Two-Channel Kondo Effect Emerging from Np and Pu Ions

On the basis of an idea of the electron-hole relation between $f^2$ and $f^4$ states in a $j$-$j$ coupling scheme, we point out a possibility that two-channel Kondo phenomena can be observed even in cubic transuranium compounds including Np and Pu ions with local $\Gamma_3$ non-Kramers doublet ground state. For the purpose, we analyze a seven-orbital impurity Anderson model hybridized with degenerate $\Gamma_8$ conduction electron bands by employing a numerical renormalization group method. From the numerical results for the case of four local $f$ electrons, corresponding to Np$^{3+}$ and Pu$^{4+}$ ions, we confirm that a residual entropy of $0.5\log 2$, a characteristic of two-channel Kondo phenomena, appears for the case with the local $\Gamma_3$ doublet ground state.


Introduction
It has been well known that the Kondo effect occurs in a dilute magnetic impurity system [1][2][3].For the case in which a single impurity spin 1/2 is embedded in a single-band conduction electron system, we observe conventional Kondo phenomena and the mechanism has been completely understood both from theoretical and experimental viewpoints [4].On the other hand, for the case in which plural numbers of electrons with spin-orbital complex degrees of freedom are hybridized with multi-channel conduction electron bands, new and rich phenomena have been actively discussed for a long time, after the clarification of the conventional Kondo effect.
When an impurity spin is hybridized with multi-channel conduction bands, an intriguing concept of multi-channel Kondo effect has been proposed [5].In particular, for the case of impurity spin 1/2 and two conduction bands, the appearance of a non-Fermi liquid ground state has been suggested.This is one of the fascinating properties of two-channel Kondo effect.Such a non-Fermi liquid state has been also pointed out in a two-impurity Kondo system [6,7].Concerning the reality of the twochannel Kondo effect, Cox has pointed out that two screening channels exist in the case of quadrupole degree of freedom in a cubic uranium compound with Γ 3 non-Kramers doublet ground state [8,9].After this proposal, the stage for the investigation of two-channel Kondo phenomena has been almost fixed as the f 2 electron system.First, cubic U compounds have been focused, but recently, cubic Pr compounds have been actively investigated [10].
Although it is important to investigate the f 2 electron system to deepen our understanding on the two-channel Kondo effect, we strongly believe that it is also meaningful to expand the research frontier of the two-channel Kondo physics to other rare-earth and actinide compounds.On the basis of our belief, we have examined the Kondo effect for the case of plural numbers of f electrons in rareearth and actinide compounds by analyzing a seven-orbital impurity Anderson model hybridized with Γ 8 conduction bands with the use of a numerical renormalization group (NRG) method [11].First we have reconfirmed the appearance of quadrupole two-channel Kondo effect for the case of n = 2 [12], where n denotes the local f electron number per ion.Then, we have moved onto the case of n = 3, corresponding to Nd ion, in which we have clarified the emergence of magnetic two-channel Kondo effect for the case with a Γ 6 ground state [13].
In this paper, we analyze the seven-orbital impurity Anderson model hybridized with Γ 8 conduction electrons by using the NRG method for the case of n = 4, corresponding to Np 3+ and Pu 4+ ions.We confirm that a residual entropy of 0.5 log 2 appears as a clear signal of the quadrupole twochannel Kondo effect for the case with the non-Kramers Γ 3 doublet ground state.In addition, we also find a quantum critical point (QCP) between local crystalline electric field (CEF) singlet and Kondo-Yosida singlet states for the case of n = 4, as has been found in the case of n = 2. Finally, we briefly discuss potential materials to observe actually the quadrupole two-channel Kondo effect for the case of n = 4. Throughout this paper, we use such units as = k B = 1.

Model and Method
To describe the local f -electron model, first we consider one f -electron state, which is the eigenstate of spin-orbit and CEF terms.Under the cubic CEF potential, Γ 7 doublet and Γ 8 quartet are obtained from j = 5/2 sextet, whereas we obtain Γ 6 doublet, Γ 7 doublet, and Γ 8 quartet from j = 7/2 octet, where j denotes the total angular momentum of f electron.By using those one-electron states as bases, we express the local f -eletctron Hamiltonian as where E f is the f -electron level to control the local f -electron number n at an impurity site, f jµτ denotes the annihilation operator of a localized f electron in the bases of (j, µ, τ ), j = 5/2 and 7/2 are denoted by "a" and "b", respectively, µ distinguishes the cubic irreducible representations, Γ 8 states are distinguished by µ = α and β, while Γ 7 and Γ 6 states are labeled by µ = γ and δ, respectively, and τ is the pseudo-spin which distinguishes the degeneracy concerning the timereversal symmetry.The definitions of λ j , B j,µ , and I will be discussed below.
As for the spin-orbit term, we obtain where λ is the spin-orbit coupling of f electron.The magnitude of λ depends on the kind of actinide atoms, but in this paper, we set λ = 0.3 eV, since we consider Np and Pu ions.Concerning the CEF potential term for j = 5/2, we obtain where B 0 4 denotes the fourth-order CEF parameter in the table of Hutchings for the angular momentum ℓ = 3 [14,15].Here we note that the sixth-order CEF potential term B 0 6 does not appear for j = 5/2, since the maximum size of the change of the total angular momentum is less than six.On the other hand, for j = 7/2, we obtain Note also that B 0 6 terms appear in this case.In the present calculations, we treat B 0 4 and B 0 6 as parameters.The matrix element of the Coulomb interaction is expressed by I. To save space, we do not show the explicit forms of I here, but they are expressed by four Slater-Condon parameters (F 0 , F 2 , F 4 , F 6 ) [16,17] and Gaunt coefficients [18,19].As for the magnitudes of Slater-Condon parameters, first we set F 0 = 10 eV by hand.Others are determined so as to reproduce excitation spectra of U 4+ ion with two 5f electrons [20].The results are F 2 = 6.4 eV, F 4 = 5.6 eV, and F 6 = 4.1 eV [21].Now we include the Γ 8 conduction electron bands hybridized with localized electrons.Since we consider the case of n < 7, the local f -electron states are mainly formed by j = 5/2 electrons and the chemical potential is situated among the j = 5/2 sextet.Thus, we consider only the hybridization between conduction and j = 5/2 electrons.The seven-orbital impurity Anderson model is given by where ε k is the dispersion of conduction electron with wave vector k, c kγτ denotes an annihilation operator of conduction electron, and we set V α = V β = V .Here V denotes the hybridization between Γ 8 conduction and localized electrons.
In this paper, we analyze the model by employing the NRG method [11].We introduce a cut-off Λ for the logarithmic discretization of the conduction band.Due to the limitation of computer resources, we keep M low-energy states.Here we use Λ = 5 and M = 2, 500.Note that the temperature T is defined as T = Λ −(N −1)/2 in the NRG calculation, where N is the number of the renormalization step.In the following calculation, the energy unit is D, which is a half of conduction band width.Namely, we assume D = 1 eV in this calculation.

Calculation Results
First we briefly discuss the f -electron configurations in a j-j coupling scheme for n = 2 and n = 4.As shown in Fig. 1(a), when we accommodate four f electrons in the j = 5/2 sextet, we find two f holes there.Of course, for the finite value of λ, the component of j = 7/2 octet should be included in the ground-state wave function.Note here that we mention only the main component of the ground and low-energy excited states for n = 2 and n = 4 [22,23].We emphasize that the electron-hole relation between n = 2 and n = 4 on the basis of the j-j coupling scheme is quite simple, but we expect to observe the quadrupole two-channel Kondo effect even for n = 4, corresponding to Np 3+ and Pu 4+ ions.Now we consider the local CEF ground-state phase diagram for n = 4, obtained from the di-  Fermi-liquid phases will be discussed elsewhere in future.
Here we remark that there exists a blue region corresponding to log 3 in the local Γ 5 state.This is quite natural, since the local Γ 5 state is triply degenerate.The Γ 5 moment is screened by Γ 8 conduction electrons and thus, it is expected that the conventional Kondo effect occurs in the Γ 5 region, although the magnitude of the Kondo temperature significantly depends on the hybridization and excitation energy.
Note also that in the region of B 0 6 < 0, we observe some blurry yellow spots along the boundary curve between Γ 1 singlet and Γ 5 triplet local ground states.The QCP is known to appear between the local CEF and Kondo singlet states for n = 2 [12].Thus, we deduce that those spots form a QCP curve, although we could not obtain enough amounts of numerical results to depict the smooth curve.
In Fig. 4, we show the curves of entropy vs. temperature for the points near B 0 4 = 0.0 and B 0 6 = −0.0002.For B 0 6 = −0.0001(CEF singlet) and −0.0003 (Kondo singlet), entropies are found to be zero around at T = 10 −4 , while we find a residual entropy of 0.5 log 2 even at T = 10 −10 for B 0 6 = −0.000190226.When B 0 6 deviates even slightly from this value, we find that 0.5 log 2 entropy immediately disappears.The present results are quite similar to those for n = 2, which we have actually found in the same model [12].In this sense, we highly expect the existence of QCP between CEF and Kondo singlet states even in the f 4 system.

Summary and Discussion
In this paper, we have analyzed the seven-orbital impurity Anderson model hybridized with Γ 8 conduction bands by using the NRG technique.For the case of n = 4, we have confirmed the emergence of two-channel Kondo effect.As for the mechanism of quadrupole two-channel Kondo effect for the case of n = 4, we deduce that it is essentially the same as that for the case of n = 2 from the viewpoint of the electron-hole relation between f 2 and f 4 states on the basis of the j-j coupling.
We have also observed the QCP curve running in the Γ 1 region near the boundary between the CEF singlet and Kondo-Yosida singlet states.Here readers may have some questions on this QCP curve.For instance, the curve seems to merge into the two-channel Kondo state in the local Γ 3 region, but the property of the state at the merging point is unclear.The details on the properties of the QCP curve will be discussed elsewhere in future.
Finally, we provide a brief comment on actual materials to observe the two-channel Kondo effect for the case of n = 4.In rare-earth ions, the case of n = 4 corresponds to Pm 3+ , but unfortunately, there exist no stable isotopes for Pm.Thus, we turn our attention to actinide ions such as Np 3+ and Pu 4+ with 5f 4 configurations.It may be difficult to synthesize new Np and Pu compounds, but we expect that Np 1-2-20 compound will be synthesized in future.

Fig. 1 .
Fig. 1.(a) Electron configurations in the j-j coupling scheme for n = 2 and n = 4.(b) Ground-state phase diagram of H loc on the (B 0 4 , B 0 6 ) plane for n = 4.

Fig. 3 .
Fig. 3. Color contour map of the entropy for n = 4 on the plane of (B 0 4 , B 0 6 ) for V =1.0 and T = 2.6 × 10 −6 .White curves denote the boundaries among local CEF ground states shown in Fig. 1.Note that yellow spots appear along the boundary curve between Γ 1 and Γ 5 regions.