The γ - α iso-structural Transition in Cerium, a Critical Element
Nicola Lanatà, Yong-Xin Yao, Cai-Zhuang Wang, Kai-Ming Ho, Jörg Schmalian, Kristjan Haule, Gabriel Kotliar
aa r X i v : . [ c ond - m a t . s t r- e l ] M a y The γ - α iso-structural Transition in Cerium, a Critical Element Nicola Lanat`a, ∗ Yong-Xin Yao, ∗ Cai-Zhuang Wang, Kai-Ming Ho, J¨org Schmalian, Kristjan Haule, and Gabriel Kotliar Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08856-8019, USA Ames Laboratory-U.S. DOE and Department of Physics and Astronomy,Iowa State University, Ames, Iowa IA 50011, USA Karlsruhe Institute of Technology, Institute for Theory of Condensed Matter, D-76131 Karlsruhe, Germany (Dated: August 28, 2018)
Below the critical temperature T c ≃ K , an iso-structural transition, named γ - α transition, can be induced in Cerium by applying pressure. This transition isfirst-order, and is accompanied by a sizable volume collapse. A conclusive theoret-ical explanation of this intriguing phenomenon has still not been achieved, and thephysical pictures proposed so far are still under debate. In this work, we illustratezero-temperature first-principle calculations which clearly demonstrate that the γ - α transition is induced by the interplay between the electron-electron Coulomb inter-action and the spin-orbit coupling. We address the still unresolved problem on theexistence of a second low- T critical point, i.e., whether the energetic effects alone aresufficient or not to induce the γ - α transition at zero temperature. The γ - α iso-structural transition in Cerium [1] was dis-covered in 1949 [2]. Since then, a lot of theoretical andexperimental work has been devoted to its understand-ing. The great interest in this phenomenon arises fromthe fact that the transition is isostructural, i.e., the lat-tice structure of the system is equal in the two phases.Furthermore, the possibility that the underlying mecha-nism lies in the electronic structure only — i.e., without itbeing necessary to involve other effects — makes Ceriuma potential theoretical testing ground for basic conceptsof correlated electron systems. Two main theoreticalpictures are still under debate to explain the volumecollapse: the Kondo volume collapse (KVC) [3, 4] andthe orbital-selective Mott transition within the Hubbardmodel (HM) [5]. According to the KVC the transition isinduced by the rapid change of the coherence tempera-ture across the transition boundaries, which affect dra-matically the structure of the conduction spd electronsthrough Kondo effect. According to the HM, instead,it is the hopping between f orbitals that changes dras-tically across the transition between the α phase (withdelocalized f electrons) and the γ phase (with localized f electrons), as for the Mott transition in the Hubbardmodel.Consistently with both the HM and the KVC pictures,the f -electrons are strongly correlated both in the α andin the γ phase. This fact is clearly indicated, e.g., bythe photoemission spectra, which is known experimen-tally [6–8] and theoretically [9–11]. Despite this similar-ity, there is a key difference between these two models:while the KVC attributes a very important role to theinterplay between the localized 4 f orbitals and the itin-erant spd conduction bands, the itinerant electrons are“spectators” in the HM picture.The development of LDA+DMFT (Local Density Ap-proximation plus Dynamical Mean Field Theory) [12] re- sults [13–15] has successfully reproduced many aspectsof this transition, and different aspects of these stud-ies can be understood in both physical pictures. Thereare still fundamental questions which have not been an-swered. (1) What is the role of the spin-orbit interaction(SOC) for the volume-collapse? (2) What is the fate ofthe pressure-temperature transition-line at very low tem-peratures [16–18]? (3) Can a first-principle based theorybe made computationally efficient so as to access boththe γ and the α Cerium at zero temperature?Due to the complexity of the problem, it is clear that,in order to be conclusive, a theoretical explanation of the γ - α transition needs to be supported by first-principlescalculations which, not only are able to take into accountboth the details of the band-structure and the strong-correlation effects, but are also able to evaluate preciselythe pressure-volume phase diagram. For this purpose,it is crucial that the computation of the total energy isessentially free of numerical error. Another key require-ment is that the two phases are treated within the sametheoretical framework. In this work, we use a combi-nation of Density Functional Theory and the GutzwillerApproximation (LDA+GA) [19–23], which satisfies all ofthese requirements. Recently, we have established for-mally [L.N. et al.] that this method can be viewed asan instance of LDA+DMFT, using Slave Bosons (or theGutzwiller method) as the DMFT impurity solver [14].This insight enabled a new efficient charge self-consistentimplementation of the LDA+GA method on top [Y.Y.X.et al.] of the LAPW DFT code Wien2k [24], which re-moves many of the approximations inherent in previousstudies. As a benchmark, in the supplementary materialwe present also LDA+DMFT calculations for Cerium.The very good agreement between the two methods givesus further confirmation that the results presented in thiswork are indeed reliable. (cid:1) ( ) (cid:4) (cid:5) i (cid:7) (cid:8) (cid:9) (cid:10) nn1l-n1-n1b-cc1l- (cid:18)(cid:19)(cid:20)(cid:21)1((cid:22)(cid:23)(cid:18)(cid:19)7(cid:9)(cid:25)(cid:26)1((cid:20)(cid:9)(cid:27)(cid:23)(cid:21)1 (cid:28) ( ) (cid:29) (cid:28) (cid:7) ncnlnsnpn(((((((((((((((((((((((((((((((((((((((((((cid:5)() (cid:1) s i(cid:7)(cid:8)(cid:9)(cid:10)0ln l- sn s- pn p- (cid:19)(cid:9)((cid:22)(cid:23)(cid:18)(cid:19)7(cid:9)(cid:25)(cid:26)1((cid:20)(cid:9)(cid:27)(cid:23)(cid:21)1 ln l- sn s- pn p- p s l csnslspsrsunn1cn1l lp lu sl sr p s l csnslspsrsunn1cn1l lp lu sl sr FIG. 1: Total energy as a function of the volume (upper pan-els) and corresponding theoretical pressure-volume curves for U = 5 , eV , J = 0 . eV , in comparison with the experimentaldata (lower panels). The experimental data relate to mea-surements at room temperature (black circles from Ref. [27],red diamonds from Ref. [28] and green squares from Ref. [29]),while our theoretical calculations are all obtained at zero tem-perature. The curves in the insets are obtained for all U ’sfrom 4 . eV to 6 . eV (from lower to higher total energies)with step of 0 . eV . Our results are shown both with (leftpanels) and without (right panels) taking into account thespin-orbit coupling. The vertical shaded line in the upper in-sets indicate the experimental volume at ambient pressure.The horizontal dotted lines in the lower-left panel and theblack diamonds in its inset indicate the pressures where thebulk-modulus K = − V dP/dV is minimum.
We employ the general Slater-Condon parametrizationof the on-site interaction, assuming a Hund’s couplingconstant J = 0 . eV [25]. Since the value U of the in-teraction strength is generally difficult to establish ac-curately (due to it’s strong sensitivity to the screeningeffect), in this work we perform calculations scanning dif-ferent values of U . Our calculations are all performed atzero temperature.In the upper panels of Fig. 1 we illustrate our theoreti-cal energy-volume diagrams for different values of U . Theresults are shown both by taking into account the SOC(left panels) and by neglecting it in the calculation (rightpanels). The corresponding pressure-volume curves, ob-tained from P = − dE/dV , are shown in the lower pan-els, in comparison with the experimental data at roomtemperature of Refs. [27, 28]. The agreement with theexperiment is good, especially for U = 6 eV , which isthe value that reproduces the experimental equilibrium- volume V eq ≃ . A / atom [26–28] (see the inset of thehigher-left panel). The small discrepancies at larger V are likely, at least in part, due to the entropy, as ourcalculations were performed at zero temperature. Notethat U = 6 eV was also previously computed within theconstrained LDA method [9, 30], which gives us furtherconfidence that this is the optimal value of the correlationstrength for Cerium.Remarkably, we observe a change of sign in the bulk-modulus K = − V dP/dV < U ≤ U c ≃ . eV (see the pressure-volume curves inthe lower-panels insets of Fig. 1); while at U = U c thetransition becomes second-order, with K = 0 minimumbulk-modulus. A signature of the transition, i.e., a localminimum of the bulk-modulus, is still present for any in-teraction strength, see the black diamonds in the insetof the lower-left panel of Fig. 1. We point out that thisfeature of the pressure-volume curve is observed only ifthe SOC is taken into account — which is a clear indi-cation of its key role in the physics underlying the γ - α transition. Note also that, for U = 6 eV , the crossoverpoint occurs at P ≃ − P exp ≃ − f entanglement entropy, S f [ ρ f ] = − Tr[ ρ f ln ρ f ] , (1)where ρ f is the reduced density matrix of the system inthe f local subspace. The value of S f is a measure of howmuch the f electrons are entangled with the rest of theenvironment. In Fig. 2 the behaviour of S f is shown as afunction of the volume for two values of U . Remarkably,if (and only if) the SOC is taken into account, a clearcrossover is visible in correspondence of the signature ofthe volume collapse, i.e., in correspondence of the mini-mum of the bulk-modulus K , which is indicated by blackdiamonds in the inset of the lower-left panel of Fig. 1.In the α phase, as expected, S f is not sensitive tothe spin-orbit splitting, indicating that the local fluctu-ations induced in the f local space by the coupling withits environment are very large. By increasing the vol-ume, the fluctuations between the J = 5 / f subspaceand the other local configurations are increasingly sup-pressed. The crossover point identifies the situation inwhich the above-mentioned fluctuations are sufficientlysmall to be hampered by the spin-orbit splitting. Thisis clearly demonstrated by the fact that in the γ phase,when the SOC is taken into account, S f & ln 6 — where6 = 2 × / / n f = 1. We point out that the above-mentioned localfluctuations are generated only by the entanglement, andso are present even if the actual temperature of the sys-tem is zero — as in our calculations. As we are going to (cid:1) (cid:2) (cid:3) (cid:4) (cid:2)(cid:5) (cid:6) (cid:7) (cid:8) (cid:7) (cid:2) (cid:3) t (cid:7) (cid:2) (cid:3) (cid:10) (cid:11)(cid:12)(cid:13) (cid:14)sα(cid:7)(cid:17)(cid:6)(cid:2)o4γp(cid:6)(cid:2)o2p (cid:23)(cid:24)))Vα66Vαγ (cid:14)s2(cid:7)(cid:17)(cid:6)(cid:2)o4γp(cid:6)(cid:2)o2p (cid:23)(cid:24)))Vα66Vαγ (cid:17)(cid:11)(cid:6)to (cid:1) ((cid:4)(cid:3)(cid:11)(cid:8)p)1 )α 61 6α γ1 γα (cid:30)(cid:2)(cid:31)(cid:6)Vt (cid:12)(cid:30)(cid:2)5(cid:11)(cid:10)"Vt(cid:31)(cid:11) FIG. 2: Local entanglement entropy of the f -electrons as afunction of the volume per atom of the system for U = 5 eV (upper panel) and U = 6 eV (lower panel) at fixed J = 0 . eV .The entanglement entropy is reported both for the case with(lines) and without (dots) the SOC. The horizontal lines cor-respond to 14, which is the dimension of the single-particlelocal space of the f -electrons, and to 6 = 2 × / / f -electrons within the single-particle local space. The vertical continue lines indicate thesignature of the transition in the pressure-volume diagram,and the dotted vertical lines indicate the boundary of the γ - α transition (which occurs only for U ≤ U c ≃ . eV ) accordingto the equal-area construction [31]. show, the main source of entanglement is the hybridiza-tion between the f and the spd electrons.A further insight of the problem can be achieved byinspecting the ground-state expectation values of thenon-local energy components of the effective Hamilto-nian ˆ H whose ground-state provides our theoretical solu-tion [12, 21]. The non-local part ˆ T of ˆ H can be conciselyrepresented as ˆ T = ˆ T ff + ˆ T fc + ˆ T cc , (2)where the symbol c represents all of the spd conductionelectrons, and ˆ T ff , ˆ T fc and ˆ T cc represent the non-local“hopping” terms between f - f , f - c and c - c electrons, re-spectively. In Fig. 3 the ground-state expectation valuesof the f - c and the f - f components of ˆ T are shown fortwo values of U . These energies represent the Kondo andthe Hubbard energy scales of the problem, respectively.In agreement with the KVC model of the transition, weobserve that the Kondo energy scale is about one order ofmagnitude bigger than the Hubbard energy scale, whichis already very small before the crossover point, as ex- (cid:1) cl(cid:4)(cid:5) (cid:1)(cid:2)(cid:1) (cid:1)(cid:2)(cid:3) (cid:6)(cid:7) (cid:8) (cid:9)(cid:10)(cid:10) (cid:11) (cid:12)(cid:13) (cid:4) (cid:12) (cid:4) (cid:15) (cid:13)(cid:16) i (cid:4) (cid:5) n (cid:19)−(.(cid:19)−(0l(cid:19)−(0(cid:19)−(−l−55555555555555555555555555555555555555555555(cid:5)(cid:9)(cid:24)5i (cid:1) )(cid:27)(cid:28)(cid:9)(cid:29)n.− .l 2− 2l 1− 1l (cid:1)c3(cid:4)(cid:5)(cid:6)(cid:7).− .l 2− 2l 1− 1l (cid:11)(cid:12) (cid:24)(5!(cid:10)(cid:11)(cid:12)y(cid:9)(cid:15) FIG. 3: Ground-state expectation values of the f - c and the f - f effective hopping energies as a function of the volumeper atom of the system. The energies are reported for U =5 eV (left panel) and U = 6 eV (right panel) at fixed J =0 . eV , both for the case with (lines) and without (dots) thespin-orbit coupling. The vertical continue lines indicate thesignature of the transition in the pressure-volume diagram,and the dotted vertical lines indicate the boundary of the γ - α transition (which occurs only for U ≤ U c ≃ . eV ) accordingto the equal-area construction [31]. pected [32]. This confirms that the main source of entan-glement between the f local space and its environmentis the hybridization between the f and the spd electrons.Note that, when the SOC is taken into account, a morerapid suppression of the Kondo energy scale is observedconcomitantly with the crossover region.We have already observed that, even though the be-haviour of the entanglement entropy (and of the Kondoenergy scale) are qualitatively the same for all U ’s, notransition can be found for U ≥ U c ≃ . eV at zerotemperature, see Figs. 1 and 2. The reason is the follow-ing. The local crossover, which induces a reduction of thebulk-modulus of the system, occurs at lower volumes forlarger U , see the black diamonds in the inset of the lower-left panel of Fig. 1. On the other hand, the bulk-modulusbecomes larger at smaller volumes (even when the SOC isnot taken into account), see the pressure-volume curvesin the lower panels of Fig. 1. For this reason, if U is largeenough, it becomes impossible for the SOC to make thebulk-modulus negative, i.e., to induce the volume col-lapse. It follows that, in principle, there are two possiblescenarios: (i) the γ - α transition exists also at zero tem-perature, or (ii) the transition line ends at a certain finitecritical temperature (at negative pressures). As we men-tioned before, U ≃ eV — which is indeed very close to U c — is a physically reasonable value for Cerium. Thissuggests that Cerium is placed essentially in the middlebetween the two above-mentioned scenarios, i.e., that the γ - α transition line ends very close to zero temperature.Note that this finding is in qualitative agreement withthe experimental results of Ref. [33]. (cid:1) tt/mt/it/ct/so (cid:1) (cid:2)(cid:3)(cid:4) (cid:1) (cid:5)(cid:3)(cid:4) (cid:9)un(cid:12)(cid:13) (cid:6) (cid:7) tt/mt/it/ct/so (cid:9)uc(cid:12)(cid:13) (cid:6) (cid:2)(cid:3)(cid:4) (cid:6) (cid:5)(cid:3)(cid:4) (cid:7)/ (cid:9) (cid:7)/ (cid:10) (cid:7)/ (cid:4) (cid:11) (cid:7) tt/mt/it/ct/so.............................................(cid:13).5 (cid:1) ) a(cid:18)(cid:19)(cid:20)(cid:21)emt mn )t )n it in mt mn )t )n it in (cid:23)(cid:24)(cid:25)(cid:26)/.(cid:27)(cid:28)(cid:23)(cid:24)3(cid:20)(cid:30)(cid:31)/.(cid:25)(cid:20) (cid:28)(cid:26)/(cid:24)(cid:20).(cid:27)(cid:28)(cid:23)(cid:24)3(cid:20)(cid:30)(cid:31)/.(cid:25)(cid:20) (cid:28)(cid:26)/ FIG. 4: Quasi-particle renormalization weights of the 7 / / f -electrons (upper panels), 7 / / f orbital popula-tions (central panels), and f configuration probabilities (lowerpanels), as a function of the volume of the system. The renor-malization weights and the configuration probabilities are re-ported both for the case with and without the spin-orbit cou-pling, for U = 5 eV (left panels) and U = 6 eV (right panels)at fixed J = 0 . eV . The vertical continue lines indicate thesignature of the transition in the pressure-volume diagram,and the dotted vertical lines indicate the boundary of the γ - α transition (which occurs only for U ≤ U c ≃ . eV ) accordingto the equal-area construction [31]. It is useful to examine how the local crossover inducedby the SOC reflects on the quasi-particle renormalizationweights and the on-site configuration probabilities. In thefirst panel of Fig. 4 are illustrated the averaged quasi-particle renormalization weights Z of the 7 / / f -electrons — which are significantly different because ofthe spin-orbit effect. As expected [32], the f -electronsare correlated ( Z is significantly smaller than 1) even inthe α phase, and the two Z ’s monotonically decrease byincreasing the volume at higher pressures. Nevertheless,they develop a qualitatively different behaviour at thecrossover point. While the 5 / / γ phase, see the second panel ofFig. 4. As shown by the f configuration probabilities inthe third panel of Fig. 4, the SOC speeds up the forma-tion of the 4 f local moment. In other words, the SOCacts as a “catalyst”, which favors the disentanglement between the 4 f electrons and the conduction electrons.In conclusion, we have performed first principle calcu-lations on Cerium using a new efficient implementationof the LDA+GA method. For the physical value of U inCerium, U ≃ eV [9, 30], a sharp crossover is observedin many physical quantities around the volume where thebulk-modulus is minimum. This finding is robust againstchanges in U , but the details are different. For U < U c ,at T = 0, there is a first-order transition at a given nega-tive value of the pressure, while for U > U c this transitionbecomes a sharp crossover. At U c there is a second-orderquantum critical point in the phase diagram. Our esti-mate for the critical interaction strength, U c ≃ . eV ,is very close to the physical value of the interactionstrength in Cerium. This finding suggests that elementalCerium is a critical element, consistently with the ex-periments [33]. Our results demonstrate the importanceof the SOC for the volume collapse in Cerium, whichis neatly captured by the rapid variation of the entan-glement entropy of the f -electrons in the region aroundthe minimum of the bulk-modulus. In the α phase, atsmall V , the f -levels are strongly hybridized with theconduction electrons, and the quasiparticle weights andthe pressure are only weakly-dependent on the spin-orbitinteraction. In this regime the system effectively behavesas if the f -level degeneracy was of the order of magnitudeof 14. In the γ phase, at large V , the spin-orbit split-ting becomes more important, and substantially reducesthe effective f -level degeneracy. The fact that the quasi-particle weight in the γ phase is much smaller when theSOC is taken into account, see Fig. 4, can be interpretedas a consequence of the above-mentioned reduction of ef-fective f -level degeneracy — a well known effect in thetheory of the single-ion Kondo impurity. As in the earlyKondo volume collapse theory [3], ∂Z/∂V contributes tothe pressure. However, some qualitative features of oursolution, such as the form of the pressure-volume phasediagram, show that other physical elements, such as thechanges in the charge density induced by the correlations,have to also be included in realistic theories of this ma-terial. CONTRIBUTIONS
N.L. and Y.X.Y. co-developed the GA code andanalyzed the data. N.L. developed the theoreticalexplanation of the iso-structural transition, wrotethe manuscript and carried out the LDA+DMFTbenchmark calculations. Y.X.Y., C.Z.W. and K.M.H.initiated the project and carried out the LDA+GAcalculations. Y.X.Y. edited the figures, and coded theinterface between Wien2k and the GA solver, which wasconstructed on the basis of the LDA+DMFT interfacedeveloped by K.H.. K.H. developed the DMFT code.G.K. and J.S. proposed the project and supervised theresearch. All the authors provided fundamental insights,and contributed to improve the manuscript.
Corresponding author:
Y.X.Yao; [email protected]
ACKNOWLEDGMENTS
N.L. and Y.X.Y. thank XiaoYu Deng and Robert Mc-Queeney for useful discussion. The collaboration wassupported by the U.S. Department of Energy throughthe Computational Materials and Chemical Sciences Net-work CMSCN. Research at Ames Laboratory supportedby the U.S. Department of Energy, Office of Basic EnergySciences, Division of Materials Sciences and Engineering.Ames Laboratory is operated for the U.S. Departmentof Energy by Iowa State University under Contract No.DE-AC02-07CH11358. ∗ Equally contributed to this work[1] Koskenmaki, D. C. & Jr., K. A. G. Chapter 4 cerium.In Karl A. Gschneidner, J. & Eyring, L. (eds.)
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In our LDA+GA code the LDA calculations are per-formed by Wien2k [1], which is an all-electron DFT pack-age based on the full-potential (linearized) augmentedplane-wave ((L)APW)+local orbitals (lo) method. A keyadvantage of this package is that it is one among themost accurate schemes for band-structure calculations,and it is free of any numerical error due to the frozen-corepseudopotentials or from the downfolding, which havebeen used in previous implementations of the LDA+GAmethod.Our new LDA+GA implementation, to be describedin a longer publication [Y.Y.X. et al.], is patterned afterthe LDA+DMFT work of Ref. [2]. Using the same in-terface, basis set, projectors onto the correlated orbitals,enables a meaningful comparison between the two meth-ods. We have employed a new numerical implementationof the GA solver [L.N. et. al, Strand Hugo U.R. et. al],which further improves the method previously proposedin Ref. [3], and is a generalization of earlier formulationsof the GA [4–8].
LDA+DMFT BENCHMARK CALCULATIONS
The purpose of this section is to benchmark ourLDA+GA calculations within the LDA+DMFT method,using the Continuous Time Quantum Monte Carlomethod (CTQMC) [9] — which is numerically-exact —as the impurity solver. We use the implementation orRef [10].Our DMFT calculations are all performed at T =58 K , while the GA calculations are done at zero tem-perature.In Fig. 5 are shown the local configuration probabilities(upper panels) and the f entanglement entropy (lowerpanels) as a function of the volume, for two different val-ues of U . The agreement between the two methods isindeed very good. In particular, we point out that the f local crossover, which is indicated by the rapid change ofthe corresponding entanglement entropy, is clearly visiblealso within the LDA+DMFT method.In Fig. 6 it is shown the evolution of the quasi-particlerenormalization weights as a function of the volume (up-per panels) and the imaginary part of the self-energyat the Fermi level (lower panels). Note that in the γ phase, i.e., after the local crossover, the 5 / Z / is not well defined at large volumes. (cid:1)(cid:2)(cid:3):(cid:5): (cid:6) (cid:1)(cid:2)(cid:3):(cid:5): (cid:7) (cid:1)(cid:2)(cid:3):(cid:5): (cid:8) (cid:1)(cid:2)(cid:3):(cid:5): (cid:9) (cid:10) (cid:5) ggtmgtrgtpgt3l (cid:11)(cid:12)(cid:13)(cid:14)(cid:3):(cid:5): (cid:6) (cid:11)(cid:12)(cid:13)(cid:14)(cid:3):(cid:5): (cid:7) (cid:11)(cid:12)(cid:13)(cid:14)(cid:3):(cid:5): (cid:8) (cid:11)(cid:12)(cid:13)(cid:14)(cid:3):(cid:5): (cid:9) (cid:8)Mo(cid:11)(cid:12)(cid:13)(cid:14) (cid:15) (cid:16) (cid:17) (cid:18) (cid:16)(cid:19) (cid:20) (cid:11) (cid:21) (cid:11) (cid:16) (cid:17) (cid:11) (cid:16) (cid:17) (cid:23) (cid:24)(cid:25)(cid:26) mmto torrto2222222222222222222222222222222222222222222222222222222222222(cid:12)2e (cid:1) a(cid:18)(cid:17)(cid:24)(cid:21)Vlo mg mo g o rg ro (cid:8)Mp(cid:11)(cid:12)(cid:31) !"lo mg mo g o rg ro FIG. 5: Local configuration weights (upper panels) and f entanglement entropy (lower panels). Comparison betweenLDA+GA (lines) and LDA+DMFT (dots) results for U =5 eV (left panels) and U = 6 eV (right panels) at fixed J =0 . eV . (cid:1) iitgit−it3iton (cid:1) (cid:2)(cid:3)(cid:4) (cid:1) (cid:5)(cid:3)(cid:4) (cid:9)7((cid:12)(cid:13) (cid:14) (cid:15)(cid:16) (cid:17)(cid:17) (cid:12) (cid:18) (cid:19) (cid:20)(cid:21) . (cid:18) (cid:16) (cid:17) (cid:12) (cid:23)ntg((cid:23)n(cid:23)it(i.......................................................(cid:13).5 (cid:1) r(cid:16)(cid:17)(cid:27)(cid:28)en( gi g( i ( −i −( (cid:9)73(cid:12)(cid:13)(cid:30)(cid:28)(cid:31) i(rg (cid:30)(cid:28)(cid:31) i/rg n( gi g( i ( −i −( FIG. 6: Quasi-particle renormalization weights of the 7 / / f -electrons (upper panels), and imaginary part of theself-energy at the Fermi level (lower panels). Comparisonbetween LDA+GA (lines) and LDA+DMFT results (dots)for U = 5 eV (left panels) and U = 6 eV (right panels) atfixed J = 0 . eV . ∗ Equally contributed to this work[1] Blaha, P., Schwarz, K., Madsen, G., Kvasnicka, D. &Luitz, J.
An augmented plane wave plus local orbitalsprogram for calculating crystal properties. University ofTechnology, Vienna (2001).[2] Haule, K., Yee, C.-H. & Kim, K. Dynamical mean-field theory within the full-potential methods: Electronicstructure of CeIrIn , CeCoIn , and CeRhIn . Phys. Rev.B , 195107 (2010). [3] Lanat`a, N., Strand, H. U. R., Dai, X. & Hellsing, B.Efficient implementation of the gutzwiller variationalmethod. Phys. Rev. B , 035133 (2012).[4] Fabrizio, M. Gutzwiller description of non-magneticMott insulators: Dimer lattice model. Phys. Rev. B ,165110 (2007).[5] Lanat`a, N., Barone, P. & Fabrizio, M. Fermi-surface evo-lution across the magnetic phase transition in the Kondolattice model. Phys. Rev. B , 155127 (2008).[6] Lanat`a, N., Barone, P. & Fabrizio, M. Superconductivityin the doped bilayer hubbard model. Phys. Rev. B ,224524 (2009).[7] Deng, X., Wang, L., Dai, X. & Fang, Z. Local den-sity approximation combined with gutzwiller method for correlated electron systems: Formalism and applications. Phys. Rev. B , 075114 (2009).[8] B¨unemann, J., Weber, W. & Gebhard, F. Multibandgutzwiller wave functions for general on-site interactions. Phys. Rev. B , 6896–6916 (1998).[9] Werner, P., Comanac, A., de’ Medici, L., Troyer, M. &Millis, A. J. Continuous-time solver for quantum impu-rity models. Phys. Rev. Lett. , 076405 (2006).[10] Haule, K. Quantum monte carlo impurity solver for clus-ter dynamical mean-field theory and electronic structurecalculations with adjustable cluster base. Phys. Rev. B75