Electronic structure and magnetism in UGa2: DFT+DMFT approach
EElectronic structure and magnetism in UGa : DFT+DMFT approach Banhi Chatterjee and Jindˇrich Kolorenˇc ∗ Institute of Physics (FZU), Czech Academy of Sciences, Na Slovance 2, 182 21 Prague, Czech Republic (Dated: February 17, 2021)The debate whether uranium 5f electrons are closer to being localized or itinerant in the ferromag-netic compound UGa is not yet fully settled. The experimentally determined magnetic moments arelarge, approximately 3 µ B , suggesting the localized character of the 5f electrons. In the same time, onecan identify signs of itinerant as well as localized behavior in various spectroscopic observations. Theband theory, employing local exchange-correlation functionals, is biased toward itinerant 5f states andseverely underestimates the moments. Using material-specific dynamical mean-field theory (DMFT),we probe how a less approximate description of electron-electron correlations improves the picture.We present two variants of the theory: starting either from spin-restricted (LDA) or spin-polarized(LSDA) band structure. We show that the L(S)DA+DMFT method can accurately describe themagnetic moments in UGa as long as the exchange interaction between the uranium 6d and 5felectrons is preserved by a judicious choice of the spin-polarized double-counting correction. Wediscuss the computed electronic structure in relation to photoemission experiments and show howthe correlations reduce the Sommerfeld coefficient of the electronic specific heat by shifting the 5fstates slightly away from the Fermi level. I. INTRODUCTION
The 5f electrons in actinides and their compounds canbe either itinerant and participating in chemical bonds, orlocalized and not contributing to cohesion. A transitionakin to Mott metal–insulator transition occurs in elemen-tal actinide metals between Pu and Am [1]. Although ele-mental uranium has itinerant 5f electrons, its compoundsdisplay both types of 5f states. A traditional way of clas-sifying uranium compounds is by placing them in the Hillplot that relates the critical temperature (magnetic orsuperconducting) to the nearest neighbor U–U spacing[2]. Small U–U distances favor superconducting behaviorat low temperatures, whereas long-range magnetic ordertakes place at spacings greater than the so-called Hilllimit (3 . , an intermetallic binary compound with ahexagonal AlB structure (space group P6/mmm, Fig. 1),the Ga atoms effectively separate the uranium atoms, in-creasing the U–U distance to 4 . T C = 125 K with the easy magnetizationaxis along the [100] direction. Experimental observationsestablish magnetic moments of approximately 3 µ B perU atom in the ferromagnetic phase, using magnetizationmeasurements [3, 4] as well as neutron diffraction [5, 6].UGa thus exhibits moments and ordering temperaturethat are larger than typical for ferromagnetic uranium in-termetallics [7], which indicates localized 5f electrons. Themagnetic behavior can indeed be accurately reproducedby a fully local crystal-field model corresponding to the5f configuration of the U ion [8]. In addition, the observedSommerfeld coefficient γ = 11 mJ/mol · K [9] is not muchenhanced compared to the analogous compound without5f electrons – LaGa , displaying γ = 5 mJ/mol · K [10], ∗ [email protected] GaU
FIG. 1. The hexagonal lattice of UGa with uranium atomsshown in blue and gallium atoms in red. The uranium magneticmoments (arrow) are aligned along the [100] direction. which testifies against a high density of electronic statesat the Fermi level in UGa , again favoring the localizedpicture of the 5f electrons. The spectroscopic evidence,on the other hand, is not conclusive about the natureof the 5f states since one can identify spectral featurescharacteristic to localized electrons as well as featurestypical to itinerant electrons [11–13]. Similarly, the Fermisurface probed by the de Haas–van Alphen effect is notcompatible with full 5f localization [9].The large spin-orbit coupling (SOC), the crystal-fieldsplitting, and the Coulomb interaction between the 5felectrons influence the magnetic moments in a non-trivialmanner. This complexity contributes to the fact thatthe electronic structure of UGa is not yet satisfactorilyunderstood. The first-principles band theory based onsemi-local approximations to the density-functional the-ory (DFT) severely underestimates the moments, yieldingabout 0.6 µ B per uranium atom [14, 15]. A more accu-rate description of the electronic correlation is apparentlyneeded. The correlated band theory incorporating an on-site Hubbard interaction term, DFT+ U , can successfullymodel the magnetically ordered states, particularly ininsulating compounds with localized 5f electrons [16–18]. a r X i v : . [ c ond - m a t . s t r- e l ] F e b In UGa , it enhances the magnetic moments up to 2.8 µ B but the spectroscopic results are not reproduced very well[15, 19].The DFT+ U approach incorporates only static correla-tions, it cannot account for the multi-reference characterof the 5f shell nor for dynamical many-body effects. Theselimitations are lifted when DFT is combined with thedynamical mean-field theory (DMFT) [20, 21], which ac-curately models both itinerant and localized electrons. Inthis paper, we investigate how the theoretical descriptionof the magnetism and of the electronic structure of UGa improves when the DFT+DMFT is applied. We estimateand discuss the effects of the 6d–5f exchange interactionson the 5f magnetic moments, and compare the computedspectral properties with the experimental valence-bandphotoemission spectra (PES). We also discuss technicalmatters pertaining to spin-polarized DFT+DMFT solu-tions. II. METHOD
In DFT+DMFT, the static Coulomb U term of DFT+ U is replaced by an energy-dependent (dynamical) potential(selfenergy) acting on the uranium 5f states [20, 22]. Thisselfenergy is computed by solving an auxiliary impuritymodel, for which we employ the exact diagonalization.We present two variants of the theory differentiated bythe selfenergy being inserted (a) into the spin-restrictedLDA band structure (we call this method LDA+DMFT),and (b) into the ferromagnetic LSDA band structure(we refer to this variant as to LSDA+DMFT). A similarcomparison of spin-restricted and spin-polarized parentband structures was performed for ferromagnetic nickelin [23]. A. General formalism
We start with determination of the first-principles bandstructure by means of the WIEN2k code [24] using pa-rameters listed in Appendix A. Scalar relativistic effectsas well as the spin-orbit coupling are included in theseWIEN2k calculations. Afterwards, the relevant valencebands are represented by a tight-binding Hamiltonian inthe basis of the maximally-localized Wannier functions[25, 26]. This Hamiltonian is then used as the parent bandstructure for the DMFT calculations.In each iteration of the DMFT self-consistency cycle,the local electronic structure around one shell of theuranium 5f Wannier functions is mapped onto a non-interacting impurity model (Appendix B),ˆ H imp = X mm σσ (cid:2) H loc (cid:3) σσ mm ˆ f † mσ ˆ f m σ + X J (cid:15) J ˆ b † J ˆ b J + X mσJ (cid:16) V Jmσ ˆ f † mσ ˆ b J + V ∗ Jmσ ˆ b † J ˆ f mσ (cid:17) , (1) where ˆ f † mσ creates an electron in the 5f shell with magneticquantum number m and spin projection σ ∈ {− / , / } (eigenvalues of ˆ s z ). The first term in Eq. (1) correspondsto the local Hamiltonian, which describes the 5f shell. Itcan be decomposed as H loc = (cid:15) f ˆ I + ζ ˆ l · ˆ s − ˆ s · ∆ X + B ˆ O + B ˆ O + B ˆ O + B ˆ O , (2)where (cid:15) f is the energy of the 5f level, ζ is the strength ofthe SOC, ∆ X gives the exchange splitting, and ˆ O kq and B kq are Stevens operators and the corresponding param-eters that characterize the D crystal-field potential atthe uranium site in UGa . In general, the parameters B kq can be spin dependent, which we briefly discuss at the endof Appendix C. Note that the decomposition introducedin Eq. (2) is only used for the analysis of H loc and has noinfluence on the DMFT calculations and results.The second term in Eq. (1) corresponds to an effectivemedium usually referred to as the bath, with which the5f shell interacts. The operator ˆ b † J creates an electron inthis effective medium. The last term in Eq. (1) accountsfor the hybridization of the 5f shell with the bath. In ourcalculations, the off-diagonal hybridization induced by thenon-commutativity of the hexagonal symmetry with theSOC is fully taken into account. The crystal-field splittingof the 5f states is partly due to the crystal-field potentialcontained in H loc and partly due to the hybridization.The full interacting impurity model, in which the self-energy is computed, is given byˆ H DMFTimp = ˆ H imp + ˆ U , (3)where ˆ H imp is the non-interacting one-electron part shownin Eq. (1) and ˆ U is the Coulomb repulsion among the 5felectrons,ˆ U = 12 X mm m m σσ U mm m m ˆ f † mσ ˆ f † m σ ˆ f m σ ˆ f m σ − X mσ ( U H − σU X ) ˆ f † mσ ˆ f mσ , (4)where U mm m m is considered in its full spherically sym-metric form parametrized by four Slater integrals F =2 . F = 7 .
09 eV, F = 4 .
60 eV, and F = 3 .
36 eV,which correspond to Coulomb U = 2 . J = 0 .
59 eV. The first integral, F , is at the upper limit,beyond which the 5f peak in the occupied LSDA+ U den-sity of states moves too far from the Fermi level to becompatible with the valence-band photoemission spectra[11, 15]. The other three parameters ( F , F , F ) corre-spond to the atomic Hartree–Fock values calculated forthe U ion (5f configuration) and then reduced to 80%to mimic screening [27, 28]. Note that the unscreenedionic F k values yield Hund J = 0 .
79 eV, which can beconsidered as the maximal value for the uranium 5f systems.The second term in Eq. (4) is the double-countingcorrection introduced to remove the static mean-fieldapproximation of the 5f–5f Coulomb interaction that isincorporated in the DFT band structure. We assume thedouble-counting correction to be spherically symmetric(neither U H nor U X depends on the magnetic quantumnumber m ), with U X = 0 for the LDA band structureand U X = 0 for the LSDA band structure. The numericalvalues of U H and U X are discussed later.The impurity model, Eq. (3), is solved using the exactdiagonalization (Lanczos) method [29, 30] as implementedin our in-house code [31]. The size of the models, whichcan be solved by this method, is limited due to unfavor-able scaling of the computational demands. The impuritymodels employed in this paper consist of 14 spinorbitalscorresponding to the 5f shell and another 42 spinorbitalsrepresenting the bath. Of the bath states, N b = 28 orbitals have (cid:15) J above the Fermilevel (they are nominally empty). Even these small modelsare too demanding unless we turn to a reduction of themany-body basis inspired by the work of Gunnarsson andSch¨onhammer [31, 32]. A cutoff M is introduced for each N -electron Hilbert space H N , and the diagonalization isperformed only in a subspace H ( M ) N = (cid:8) | f N − N
We investigated several tight-binding models ˆ H k ofincreasing size. As the minimal model, we considered onethat contains gallium 4s and 4p, and uranium 5f and 6dstates. Then we included uranium 7s and finally also 7pstates. Various characteristics of these models are listedin Table I. Although the uranium 7p states are relativelyhigh above the Fermi level, their inclusion makes a sizabledifference, in particular to the crystal-field parameters in H loc and to the filling of the gallium states.On the top of that, we found that the LDA+DMFTcalculations without the U 7p states converge to the out-of-plane [001] ferromagnetic state, whereas the calculationswith the U 7p states predict an in-plane ferromagneticstate. Since the experiments determine UGa to be anin-plane ferromagnet [3, 4], all results presented in the fol-lowing sections were obtained in the tight-binding modelsthat include uranium 7s and 7p states. C. LDA+DMFT
When the parent band structure is spin-restricted(LDA), we induce the ferromagnetic solution by intro-ducing a small symmetry-breaking magnetic field intothe impurity model, Eq. (1), in the first few iterations ofthe DMFT self-consistency cycle. Afterwards, this fieldis removed again. Since we do not implement any chargeself-consistency, the tight-binding Hamiltonian ˆ H k re-mains unchanged during the whole LDA+DMFT cycleand the spin (and orbital) polarization is introduced onlyby means of the polarized selfenergy applied to the 5fstates. This method very likely results in an underesti-mated spin polarization of the 6d bands. Moreover, thelocal Hamiltonian H loc stays non-polarized as demon-strated in Appendix C, that is, no exchange field ∆ X isinduced in H loc by the polarized selfenergy. Nevertheless, TABLE I. Characteristics of several tight-binding models derived from the DFT band structure. All models contain gallium 4sand 4p orbitals, the included uranium orbitals are listed in the first column. The quantities ζ , (cid:15) f and ∆ X are shown in eV, thecrystal-field parameters B kq in meV.model orbital occupations local Hamiltonian H loc U 5f U 5f ↑ U 5f ↓ U 6d ↑ U 6d ↓ U 7s U 7p Ga 4s Ga 4p ζ (cid:15) f ∆ X B B B B nonmagnetic solutiond,f 2.79 0.94 0.94 – – 1.51 2.18 0.248 0.634 0 − . − .
14 0 . − . − . − . − . − . − . − .
01 0 . − . . − . − . − . .
71 0 .
01 0 . − . .
92 0 .
03 0 . − . there should be some exchange field present in H loc dueto the partially filled and partially polarized 6d bands,and neglecting this exchange certainly means underes-timated 5f moments (which is indeed what we observein Sec. III A). We fix this deficiency by introducing anempirical exchange field ∆ fd analogously to the earliercomputational studies of rare-earth systems [34, 35]. Themagnitude of this field is estimated as ∆ fd ≈ I fd m d [34],where m d is the magnetic moment due to the 6d electronsand I fd is intra-atomic exchange integral. The magneticmoment is approximated by its LSDA value, m d ≈ .
24 µ B (see Table I for the spin-resolved filling of the 6d bands),the exchange integral is estimated by atomic calculations, I fd ≈ .
15 eV/µ B [36]. This yields I fd m d ≈
36 meV andwe explore the LDA+DMFT solutions for ∆ fd variedaround this value.The absence of ∆ X is a disadvantage of the spin-restricted parent band structure. Its advantage, on theother hand, is that the double-counting correction inEq. (4) reduces to a single number, U H , since the spin-dependent part, U X , vanishes. One possible approxima-tion to the double counting is the so-called fully localizedlimit (FLL), U FLL H = U ( n f − / − J ( n f − / , (11)where n f is the self-consistently determined number of5f electrons [16, 37]. In our calculations, it turned outthat this U FLL H severely overestimates the number of 5felectrons, resulting in n f ≈
4. We hence employ an alter-native strategy: we choose U H such that the number of5f electrons remains close to its LDA value ( n f = 2 . U H ≈ n f = 2 .
72, 4.41 eV for n f = 3,and 2.71 eV for n f = 2. D. LSDA+DMFT
As discussed above, using spin-restricted LDA as theparent band structure has two deficiencies: underesti-mated spin polarization of the 6d (and other) bands, andmissing exchange field due to 6d moments acting on the 5felectrons. We dealt with the second issue empirically, butwe did not address the first one yet. We attempt to do soby using the spin-polarized (LSDA) solution as the parentband structure. This way, all non-5f bands are potentiallyspin-polarized, which enhances the polarization of thebath and of the bath–5f hybridization in the auxiliaryimpurity model, Eq. (1), when compared to LDA+DMFTdescribed in Sec. II C.Although it may seem that the LSDA parent bandstructure also provides an improved estimate of the localexchange field ∆ X , it is not so, since the LSDA exchangefield combines the 6d–5f exchange (tens of meV) withthe 5f–5f exchange (about 1 eV). The latter has to beremoved by the double-counting correction U X , which weknow only approximately. The FLL ansatz for the doublecounting U X reads as [16] U FLL X = E ↓ FLL − E ↑ FLL = J ( n ↑ f − n ↓ f ) , (12)where E σ FLL = U ( n f − / − J ( n σf − / , (13)which we find to overcorrect the LSDA 5f–5f exchange.With the LSDA occupation numbers (Table I) and with J = 0 .
59 eV, the double counting U FLL X becomes 1.19 eVwhereas the LSDA exchange is only ∆ X = 0 .
98 eV (Ta-ble I).Instead of using Eq. (12) or any other similar formula,we again employ the approach introduced in Sec. II C, thatis, we select U X such that ∆ fd = ∆ X − U X ≈ I fd m d ≈
36 meV. Since ∆ X is a parameter of the local Hamiltonian,it remains constant during the DMFT self-consistency m o m e n t ( µ B ) a l ong [ ] ∆ fd (meV) m tot m L − m S FIG. 2. The total magnetic moment (red), the orbital (black)and spin (blue) contributions to the magnetic moment of the 5fshell as functions of the exchange field ∆ fd applied along the[210] direction in the LDA+DMFT calculations. The realisticvalue of ∆ fd is marked by the orange stripe, the experimentalmagnetic moment is indicated by the dashed line [4]. iterations as follows from the derivation presented inAppendix C, hence ∆ fd and U X remain constant as well.For the spin-independent part of the double-countingcorrection, we choose U H = 3 . . . III. RESULTSA. Magnetic moments
The method outlined in the preceding sections is notentirely self-contained – there are several semi-empiricalparameters, such as the Coulomb parameters F k , thedouble-counting correction U H , and the exchange field∆ fd . Especially the exchange field was estimated onlyroughly and hence we decided to explore a range of valuesaround this estimate.In Figure 2 we show how the magnetic moments dependon ∆ fd in the LDA+DMFT calculations when ∆ fd is ap-plied in plane, along the [210] direction, which correspondsto the [210] ferromagnetic state [40]. The orbital and spincontributions to the magnetic moment are antiparallel asexpected for 5f filling smaller than 7. At ∆ fd = 0 meV,the total magnetic moment is clearly underestimated(1.76 µ B ), which confirms our earlier reasoning that someexchange field has to be introduced. As the exchange fieldincreases, the magnetic moment quickly increases too, itreaches 2.75 µ B at ∆ fd = 35 meV, at which point it isalready very close to the saturation value ≈ .
88 µ B . The m o m e n t ( µ B ) ∆ fd (meV) m tot[001] m tot[210] FIG. 3. Projection of the total magnetic moment to the [210]direction (red) and to the [001] direction (green) when theexchange field ∆ fd is applied along the [001] direction, and theLDA+DMFT calculations are started from the LDA solutionwith Σ( z ) = 0. The orbital and spin moments (not shown)behave similarly as in Fig. 2. quick saturation of the moments is a convenient feature– an inaccuracy in estimating the realistic value of ∆ fd translates to only a minor uncertainty of the computedmagnetic moments. The moments and 5f filling at therealistic value of ∆ fd are compared to the LSDA solutionand to experiments in Table II.Analogous calculations were performed also for theexchange field ∆ fd applied along the out-of-plane [001]direction. In this case, the ferromagnetic state parallel tothe exchange field is stable only above some critical valueof ∆ fd , see Fig. 3. Above this value, the magnetic momentvery quickly saturates, much faster that in Fig. 2. Forsmaller values of ∆ fd , the DMFT iterations converge to TABLE II. The orbital and spin magnetic moments in uranium5f shells, m S and m L (in µ B ), the total magnetic moment inthe unit cell m tot (in µ B ), the occupation of the 5f shells n f ,and the Sommerfeld coefficient γ (in mJ/mol · K ). The mo-ments and the 5f filling correspond to the maximally localizedWannier functions. The experimental m tot is taken from [4],the experimental γ from [9]. U H ∆ fd dir. m S m L m tot n f γ LSDA – – [210] − .
96 2 .
79 0 .
65 2.72 24.5LSDA – – [001] − .
00 2 .
89 0 .
70 2.72 21.2LDA+DMFT 3.0 35 [210] − .
85 4 .
60 2 .
75 2.76 8.2LDA+DMFT 3.0 35 [001] − .
91 4 .
87 2 .
96 2.74 7.7LSDA+DMFT 3.3 35 [210] − .
66 4 .
15 2 .
30 2.82 7.2LSDA+DMFT 3.3 35 [001] − .
59 3 .
89 2 .
12 2.80 7.5experiment 3.07 11.0 m o m e n t ( µ B ) a l ong [ ] ∆ fd (meV) m tot m L − m S FIG. 4. Variation of magnetic moments with ∆ fd , computedusing the LSDA+DMFT method when the parent band struc-ture is polarized in the [210] direction. Compare with Fig. 2. a nearly in-plane state with just a small out-of-plane tiltof the magnetic moments. For a range of ∆ fd values weget two stationary solutions, one nearly in-plane and theother out-of-plane, depending on the starting point of theDMFT iterations. Figure 3 shows calculations that werestarted at a given ∆ fd from the LDA state with Σ( z ) = 0.The transition from the in-plane to out-of-plane statethen occurs at ∆ fd ≈ . fd > ∼ . fd between 0.6 meV and2.2 meV is the ground state because we cannot reliablyevaluate the total energy in our LDA+DMFT implemen-tation. For the same reason, we cannot estimate the mag-netocrystalline anisotropy energy. We can, however, con-clude that the response of the magnetic moments to ∆ fd as observed in LDA+DMFT is consistent with the ex-perimental finding that the easy axis is oriented in plane.Starting from the paramagnetic state (∆ fd = 0) and cool-ing down, the system always ends up in the in-plane state,since the moments exhibit an instability toward in-planedirection. Increasing in-plane moment increases in-plane∆ fd , which stabilizes the in-plane state further.The magnetic moments computed using LSDA+DMFT,with the spin-dependent part of the double-counting cor-rection U X varied to reproduce the same range of ∆ fd as explored above, are presented in Figs. 4 and 5 for thein-plane and out-of-plane orientation of the LSDA po-larization. As in the LDA+DMFT, the total magneticmoments relatively quickly saturate with increasing ∆ fd ,and the saturation is again faster in the [001] state thanin the [210] state. Surprisingly, the saturated values ofthe total moments are noticeably smaller than in thecorresponding LDA+DMFT calculations, by 15% in thecase of the [210] ferromagnet and by 30% in the case m o m e n t ( µ B ) a l ong [ ] ∆ fd (meV) m tot m L − m S FIG. 5. The same plot as in Fig. 4, only the parent LSDA bandstructure is polarized in the [001] direction. To be comparedwith Fig. 3. of the [001] ferromagnet (compare with Figs. 2 and 3).We expected the opposite, since the LSDA parent bandstructure is certainly more polarized than the LDA parentband structure – besides ∆ fd that is the same in bothapproaches by construction, the LSDA has all non-5fbands spin split, which results in an enhanced polariza-tion of the hybridization function. Intuitively, this shouldhave induced a larger polarization in the 5f shell but thecalculations show that it does not.The difference in the computed moments could in princi-ple be due to a difference in fillings of the 5f shell betweenthe LDA+DMFT and LSDA+DMFT states, but this isnot the case either. The 5f filling in both methods is veryclose as can be checked in Table II where we summarizeour results for the realistic setting of the exchange field∆ fd . We speculate that the inaccurate LSDA+DMFTmoments come from some artifact of the static LSDAapproximation. One suspect feature is the strong spin de-pendence of the crystal-field parameters B kq in the localHamiltonian shown in Appendix C. Another feature, forwhich we do not have a clear explanation and which islikely to be connected to the LSDA solution as well, isthe jump in magnetic moments near ∆ fd = 30 meV inFig. 4. B. Valence band spectroscopy
Two measurements of valence-band photoemission spec-tra of UGa can be found in the literature, the ultravioletphotoemission spectrum [11] (UPS, shown in the left panelof Fig. 6) and the soft-x-ray photoemission spectrum [12](SX-PES, shown in the middle panel of Fig. 6). The UPSwas measured on sputter-deposited films at room temper-ature, that is, in the paramagnetic phase. The maximumintensity was observed just below the Fermi level with a DO S , P E S E (eV) PES (40.8 eV)DMFT PESDMFT DOS−8 −7 −6 −5 −4 −3 −2 −1 0 E (eV)PES (800 eV)DMFT PESDMFT DOS−3 −2 −1 0 E (eV)PES (800 eV)DMFT f DOSDMFT spd DOS−3 −2 −1 0 FIG. 6. Experimental photoelectron spectra (black line) from [11] (left panel) and from [12, 41] (middle panel) are comparedto the LDA+DMFT estimate of the spectra (green line). A Gaussian broadening (FWHM 0.2 eV) is added to simulate theinstrument resolution. The LDA+DMFT total DOS, subject to the same broadening, is shown for comparison (dashed line). Inthe right panel, we plot the orbital-resolved DOS without broadening (5f in red, sum of all others in blue). All theoretical linescorrespond to the [210] ferromagnet (∆ fd = 35 meV). long tail extending toward higher binding energies. TheSX-PES was measured on a freshly cleaved single crystalat T = 20 K, that is, well below the Curie temperature.The spectrum shows a narrow peak slightly below theFermi level accompanied with two broader features at − . − . − . fd = 35 meV, but the spectra are not sensitive tovariations of the 6d–5f exchange field). The spectra areconstructed as linear combinations of the orbital-resolved densities of states (DOS) weighted with photoionizationcross sections listed in [42]. According to these cross sec-tions, the 5f DOS has by far the largest weight for both40.8 eV and 800 eV photon energies, and hence thesephotoemission measurements probe mainly the 5f states.The computed spectra display a main peak at − .
15 eVand a satellite at − . − . − . total DOS (Fig. 6). They originate from orbitals that havesmall photoionization cross sections. These peaks are dueto hybridized U 6d and Ga 4p bands at − . − . − . edges in UGa elsewhere [13]). In Figure 7, we analyze the complete(occupied and unoccupied) 5f DOS from a theoreticalperspective. We compare the LDA+DMFT result withthe DOS computed for a spherically symmetric 5f ion.We can achieve a very close correspondence of these twodensities of states when the Coulomb U in the ionic modelis reduced to 1 .
55 eV compared to 2 . F k and the spin-orbit pa-rameter ζ are identical. This observation indicates thatthe 5f states in the LDA+DMFT are very close to beingfully localized, only their Coulomb repulsion is screenedmore than it would be in the fully localized Hubbard-Iapproximation. In addition, Fig. 7 also shows the j = 5 / ion U = 1.55 eV E (eV)−2 −1 0 1 2 3 4 5 DMFT U = 2.0 eV f DO S ( s t a t e s / e V ) total 5f j = 5/202468 FIG. 7. The uranium 5f DOS in the [210] ferromagnet fromthe LDA+DMFT method (∆ fd = 35 meV) in the top panel iscompared to the DOS from an atomic calculation (5f state)in the bottom panel (black lines). The parameter F = U was reduced in the atomic calculation to mimic the screeningeffects incorporated in the LDA+DMFT method. The j = 5 / component of the 5f DOS to be compared with the shapeof the M absorption line [13].Finally, in Fig. 8 we present the momentum-resolved5f spectral density along high-symmetry directions inthe Brillouin zone. We compare different models forthe electronic correlations, namely LSDA, LSDA+ U andLSDA+DMFT, in the ferromagnetic state with magneticmoments pointing along the [210] direction. The [001] fer-romagnetic state differs only in minor details. When theHubbard term is included (LSDA+ U and LSDA+DMFT),a gap between the occupied and unoccupied 5f bands ap-pears and the occupied 5f states move slightly away fromthe Fermi level. Given the same interaction parameters( U and J , or F k ), this gap is larger in LSDA+ U , whichindicates that the screening of the Coulomb parametersis stronger in LSDA+DMFT than in LSDA+ U . The situ-ation is analogous to Fig. 7 since the U -induced potentialin LSDA+ U has the form of a ionic Hartree–Fock approx-imation. Another difference between the LSDA+DMFTand LSDA+ U electronic structure is the incoherent char-acter of the 5f states visible in the LSDA+DMFT solution,starting approximately 2 . C. Sommerfeld coefficient
Figure 8 illustrates that the Fermi level cuts rightthrough the 5f bands in LSDA, which is accompaniedby a high density of states at the Fermi level and, subse-quently, by a large Sommerfeld coefficient of the electronicspecific heat γ . Indeed, LSDA predicts γ >
20 mJ/mol · K FIG. 8. Momentum-resolved 5f spectral density. The electroniccorrelations are described with increasing level of sophisticationfrom top to bottom: LSDA, LSDA+ U (with U = 2 . J = 0 .
59 eV and the FLL double counting), and LSDA+DMFT(with the same interaction parameters, and U H = 3 . fd = 35 meV). The ferromagnetic state with moments alongthe [210] direction is shown in all three panels. The same U and J produce a larger gap between the occupied and unoccupied5f states in LSDA+ U than in LSDA+DMFT. (Table II), which is at odds with the experimental value11 mJ/mol · K [9]. In DFT+DMFT (and in LSDA+ U as well), the 5f states move away from the Fermi leveltoward higher binding energies, and the coefficient γ is re-duced to approximately 8 mJ/mol · K (Table II), yieldinga considerably better agreement with experiments. Thecomputed Sommerfeld coefficient should be smaller thanobserved in experiments since we do not take into accountany enhancement due to phonons. We do not observemuch variation of γ when changing the orientation of themagnetic moments or when alternating the parent bandstructure (Table II). IV. CONCLUSIONS
We have studied the electronic structure and magneticproperties of the ferromagnetic compound UGa usingthe DFT+DMFT method, and compared our resultswith more approximate electronic-structure methods. Wehave found that our implementation of the DFT+DMFTmethod reproduces the experimentally observed largemagnetic moments as well as the sign of the magnetocrys-talline anisotropy energy, when the exchange interactionbetween uranium 6d and 5f states is included in a semi-empirical manner. This is done either in the form of anextra potential acting on the 5f states or in the formof a spin-polarized double-counting correction. We havecompared two formulations of the DFT+DMFT method,one keeping the non-5f states spin restricted (LDA), andthe other allowing their spin polarization (LSDA). Of thetwo, the LDA-based variant was found to provide moreconsistent results. It is a future work to investigate howthe semi-empirical approach to the 6d–5f exchange couldbe improved toward a fully first-principles method.Besides the magnetic properties, we have also modeledthe valence-band photoemission spectra on the basis ofthe DFT+DMFT density of states. We were not able tofully explain the differences between the two publishedphotoemission experiments [11, 12] but we could under-stand how the electron-electron correlations move the 5fstates slightly away from the Fermi level, which is in ac-cord with both photoemission spectra as well as with theobserved small Sommerfeld coefficient of the electronicspecific heat. With the aid of the DFT+DMFT method,it is thus possible to reconcile large magnetic momentsand a small Sommerfeld coefficient with the 5f spectraldensity in the close vicinity of the Fermi level.Our calculations indicate a close-to-localized uranium5f states in UGa . From the comparison to the experimen-tal photoemission spectra we deduce that the tendencyto localization is probably slightly overestimated in ourtheoretical description. Such a tendency is to be expectedfor the employed impurity solver that implements a formof expansion around the atomic limit. TABLE III. The orbital and spin magnetic moments in ura-nium 5f shell, m S and m L (in µ B ), the total magnetic momentin the unit cell m tot (in µ B ), the occupation of the 5f shell n f , and the Sommerfeld coefficient γ (in mJ/mol · K ). The 5fmagnetic moments and the 5f filling correspond to the atomic(muffin-tin) spheres.direction m S m L m tot n f γ LDA – – – – 2.45 43.9LSDA [100] − .
82 2 .
67 0 .
57 2.51 24.7LSDA [210] − .
82 2 .
64 0 .
54 2.50 26.7LSDA [001] − .
86 2 .
72 0 .
57 2.50 22.5
ACKNOWLEDGMENTS
The work was supported by the Grant agency of theCzech Republic under the grants No. 18-02344S andNo. 21-09766S. We thank S.-i. Fujimori for experimentaldata, and L. Havela, J. Kuneˇs and A. B. Shick for fruit-ful discussions. Computational resources were partiallysupplied by the project “e-Infrastruktura CZ” (e-INFRALM2018140) provided within the program Projects ofLarge Research, Development and Innovations Infrastruc-tures.
Appendix A: Parameters of DFT calculations
To perform all DFT calculations presented in this pa-per, we employed the WIEN2k package [24] that imple-ments linearized augmented plane-wave method and itsextensions. It combines a scalar-relativistic descriptionwith spin-orbit coupling. All calculations were performedat the experimental lattice constants a = 4 .
213 ˚A and c = 4 .
020 ˚A, reported in [3], with the following parameters:the radii of the muffin-tin spheres were R MT (U) = 2 . a B for uranium atoms and R MT (Ga) = 2 . a B for galliumatoms, the Brillouin zone was sampled with 6137 k points(900 k points in the irreducible wedge), and the basis-setcutoff K max was defined with R MT (Ga) × K max = 10 . ≈ − . B ) and in the interstitial( ≈ − . B ). The moments induced at Ga atoms arenegligible.The maximally localized Wannier functions for theDMFT calculations were found with the Wannier90code [26]. The spread minimization was performed on16 × ×
16 mesh of k points. Since there are no gaps inthe spectrum above the Fermi level, disentanglement wasnecessary [43]. We used 62 Bloch states on input, whichcorresponds to the energy window from −
10 eV to 24 eV.(Our largest tight-binding models, that is, those actuallyused for the DMFT calculations, have 48 Wannier func-tions). The frozen inner window extended to 6 eV (3 eV forthe smallest model listed in Table I), going higher meantthat the centers of the Wannier functions started driftingaway from the atomic centers, which is undesirable inour application that assumes the Wannier functions to beatomic-like. In the model used for the DMFT calculations,0the original WIEN2k bands were represented perfectlyup to 6 eV above the Fermi level, the match was stillvery good up to approximately 12 eV, and above that thecorrespondence quickly deteriorated.
Appendix B: Construction of the impurity model
In this appendix we discuss how the parameters ofthe finite impurity model, Eq. (1), are found so that themodel matches the effective medium (the bath) as closelyas possible. The impurity Hamiltonian has the form ablock matrix H imp = H loc V V V · · · V † H (1)bath · · · V † H (2)bath · · · V † H (3)bath · · · ... ... ... ... . . . , (B1)where all blocks are 14 ×
14 square matrices. The localHamiltonian H loc contains a strong spin-orbit couplingthat does not commute with the hybridization functionthat follows the crystal symmetry. Therefore, the problemcannot be simplified to diagonal matrices.If there is only one H bath block, all three matrices H loc , H bath and V can be determined by comparing the large z asymptotics of the local block of the impurity Green’sfunction, G loc ( z ) = h z I − H loc − X i V i (cid:0) z I − H ( i )bath (cid:1) − V † i i − , (B2)to the asymptotics of the bath Green’s function definedas G ( z ) = (cid:2) G − f ( z ) + Σ( z ) (cid:3) − . (B3)Here G f ( z ) is the 5f block of the local Green’s function G ( z ) from Eq. (6). We refer the reader to [31] for details.For larger impurity models, like Eq. (B1), this strategyleads to an unsolvable set of polynomial equations for the14 ×
14 square matrices. To overcome the problem, wecombine two shorter asymptotic expansions, one for theGreen’s function as before, and one for the hybridizationfunction.The asymptotic expansion of the local block of theimpurity Green’s function G loc ( z ) starts as G loc ( z ) = I z + H loc z + O ( z − ) , (B4)and the analogous expansion of the hybridization function∆ imp = z I − H loc − G − ( z ) (B5)starts as∆ imp = X i V i (cid:0) z I − H ( i )bath (cid:1) − V † i = X i (cid:20) V i V † i z + V i H ( i )bath V † i z (cid:21) + O ( z − ) . (B6) From the other side, the bath Green’s function, Eq. (B3),reads in the spectral representation as G ( z ) = Z A ( (cid:15) ) z − (cid:15) d (cid:15) , (B7)where we introduced the spectral density A ( (cid:15) ) = G ( (cid:15) − i0) − G ( (cid:15) + i0)2 π i . (B8)The asymptotic expansion of the bath Green’s functionis obtained by expanding the denominator in Eq. (B7), G ( z ) = ∞ X n =0 M n z n +1 , M n = Z (cid:15) n A ( (cid:15) ) d (cid:15) , (B9)where M n are moments of the spectral density. The spec-tral density A ( (cid:15) ) is a hermitian matrix and hence itsmoments are hermitian matrices as well. We immediatelysee that H loc = M . (B10)The spectral representation of the hybridization func-tion corresponding to G ( z ), that is, of ∆( z ) = z I − M − G − ( z ), can be written as∆( z ) = Z B ( (cid:15) ) z − (cid:15) d (cid:15) , (B11)where the spectral density is defined as B ( (cid:15) ) = ∆( (cid:15) − i0) − ∆( (cid:15) + i0)2 π i . (B12)Now we split the support of B ( (cid:15) ) to as many segments asmany H ( i )bath blocks we wish (or can afford) to have,∆( z ) = X i ∆ i ( z ) , where ∆ i ( z ) = Z (cid:15) i +1 (cid:15) i B ( (cid:15) ) z − (cid:15) d (cid:15) (B13)with (cid:15) i < (cid:15) i +1 , and we pair each ∆ i with one summandin Eq. (B6). The splitting can be arbitrary or it can beguided by an insight into the structure of the hybridizationfunction – the individual H ( i )bath blocks can be aligned withgroups of bands. In UGa , the hybridization below theFermi level comes mainly from Ga 4s and 4p bands, andin the first ≈ i ( z ) = ∞ X n =0 N ( i ) n z n +1 , N ( i ) n = Z (cid:15) i +1 (cid:15) i (cid:15) n B ( (cid:15) ) d (cid:15) . (B14)Comparing Eqs. (B6) and (B14), the blocks of H imp canbe written in terms of the moments N ( i ) n as V i = V † i = q N ( i )0 , (B15a) H ( i )bath = V − N ( i )1 (cid:0) V † (cid:1) − , (B15b)1 + C + L − C − LL − (1) L + (1) C − (1) C + (1) C − (2) + (2) CL + L − (2)(2) FIG. 9. (Color online) Contours in the complex plane usedfor integration of the moments M (blue) and N ( i ) n (red). Linesegments are denoted as L ± , half circles as C ± . which, together with Eq. (B10), concludes the construc-tion of the impurity model H imp from the local Green’sfunction G ( z ). Optionally, we can diagonalize the blocks H ( i )bath to make their interpretation more straightforwardand to arrive at the form of the impurity model used inEq. (1). The corresponding transformations are H ( i )bath → C − i H ( i )bath C i , V i → V i C i , (B16)where C i are the appropriate unitary matrices and thenew V i are no longer hermitian. By construction, theeigenvalues of H ( i )bath are confined to intervals ( (cid:15) i , (cid:15) i +1 ).For the purpose of their actual evaluation, the momentsare expressed in terms of contour integrals in the complexplane. Using the path segments sketched in Fig. 9, wehave M = 12 π i (cid:20)Z − L − − Z L + (cid:21) z G ( z ) d z = 12 π i (cid:20)Z C − + Z C + (cid:21) z G ( z ) d z , (B17) N ( i ) n = 12 π i (cid:20)Z − L ( i ) − − Z L ( i )+ (cid:21) z n (cid:2) z I − M − G − ( z ) (cid:3) d z = 12 π i (cid:20)Z C ( i ) − + Z C ( i )+ (cid:21) z n (cid:2) z I − M − G − ( z ) (cid:3) d z , (B18)where the integral over the (blue) circle C = C − ∩ C + encloses the entire support of A ( (cid:15) ) and the integrals overthe (red) circles C ( i ) = C ( i ) − ∩ C ( i )+ enclose the intervals( (cid:15) i , (cid:15) i +1 ). During the DMFT calculations, the selfenergy is thus evaluated along the circles C and C ( i ) , and alsoalong one additional semicircle in the upper half plane tocompute the number of electrons in the primitive cell andto adjust the Fermi level. An alternative to the circle C ,which serves for evaluation of H loc = M , is described inAppendix C.In the DFT+DMFT calculations of UGa discussedin the paper, we used three intervals ( (cid:15) i , (cid:15) i +1 ), namely( − ,
0) eV, (0 ,
6) eV and (6 ,
12) eV. The hybridizationabove 12 eV was discarded, since our tight-binding Hamil-tonians do not accurately represent the original DFTbands that far above the Fermi level (Appendix A).
Appendix C: Asymptotics of the bath Green’sfunction and the local Hamiltonian
At each k point, the tight-binding Hamiltonian ˆ H k canbe divided into four blocks,ˆ H k = ˆ H f k ˆ T k ˆ T † k ˆ H spd k ! , (C1)and the 5f block of the lattice Green’s function can bewritten asˆ G f k ( z ) = h z ˆ I − ˆ H f k − ˆΣ( z ) − ˆ T k (cid:0) z ˆ I − ˆ H spd k (cid:1) − ˆ T † k i − . (C2)Its asymptotic expansion reads asˆ G f k ( z ) = ˆ Iz + ˆ H f k + ˆΣ( ∞ ) z + O ( z − ) , (C3)where ˆΣ( ∞ ) is the static part of the selfenergy, which isthe leading term of the expansion ˆΣ( z ) = ˆΣ( ∞ ) + O ( z − ).For the bath Green’s function, Eq. (B3), we need onlythe local element,ˆ G f ( z ) = 1 N X k ˆ G f k ( z )= ˆ Iz + N − P k ˆ H f k + ˆΣ( ∞ ) z + O ( z − ) , (C4)respectively its inverse,ˆ G − f ( z ) = z ˆ I − N X k ˆ H f k − ˆΣ( ∞ ) + O ( z − ) . (C5)Inserting this expression into the definition of the bathGreen’s function, Eq. (B3), yields G ( z ) = ˆ Iz + 1 z N X k ˆ H f k + O ( z − ) . (C6)The selfenergy cancels out from the first moment of thecorresponding spectral density, and the moment thusequals to the local block of the tight-binding Hamiltonian, M = 1 N X k ˆ H f k = H loc , (C7)2 TABLE IV. Crystal-field parameters B σkq , Eq. (C8e), derivedfrom the LSDA tight-binding Hamiltonian (s,p,d,f model).Spin-restricted parameters B kq computed from Eq. (C8d) arethe same as shown in Table I. B B B B ferromagnetic solution [001]restricted 3 .
72 0 . − . − . ↑ . − . . − . ↓ .
68 0 . − . . .
92 0 . − . − . ↑ .
54 0 . − . − . ↓ .
30 0 . . − . throughout the whole DMFT self-consistency loop.To extract the individual contributions to the Hamilto-nian shown in Eq. (2), we can exploit the orthogonalityof operators ˆ I , ˆ l · ˆ s , ˆ s and ˆ O kq as 14 ×
14 matrices. Wecan write (cid:15) f = Tr( H loc ) / , (C8a)∆ αX = Tr(ˆ s α H loc ) / Tr(ˆ s α ˆ s α ) , α = x, y, z , (C8b) ζ = Tr(ˆ l · ˆ s H loc ) / Tr(ˆ l · ˆ s ˆ l · ˆ s ) , (C8c) B kq = Tr( ˆ O kq H loc ) / Tr( ˆ O kq ˆ O kq ) . (C8d)In the case of spin-polarized electronic structure, spin-dependent crystal-field parameters can be introduced as B σkq = Tr( ˆ O kq ˆ P σ H loc ) / Tr( ˆ O kq ˆ P σ ˆ O kq ˆ P σ ) , (C8e)where ˆ P σ is a projector to spin σ . Since the operator ˆ O kq is spin-independent, it commutes with ˆ P σ and we cansimplify the denominator asTr( ˆ O kq ˆ P σ ˆ O kq ˆ P σ ) = Tr( ˆ O kq ˆ O kq ˆ P σ ˆ P σ )= Tr( ˆ O kq ˆ O kq ˆ P σ ) = 12 Tr( ˆ O kq ˆ O kq ) . (C9)Consequently, the parameters B kq are averages of thespin-dependent parameters B σkq , B kq = 12 X σ B σkq . (C10)The spin dependence of the crystal-field parameters de-rived from the LSDA band structure is substantial, whichis illustrated in Table IV. Note that we do not attemptto remove the 5f self-interaction from the crystal-field po-tential [44, 45]. Nevertheless, the spin dependence wouldnot disappear even if we did [45]. [1] K. T. Moore and G. van der Laan, Nature of the 5 f states in actinide metals, Rev. Mod. Phys. , 235 (2009),arXiv:0807.0416 [cond-mat.str-el].[2] H. H. Hill, in Plutonium 1970 and other Actinides , editedby W. N. Miner (The Metallurgical Society of the AIME,New York, 1970) p. 2.[3] A. V. Andreev, K. P. Belov, A. V. Deryagin, Z. A. Kazel,R. Z. Levitin, A. Menovsky, Y. F. Popov, and V. I.Silant’ev, Crystal structure, and magnetic and magne-toelastic properties of UGa , Sov. Phys. JETP , 1187(1978).[4] A. V. Kolomiets, J.-C. Griveau, J. Prchal, A. V. An-dreev, and L. Havela, Variations of magnetic propertiesof UGa under pressure, Phys. Rev. B , 064405 (2015),arXiv:1502.04948 [cond-mat.str-el].[5] A. C. Lawson, A. Williams, J. L. Smith, P. A. Seeger,J. A. Goldstone, J. A. O’Rourke, and Z. Fisk, Magneticneutron diffraction study of UGa and UGa , J. Magn.Magn. Mater. , 83 (1985).[6] R. Ballou, A. V. Deriagin, F. Givord, R. Lemaire, R. Lev-itin, and F. Tasset, U form factor in UGa , J. Phys.Colloques , C7–279 (1982).[7] J.-M. Fournier and R. Troc, in Handbook on the Physicsand Chemistry of the Actinides , Vol. 2, edited by A. Free-man and G. Lander (North Holland, Amsterdam, 1985)p. 35.[8] R. J. Radwa´nski and N. H. Kim-Ngan, The crystal-fieldand exchange interactions in UGa , J. Magn. Magn. Mater. , 1373 (1995).[9] T. Honma, Y. Inada, R. Settai, S. Araki, Y. Tokiwa, T. Takeuchi, H. Sugawara, H. Sato, K. Kuwahara,M. Yokoyama, H. Amitsuka, T. Sakakibara, E. Yamamoto,Y. Haga, A. Nakamura, H. Harima, H. Yamagami, andY. ¯Onuki, Magnetic and Fermi surface properties of theferromagnetic compound UGa , J. Phys. Soc. Japan ,2647 (2000).[10] L. M. Da Silva, A. O. Dos Santos, A. N. Medina, A. A.Coelho, L. P. Cardoso, and F. G. Gandra, A study ofpressure and chemical substitution effects on the magne-tocaloric properties of the ferromagnetic compound UGa ,J. Phys. Condens. Matter , 276001 (2009).[11] T. Gouder, L. Havela, M. Diviˇs, J. Rebizant, P. M. Op-peneer, and M. Richter, Surface electronic structure ofUGa x films, J. Alloys Compd. , 7 (2001).[12] S.-i. Fujimori, M. Kobata, Y. Takeda, T. Okane, Y. Saitoh,A. Fujimori, H. Yamagami, Y. Haga, E. Yamamoto, andY. ¯Onuki, Manifestation of electron correlation effect in 5 f states of uranium compounds revealed by 4 d –5 f resonantphotoelectron spectroscopy, Phys. Rev. B , 035109(2019), arXiv:1901.00956 [cond-mat.str-el].[13] A. V. Kolomiets, M. Paukov, J. Valenta, B. Chatterjee,A. V. Andreev, K. Kvashnina, F. Wilhelm, A. Rogalev,D. Drozdenko, P. Minarik, J. Kolorenˇc, M. Richter, andL. Havela (2021), unpublished.[14] M. Diviˇs, M. Richter, H. Eschrig, and L. Steinbeck, Abinitio electronic structure, magnetism, and magnetocrys-talline anisotropy of UGa , Phys. Rev. B , 9658 (1996).[15] B. Chatterjee and J. Kolorenˇc, Magnetism and magneticanisotropy in UGa , MRS Advances , 2639 (2020).[16] V. I. Anisimov, F. Aryasetiawan, and A. I. Lichtenstein, First-principles calculations of the electronic structureand spectra of strongly correlated systems: the LDA + U method, J. Phys.: Condens. Matter , 767 (1997).[17] M.-T. Suzuki, N. Magnani, and P. M. Oppeneer, Micro-scopic theory of the insulating electronic ground states ofthe actinide dioxides AnO (An = U, Np, Pu, Am, andCm), Phys. Rev. B , 195146 (2013), arXiv:1305.5627[cond-mat.str-el].[18] R. Qiu, B. Ao, and L. Huang, Effective Coulomb inter-action in actinides from linear response approach, Comp.Mater. Sci. , 109270 (2020), arXiv:1810.05859 [cond-mat.mtrl-sci].[19] V. N. Antonov, B. N. Harmon, and A. N. Yaresko, Elec-tronic structure and magneto-optical Kerr effect in UGa ,J. Appl. Phys. , 7240 (2003).[20] A. I. Lichtenstein and M. I. Katsnelson, Ab initio calcu-lations of quasiparticle band structure in correlated sys-tems: LDA++ approach, Phys. Rev. B , 6884 (1998),arXiv:cond-mat/9707127 [cond-mat.str-el].[21] G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko,O. Parcollet, and C. A. Marianetti, Electronic structurecalculations with dynamical mean-field theory, Rev. Mod.Phys. , 865 (2006), arXiv:cond-mat/0511085 [cond-mat.str-el].[22] A. Georges, G. Kotliar, W. Krauth, and M. J. Rozen-berg, Dynamical mean-field theory of strongly correlatedfermion systems and the limit of infinite dimensions, Rev.Mod. Phys. , 13 (1996), arXiv:cond-mat/9510091.[23] M. I. Katsnelson and A. I. Lichtenstein, Electronic struc-ture and magnetic properties of correlated metals, Eur.Phys. J. B , 9 (2002), arXiv:cond-mat/0204564 [cond-mat.str-el].[24] P. Blaha, K. Schwarz, F. Tran, R. Laskowski, G. K. H.Madsen, and L. D. Marks, WIEN2k: An APW+lo pro-gram for calculating the properties of solids, J. Chem.Phys. , 074101 (2020).[25] J. Kuneˇs, R. Arita, P. Wissgott, A. Toschi, H. Ikeda, andK. Held, Wien2wannier: From linearized augmented planewaves to maximally localized Wannier functions, Comput.Phys. Commun. , 1888 (2010), arXiv:1004.3934 [cond-mat.mtrl-sci].[26] A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Van-derbilt, and N. Marzari, wannier90: A tool for ob-taining maximally-localised Wannier functions, Comput.Phys. Commun. , 685 (2008), arXiv:0708.0650 [cond-mat.mtrl-sci].[27] R. D. Cowan, The theory of atomic structure and spectra (University of California Press, Berkeley, 1981).[28] H. Ogasawara, A. Kotani, and B. T. Thole, Calculation ofmagnetic x-ray dichroism in 4d and 5d absorption spectraof actinides, Phys. Rev. B , 2169 (1991).[29] H.-D. Meyer and S. Pal, A band-Lanczos method forcomputing matrix elements of a resolvent, J. Chem. Phys. , 6195 (1989).[30] A. Liebsch and H. Ishida, Temperature and bath size in ex-act diagonalization dynamical mean field theory, J. Phys.:Condens. Matter , 053201 (2012), arXiv:1109.0158[cond-mat.str-el]. [31] J. Kolorenˇc, A. B. Shick, and A. I. Lichtenstein, Electronicstructure and core-level spectra of light actinide dioxidesin the dynamical mean-field theory, Phys. Rev. B ,085125 (2015), arXiv:1504.07979 [cond-mat.str-el].[32] O. Gunnarsson and K. Sch¨onhammer, Electron spectro-scopies for Ce compounds in the impurity model, Phys.Rev. B , 4315 (1983).[33] L. V. Pourovskii, G. Kotliar, M. I. Katsnelson, and A. I.Lichtenstein, Dynamical mean-field theory investigationof specific heat and electronic structure of α - and δ -plutonium, Phys. Rev. B , 235107 (2007), arXiv:cond-mat/0702342 [cond-mat.str-el].[34] L. Peters, I. Di Marco, P. Thunstr¨om, M. I. Katsnelson,A. Kirilyuk, and O. Eriksson, Treatment of 4 f states ofthe rare earths: The case study of TbN, Phys. Rev. B ,205109 (2014), arXiv:1605.09538 [cond-mat.str-el].[35] A. Shick and A. Lichtenstein, Electronic structure andmagnetic properties of Dy adatom on Ir surface, J. Magn.Magn. Mater. , 61 (2018).[36] M. S. S. Brooks and B. Johansson, Exchange integralmatrices and cohesive energies of transition metal atoms,J. Phys. F: Met. Phys. , L197 (1983).[37] I. V. Solovyev, P. H. Dederichs, and V. I. Anisimov, Cor-rected atomic limit in the local-density approximationand the electronic structure of d impurities in Rb, Phys.Rev. B , 16861 (1994).[38] B. Amadon, F. Lechermann, A. Georges, F. Jollet, T. O.Wehling, and A. I. Lichtenstein, Plane-wave based elec-tronic structure calculations for correlated materials usingdynamical mean-field theory and projected local orbitals,Phys. Rev. B , 205112 (2008), arXiv:0801.4353 [cond-mat.str-el].[39] L. Havela, S. Maˇskov´a, J. Kolorenˇc, E. Colineau, J.-C.Griveau, and R. Eloirdi, Electronic properties of Pu Ossimulating β -Pu: the strongly correlated Pu phase, J.Phys.: Condens. Matter , 085601 (2018).[40] We choose the [210] in-plane direction of magnetizationinstead of the experimental [100] direction due to tech-nical limitation of our impurity solver. We have checkedthat this change does not significantly affect the orderedmagnetic moments or the spectra in LSDA (Table III).[41] S.-i. Fujimori, private communication.[42] J. Yeh and I. Lindau, Atomic subshell photoionizationcross sections and asymmetry parameters: 1 ≤ Z ≤ , 1 (1985).[43] I. Souza, N. Marzari, and D. Vanderbilt, Maximally local-ized Wannier functions for entangled energy bands, Phys.Rev. B , 035109 (2001), arXiv:cond-mat/0108084 [cond-mat.mtrl-sci].[44] P. Nov´ak, K. Kn´ıˇzek, and J. Kuneˇs, Crystal field parame-ters with Wannier functions: Application to rare-earth alu-minates, Phys. Rev. B , 205139 (2013), arXiv:1303.1281[cond-mat.str-el].[45] P. Delange, S. Biermann, T. Miyake, and L. Pourovskii,Crystal-field splittings in rare-earth-based hard magnets:An ab initio approach, Phys. Rev. B96