenberg model [34,35] described now by where I ↔ is a cartesian uniaxial inter-site exchange tensor (counted per n.n.bond ij ) with only diagonal components I x,y ij ≡ I ⊥ ij and I z ij .Note that for consistency we use the same sign convention for both on-site Kondo and inter-site exchange, i.e. negative for FM and positive for AF coupling.For reasons mentioned before we use a phenomenological Lorentzian model for the inter-site exchange of the form where q 0 = (0, 0) is a zone center (FM, I µ 0 < 0) or zone boundary (AF, I µ 0 > 0) q 0 = (π, π) ≡ Q wave vector, respectively and adjustable parameters I µ 0 , Γ characterize height and sharpness of the maximum in I(q) around q 0 , respectively.
In RPA approximation the collective dynamical susceptibility components (µ =⊥, z) due to the last term in the above equation are then represented by χµ RP A (q, iν l ) = 1 − I µ (q) χµ (q, iν l ) −1 χµ (q, iν l ), (28) where the bare magnetic susceptibilities χµ (q, iν l ) of heavy quasiparticle bands have been evaluated in the previous section (Eq.( 18)).The magnetic excitation spectrum of the underscreened KL as accessible by INS is then finally obtained as being proportional to the dynamical structure function where qµ = q µ /|q| is normalized momentum transfer component.

VI. MIXED CEF-KONDO SPIN EXCITONS AND THEIR TEMPERATURE DEPENDENCE FROM RPA RESPONSE
We first discuss the behaviour of bare magnetic response functions in Eq. ( 18) which is shown in Fig. 3 for (a,b) and ⊥ directions (c,d), respectively and for zone center q = (0, 0) and zone boundary q = (π, π) wave vectors.As in the case of optical conductivity (q = 0) one can clearly identify the two peak structure originating now from the magnetic transitions between lower band (n = 2) and central (n = 3) as well as upper (n = 1) bands (Fig. 1).They are now of comparable intensity because the central band has mainly f-electron content leading to large magnetic matrix elements.The peaks are sharper for the direction whereas the spectrum (imaginary part) is more spread out for the ⊥ direction.Whether they appear directly in the RPA spectrum and INS structure function S(q, ω) depends strongly on the type and strength of quasiparticle interactions described phenomenologically in Eq. (27).For small |I µ 0 | the bare spectrum is hardly changed.However it is clear from Eq. ( 31) that for sufficiently large interaction when is first fulfilled for a frequency ω r a collective magnetic resonance mode appears inside the hybridisation gaps (ω r < ∆ d h1 , ∆ d h3 ) [Fig.3(e,f)] that absorbs almost all the intensity while only small features are left at the bare peak positions which are prominent in Fig. 3(a-d).Note that here the resonance is most pronounced at q = 0 connected with the direct magnetic transitions.In the conventional KL model with a single hybridization gap the bare susceptibility exhibits singular behaviour as function of frequency around the indirect gap threshold and the spin exciton resonance mode evolves at the zone boundary and inside the gap of order T * [19].In contrast, in the present underscreened KL model with more realistic band structure involves both the CEF and Kondo energy scales in direct and indirect hybridization gaps (Sec.III) and therefore the resonance may also appear at a zone-center wave vector.This depends, however, on the FIG. 4. Frequency and momentum dependence of the intensity plots of bare and RPA susceptibilities along the q = (0, 0) to q = (π, π) (ΓM) direction.The left panel represents the (Jz) magnetic response, χ (q, T ), and the second panel shows the ⊥ (Jx,y) magnetic response, χ⊥ (q, T ).In each panel, the first and the second sub-panels indicate the real and imaginary part of the bare susceptibilities, respectively.The last sub-panels show the RPA result but in the logarithmic scale.Parameters in Iµ(q) same as in Fig. 3.The spectrum of Imχ ⊥ RP A shows an incipient soft mode (q → 0) of the hybrid CEF-Kondo collective magnetic exciton.
precise form of I µ (q) and its maximum position.The lowest hybridization gap scale is of order . Therefore a collective mode inside this gap as seen in Fig. 3(e,f) may be termed a hybrid CEF-Kondo magnetic exciton.In the limit T * → 0 it becomes the conventional CEF magnetic exciton which is the bare CEF excitation at ∆ 0 dispersing due to non-diagonal intersite exchange matrix elements (∼ c 12 ) of the bare localized two level (τ = 1, 2) system (Sec.VII).
We also show the magnetic response and spectrum in the (q, ω) plane for q along ΓM direction (Fig. 4).For moment the bare and RPA spectrum are rather similar, meaning one is far from the resonance condition in Eq. (30).While for ⊥ direction the comparison of bare and RPA spectrum clearly shows that a resonance mode at low energy has evolved around q = 0. by properly tuning I ⊥ (q) to achieve the condition in Eq. (30).For this direction the strength |I µ 0 | needs to be much less than for direction due to the difference in the low frequency bare susceptibilities (real parts) as seen in Fig. 3.
For the parameters used the hybrid CEF-Kondo magnetic exciton in Fig. 4(f) shows an incipient soft-mode behaviour with ω r (q → 0) approaching zero.This is the precursor of an induced moment FM phase transition that will appear for slightly larger coupling strength.We stress that this type of excitonic KL magnetism induced by off-diagonal exchange matrix elements ∼ c 12 connecting different split CEF states is fundamentally different from the usual KL magnetism [36][37][38][39] with fully degenerate f-states.The soft mode behaviour of the zone-center ω r (q = 0, T) is also observed as a function of temperature (Fig. 5).Note that within the underlying slave-boson theory for the heavy bands the T-dependence has to be restricted to the range kT < (T * 2 + ∆ 2 0 ) 1 2 .The induced moment transition is discussed using a qualitative analytical approach in the following section.

VII. INDUCED MAGNETISM CRITERION WITH KONDO SCREENING EFFECT
In the case when the nondiagonal exchange dominates due to c 2 12 c 2 τ τ the softening of magnetic exciton mode for T > T c indicates an induced magnetic phase transition.This is well known in fully localized f-systems, e.g. in various Pr [40] and U [41] compounds with lowest singlet-singlet CEF level scheme.The condition for the critical temperature is obtained from the the divergence of static (iν l = 0) RPA susceptibilities [µ = z, ⊥ (x, y)], i.e. χµ RP A (q) −1 = 1 − I µ (q) χµ (q, T ) / χµ (q, T ) = 0. (31) For the simple TB band structure and effective inter-site exchange model used here we can restrict to FM (q = 0) transition at T c or AF (q = Q) transition at T N .Naturally for the itinerant Kondo model the above equation can only be treated numerically using Eqs.(18,25).First we recapitulate the result within the completely localized model without Kondo term but finite inter-site interaction.We consider the case c 2 12 c 2 τ τ when the non-diagonal vanVleck terms dominate (Appendix B).Then we have (β = 1/kT ) where we defined m µ2 τ τ = 1 12 and m z2 12 = 0 according to Eqs. (15,16).The solution of Eq. ( 31) is then given by where T m = T c for FM (q = 0) or T m = T N for AF (q = Q) case, respectively.Here ξ q is the control parameter for induced moment magnetism which is not due to the ground state polarization alone but primarily (c 2 c 2 τ τ ) due to the admixiture of the excited state into the ground state by the inter-site exchange.The induced moment ground state appears only when the critical parameter fulfils ξ q > 1.This mechanism is preceded by the magnetic exciton (the bare dispersive CEF excitation) softening above T cr .It is obtained from the pole of Eq. ( 28) (for µ =⊥) as This mode becomes becomes soft, i.e. ω(q 0 ) = 0 at the ordering temperature T cr and wave vector q 0 = 0 or q 0 = Q.
When the on-site Kondo coupling I ex to conduction electrons is included in Eq. ( 1) the localized doublets will turn into the (partly) heavy itinerant quasiparticle bands of Fig. 1.The excitation spectrum and critical temperature then requires the numerical evaluation of Eqs.(28,31) using Eqs.(18,25) as in the previous section.It is worthwhile, however, to have at least a qualitative understanding how the criticality condition for induced moment magnetism is modified under the presence of the Kondo screening and resulting 4f quasiparticle itineracy.This can be achieved for the FM case by using a simple analytical estimate for χ⊥ (q = 0, T ) including only the transitions between bands n = 2, 3.This leads approximately to where we defined the average ∆ * 0 = (T * 2 + ∆ 2 0 ) 1 2 with T * denoting the Kondo temperature of Eq. (6).Furthermore ∆ e is the effective dominating low energy scale for the vanVlecktype susceptibility contribution of transitions between occupied and empty band states n = 2, 3. Then from Eq. ( 31) we obtain the modified instability criterion for the FM case T * m = T * c which includes the effect of Kondo screening as where ξ * 0 is the new critical parameter for induced magnetism (ξ * 0 > 1) renormalized by the Kondo effect.It is instructive to consider two limiting cases: ∆0 leading to ∆ e → ∆ and therefore ξ * 0 → ξ 0 .This means in the limit of vanishing Kondo coupling T * → 0 we recover the bare CEF expression in Eq. ( 33) for T c .
• Dominating Kondo coupling T * ∆ 0 : In this case T * and therefore ∆ e → 2T * 2 /∆ 0 .This leads to ξ * 0 → 1 2 (∆ 0 /T * ) 2 ξ 0 ξ 0 .Therefore the effective control parameter ξ * 0 is much reduced and unless the bare parameter ξ 0 is very large the Kondo screened ξ * 0 may fall below the critical value ξ * 0 = 1 preventing the magnetic instability.These limits imply that for all ratios of T * /∆ 0 we have ξ * 0 < ξ 0 and the Kondo screening effect will reduce or suppress completely the appearance of the induced magnetic ordering temperature T c .This behaviour is illustrated in Fig. 6.

VIII. DISCUSSION AND CONCLUSION
For Kondo compounds with CEF splitting the underscreened quasi-quartet KL model has more realistic features than the conventional SU(N) model.It shows a richer structure of quasiparticle bands around the Fermi energy that encompasses the Kondo-as well as CEF energy and effective hybridisation scales and their mutual influence.In particular it was found that the optical conductivity which envolves only direct transitions has a distinct two-peak structure.The lower peak (ω l ) is dominated by the smaller scales (T * , ∆) while the upper peak (ω u ) by the larger effective hybridisation scale 2 V .Only the latter is present in the conventional KL model and therefore within this model the Kondo scale T * is not directly visible in the optical conductivity.Due to the presence of the third heavy band inside the main (large) hybridization gap 2 V leaves a direct signature in σ(ω) around ω l at the Kondo/CEF scales.This may qualitatively explain some puzzling features observed previously in the optical conductivity of YbB 12 [31].In this compound experiment showed not only a peak at the main hybridisation gap energy, as expected from the conventional KL model.It also exhibits a clear onset in σ(ω) at the much lower Kondo scale given by T * which can in the conventional KL picture only be associated with large momentum transitions accross the indirect hybridization gap normally not accessible for the optical response.Therefore it was suggested that this low-frequency onset must be due to phonon-assisted indirect transitions which are possible for zone-boundary momentum transfer carried by phonons.The present investigation suggest an alternative mechanism: The low frequency onset of σ(ω) at T * in YbB 12 can be due to direct transitions to the central heavy band present in the underscreened KL model.In this context there is no need to resort to indirect phonon assisted transitions.One must add, however, that cubic YbB 12 has a quartet ground state and two closeby doublet (a quasi-quartet) excited states [16,42] and so the details may be different from the model investigated here.Evidence for multi-peak hybridization gap structure in σ(ω) has also been found in Ce-compounds [43].
The intricate quasiparticle band structure of the quasiquartet KL model also shows up in the inelastic magnetic response functions probed by INS.We found that the basic ingredients of the two-peak structure due to small and large hybridization gap scales should also be present.The details depend considerably on the CEF parameters that enter as weights in the dynamic susceptibilities and on the form of the phenomenological intersite exchange.The combined itinerant Kondo/CEF magnetic exciton spectrum may exhibit a softening as function of temperature at the wave vector where the exchange has a maximum.This is a precursor for an induced magnetic phase transition due to dominating nondiagonal exchange between the CEF-split doublets.In the FM case (q = 0) this may be described by a simplified quasilocalized model where the control parameters a modified due to the presence of the Kondo screening.The further development of this hybrid localized-itinerant picture for CEF-Kondo magnetic excitons and induced magnetism needs an inspiration from INS and other experiments preferably on Ce-and Yb-based Kondo lattice compounds.

FIG. 3 .
FIG. 3. (a-d) Frequency dependence of bare susceptibilities: (a,b) show the in plane ( )and (c,d) the out of plane (⊥) magnetic response functions, at q = (0, 0) and q = (π, π), respectively.Peak features at ω l and ωu for both q = 0, Q stemming from large and small hybridization gaps in Fig.1can be discerned.(e,f) The frequency dependence of the perpendicular RPA susceptibilities, at q = (0, 0) and q = (π, π) for exchange function parameters I

FIG. 5 .
FIG.5.Frequency and temperature dependence of the spectral intensity plots for bare and RPA out of plane susceptibilities (⊥) at q = (0, 0).The last panel (RPA) is in the logarithmic scale.It demonstrates the softening of the hybrid CEF-Kondo magnetic exciton with decreasing temperature.

FIG. 6 .
FIG. 6. (top panel) Dependence of the Kondo-screened control parameter ξ * 0 (normalized to the bare CEF value ξ0) for induced magnetic instability on the ratio of Kondo temperature T * to CEF splitting ∆0.(bottom panel) Suppression of induced magnetic ordering temperature T * c due to the strong decrease of ξ * 0 (top) with increasing Kondo energy scale T * , plotted for several values of bare abovecritical control parameter ξ0 > 1.