Multiple Quantum Phase Transitions of Plutonium compounds
Munehisa Matsumoto, Quan Yin, Junya Otsuki, Sergey Yu. Savrasov
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Multiple Quantum Phase Transitions of Plutonium compounds
Munehisa Matsumoto , Quan Yin , , Junya Otsuki , Sergey Yu. Savrasov Department of Physics, University of California, Davis, California 95616, USA Department of Physics & Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA, Department of Physics, Tohoku University, Sendai 980-8578, Japan (Dated: December 13, 2018)We show by quantum Monte Carlo simulations of realistic Kondo lattice models derived fromelectronic–structure calculations that multiple quantum critical points can be realized in Plutonium–based materials. We place representative systems including PuCoGa on a realistic Doniach phasediagram and identify the regions where the magnetically mediated superconductivity could occur.Solution of an inverse problem to restore the quasiparticle renormalization factor for f -electrons isshown to be sufficiently good to predict the trends among Sommerfeld coefficients and magnetism.Suggestion on the possible experimental verification for this scenario is given for PuAs. PACS numbers: 71.27.+a, 75.30.Kz, 75.40.Mg
Motivation
Discovery of unconventional supercon-ductivity in PuCoGa [1, 2] opened a new arena for thestudies of strongly-correlated materials. It has the high-est superconducting transition temperature T c = 18 . f -electron-based materials and it has been dis-cussed to reside somewhere in between the Cerium-based heavy fermion (HF) superconductors and high- T c cuprates, where the latter still challenges theoretical con-trol from first-principles.In the present work we make predictions on themagnetism and HF behavior of several Pu compoundsincluding Pu-115’s where the mechanism of possiblemagnetically-mediated superconductivity is discussed tobe more complicated than their Cerium counterparts [3].Furthermore, with experimental challenges such as theself-heating of samples due to the radiative nature of Punuclei, a computational guide should be of help regardingthe determination of the linear coefficient of electronicheat capacity, so-called Sommerfeld coefficient γ . Ourcomputational method is based on a recently-developedscheme for realistic Kondo lattice simulations [4, 5] whichenabled us to predict the location of magnetic quantumcritical point (QCP) [6] from electronic structure calcu-lations [7] for HF materials.Our main results are shown in Fig. 1 in the format ofDoniach phase diagram [8] plotted with realistic settingsfor the target materials. Striking double-dome structureis seen both in the magnetic Doniach phase diagram ofPuCoGa plotted on ( J K , T N )-plane and an analogousplot on ( J K , γ )-plane, where J K is the Kondo couplingand T N is the Neel temperature. For Pu-115’s there areat least two antiferromagnetic phases with at least threeQCP’s. We see that Pu-115’s are indeed separated fromthe first i.e. lowest-energy magnetic QCP, being consis-tent with the situation discussed in Ref. [3]. However wefind that the second and third QCP’s are encountered on J K -axis and the realistic point for Pu-115 is actually inthe proximity to the latter. γ [ m J / m o l/ K ] J K [K] PuCoGa PuRhGa T N [ K ] J K [K] FIG. 1: (Color online) Summary of our results for Pu-115’son realistic Doniach phase diagram. The realistic data pointis indicated by the symbol on the line for each target ma-terial. The main panel show the trend among Sommerfeldcoefficients and the inset show the N´eel temperatures.
Methods
The electronic structure calculation basedon local density approximation (LDA) combined with dy-namical mean-field theory (DMFT) (LDA+DMFT) hasbeen successful in addressing interesting properties ofstrongly-correlated materials [7]. What motivates us forthe Kondo-lattice model (KLM) description of HF ma-terials is that efficient and exact quantum Monte Carlo(QMC) simulations in low-temperature region are possi-ble [9], typically around O (10) [K] and down to O (1) [K].This advantage is due to having only f -spins and elim-inating the f -electron charge degrees of freedom viaSchrieffer-Wolff transformation [10] implemented in a re-alistic way [4]. This is in contrast to the fact that thestandard LDA+DMFT based on solving the Andersonmodel, which was used e.g. in Ref. [11] for δ -Pu, typi-cally can reach the temperatures down to O (100) [K] ifthe core impurity problem is to be exactly solved by QMCmethod. Here, some basis-cutoff schemes have been im-plemented [12] to reduce the computational cost.One of the reasons Plutonium compounds have beeninteresting and difficult to address is that they reside on material N tot (0) N f (0) − Tr ℑ ∆(0) /π [eV]PuCoGa δ -Pu 20.40. . . 2.781. . . 1.13PuSe 17.30. . . 4.342. . . 0.619PuTe 13.45. . . 0.7729. . . 0.360PuAs 8.351. . . 0.6068. . . 0.255PuSb 10.01. . . 0.2699. . . 0.193PuBi 7.723. . . 0.2534. . . 0.133TABLE I: Summary of LDA+Hubbard-I results for targetmaterials. The unit of density of states, N tot (0) for allelectrons and N f (0) for f -electrons on the Fermi level, is[states/Ry/cell]. the border of itinerancy and localization of 5 f -electronsamong actinides [13, 14]. At least for Pu-chalcogenidesand pnictides, experimental evidence for localized 5 f -electrons was revealed [15] and a recent theoreticalwork [16] agrees with that so these can be benchmarkcases for the realistic KLM simulations. For Pu-115’sit has been known that the Curie law persists down tothe superconducting temperature for PuCoGa [1] whichsupports the presence of localized 5 f -electrons, but someattention must be reserved for a possible sample depen-dence: radiative Pu decays into U, which can introducemagnetic impurities. It is thus controversial whether theCurie-Weiss law is intrinsic or not [17]. Analyses of ex-periments point to n f = 5 .
03 for δ -Pu [13] which webelieve is sufficient for the KLM to work. The valence de-duced from the calculations shows a much larger spread(between 4 [18] and 6 [19]) with the values of 5.2 for δ -Pu [20] and 5.26 for PuCoGa [21] from most accurateCT-QMC calculations.Our realistic KLM framework for the above-mentionedPu compounds goes as follows. At the first stage LDA for s , p , and d -conduction electrons and Hubbard-I approx-imation [22] for the self-energy of localized f -electronsgives us the partial densities of states and hybridiza-tion functions as prescribed by LDA+DMFT frame-work [7]. The data are summarized in Table I. Itis clear that Pu-115’s have much higher energy scalesthan the Cerium ones [5]. At the second stage wesolve the low-energy effective KLM Hamiltonian withdynamical-mean field theory [23, 24], utilizing state-of-the-art continuous-time quantum Monte Carlo (CT-QMC) impurity solver [9, 12, 25]. For the 5 f -orbitals ofPu, it is known that there is a big spin-orbit splittingof 1 [eV] and the five possibly localized electrons fill inthe lower j = 5 / f level. Solving an inverse problem to restore f -electrons Even if we eliminated f -electrons and kept only f -spinsin our KLM, part of the information for the localized f -electrons can be restored from the relation Σ c ( iω n ) ≡ V / [ iω n − ǫ f − Σ f ( iω n )] , where Σ c is our conduction-electron self energy, iω n = (2 n + 1) πT is the Matsubarafrequency, ǫ f = − f is the f -electron self-energy which we do nothave explicitly in our KLM calculations. Provided thatwe reach the temperature for a given target material to bein a Fermi-liquid region concerning its f -electrons, whichis mostly the case for Pu compounds, the quasiparticlerenormalization factors are well defined and written as z x = (1 − ∂ ℑ Σ x ( iω n ) /∂ ( iω n )) − , with x = c and f forconduction electrons and f -electrons, respectively. Herethe derivative is taken at iω n = 0. We get from theabove definition of Σ c the following inversion relation : z f = [ | Σ c (0) | /V ] z c / (1 − z c ) . Because z ’s are written interms of the derivative of the corresponding self-energyat the lowest frequency, our effective low-energy descrip-tion based on KLM enables a good solution of this in-verse problem as far as z f is concerned. The Sommerfeldcoefficient γ = (1 / π N eff (0), where N eff (0) is the effec-tive total density of states (DOS) on the Fermi level, canbe estimated by N eff (0) = N c (0) /z c + N f (0) /z f , where N c (0) is DOS of s , p , d -conduction electrons and N f (0)is that of localized f -electrons in our LDA+Hubbard-Icalculations. With a given KLM, we extract z c from Σ c obtained after DMFT, invert it to z f , and get the Som-merfeld coefficient γ with the above formula. In this waywe can restore an analogue of Doniach phase diagram for γ as was shown in Fig. 1 for Pu-115’s. The results onthe realistic data point for each target material are sum-marized in Table II together with the experimental datataken from the literature. Our prediction follows the ex-perimental trend among γ semi-quantitatively. We notethat γ is sensitive to the estimate of the realistic point of J K especially around QCP’s, considering the sharp peakstructure as seen in Fig. 1 for the plot of γ vs J K . Sothe overall trend among materials is the most importantresult. Magnetism and quasiparticle renormalizations
Theresults for magnetism are schematically summarized inFig. 2 for all target materials in the format of a rescaledDoniach phase diagram. It illustrates how we understandthe results in the inset of Fig. 1 for Pu-115’s. Strikingmulti-dome structure shows up together with multipleQCP’s for materials with strong Kondo coupling. We findthat Pu-115’s are located in a region where antiferromag-netic long-range order is suppressed, possibly near a hid-den or pseudo-QCP, within some numerical noise at thelowest reachable temperatures at present. Inspecting thedistribution of materials around the QCP’s in Fig. 2, wehave pnictides on the left-hand side and chalcogenides onthe right-hand side of the antiferromagnetic QCP. Thisis consistent with what has been known experimentally, material z f z c our γ experimental γ PuCoGa a -116 b PuRhGa a -80 c δ -Pu 0.0625 0.0738 49 50-64 d PuSe 0.0480 0.0232 110 90 e PuTe 0.00883 0.0313 85 30 f -60 e PuAs 0.0588 0.00456 295PuSb 0.0231 0.0168 102 6 b -20 g PuBi 0.00202 0.0965 35 a Ref. [30] b Ref. [31] c Ref. [32] d Ref. [33] e Ref. [29] f Ref.[34] g Ref.[35]TABLE II: Summary of our data obtained with realisticKondo lattice simulations and our prediction for γ based onthem. The unit of γ is [mJ/mol/K ]. Experimentally knownresults are taken from the literature. t=(J K − J K,QCP )/ J K,QCP T N [K] AF~100 PuAsPuBi PuSbPuTe PuSe PM0 0T N [K] AF δ -PuPuCoGa PuRhGa ~10 AF t=(J K − J K,QCP )/ J K,QCP
00 PM (a)(b)-0.5-0.5 0.5 0.5
FIG. 2: (Color online) Schematic summary of our magneticphase diagrams for Pu compounds plotted on the ( t, T N )-planewhere t ≡ ( J K − J K , QCP ) /J K , QCP is the rescaled Kondo cou-pling with J K , QCP being the first QCP in (a) and the thirdQCP in (b). that is, pnictides such as PuAs, PuSb [27], and PuBi [28]are magnets and chalcogenides such as PuSe and PuTeare paramagnets [29]. The actual magnetism is stronglyspatially anisotropic [14, 28] whose treatment is unfortu-nately beyond the level of single-site DMFT description.For now we will leave the issue of ordering wavevectorsfor future projects and focus on the trends across targetmaterials spanning between magnetism and HF behav-ior. The characteristic energy scales of Kondo-screeningand magnetic ordering have been captured by fully incor-porating the frequency-dependence of the hybridization.Multi-dome structure together with multiple QCP’sshown in Fig. 1 for Pu-115’s can be understood in termsof strong-coupling nature of the Kondo lattice [36] based (a) T K [ K ] J K [K]realistic data point (b) z J K [K] z f z c FIG. 3: (Color online) Analogue of Doniach phase diagramfor (a) Kondo temperature and (b) quasiparticle renormal-ization factors for PuCoGa . The inset in (b) is a zoom-uppicture around the first QCP with the vertical axis plotted inlogarithm. on the growth of characteristic Kondo energy scale T K with respect to J K as shown in Fig. 3 (a) which is ob-tained from our local susceptibility data. Simple pertur-bative arguments also prompt for two crossover pointsbetween T RKKY ∼ J ρ and T K ∼ exp[ − / ( J K ρ )] /ρ tem-perature scales, where the latter can saturate at somepoint with respect to large J K while the former keeps ongrowing. Here ρ is the characteristic DOS.We demonstrate our predictive power regarding T K also for the case of δ -Pu where we get T K ∼ [K] fromour local susceptibility data which is seen to be closeto the previous results, T K ∼
700 [K] in Ref. [20]. Wenote that the Kondo screening energy scale is approach-ing a comparative scale to the characteristic bandwidth,or the kinetic energy of the conduction electrons whichis O (1) [eV]. Such situation had been discussed in theliterature [36, 37] in the context of models, which is nowfound to be realized in Plutonium heavy-fermion materi-als.The behavior of quasiparticle renormalization factors z x ( x = c or f ) as shown in Fig. 3 (b) further givesthe physical picture: starting from J K = 0 where thereare free conduction electrons and completely localized f -electrons with ( z c , z f ) = (1 , f -electrons gets“delocalized” in the sense that they start to take partin Fermi surface (FS) [38, 39]. Passing the first QCP,heavy quasiparticles composed both of conduction elec-trons and f -electrons evolve together after z c has shown adip around the first QCP, letting f -spins show up againwith the underscreening effects that correspond to theslightly elevated z c . Thus after the revival of magnetismthe same thing can happen again and could repeat it-self, with the re-defined much smaller energy scales everytime, all the way to the J K → ∞ limit, being consistentwith the phase diagram obtained in Ref. [36].This strong-coupling KLM scenario for Pu compoundscan in principle be checked by de Haas-van Alphen ex-periments which would measure the size of the FS tosee if it counts the number of “localized” f -electrons.The ferromagnetic phase of PuAs should be the onewith the “large” FS including the spins of localized 5 f -electrons. This phase would be in contrast to the typi-cal magnetic phases in HF compounds with the “small”FS, being located in the weak-coupling region. Followingthe method of Ref. [39], we can track the evolution oflarge FS for representative Pu compounds obtained from −ℜ Σ c ( iω n ) | iω n =0 and we find that PuAs shows a remark-able evolution of “large” FS, which is to be comparedwith experiments to see if the strong-coupling KLM pic-ture can hold. Conclusions
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