Phase Diagram and Electronic Structure of Praseodymium and Plutonium

PHYSICAL REVIEW X 5, 011008 (2015)

† Nicola Lanatà,1,* Yongxin Yao,2, Cai-Zhuang Wang,2 Kai-Ming Ho,2 and Gabriel Kotliar1 1Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08856-8019, USA 2Ames Laboratory-U.S. DOE and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA (Received 26 May 2014; revised manuscript received 18 September 2014; published 29 January 2015) We develop a new implementation of the Gutzwiller approximation in combination with the local density approximation, which enables us to study complex 4f and 5f systems beyond the reach of previous approaches. We calculate from first principles the zero-temperature phase diagram and electronic structure of Pr and Pu, finding good agreement with the experiments. Our study of Pr indicates that its pressure-induced volume-collapse transition would not occur without change of lattice structure—contrarily to Ce. Our study of Pu shows that the most important effect originating the differentiation between the equilibrium densities of its allotropes is the competition between the Peierls effect and the Madelung interaction and not the dependence of the electron correlations on the lattice structure.

DOI: 10.1103/PhysRevX.5.011008 Subject Areas: Condensed Matter Physics, Materials Science, Strongly Correlated Materials

I. INTRODUCTION the projector-augmented wave method [15], and linearized muffin-tin orbitals [16]. Another important approach— There has been a renewed interest in first-principles which is not as accurate as DMFT but has the advantage approaches to the electronic structure of strongly correlated — materials. Density functional theory (DFT)—and, in par- to be less computationally demanding is the Gutzwiller – ticular, the local density approximation (LDA)—proved to approximation (GA) [17 19], which was first implemented be a good starting point for deriving model Hamiltonians to study real solids in Ref. [20]. The GA approximation – [1,2] that can be studied with more elaborate methods that was, thereafter, extensively developed [21 27], and it are able to treat correlations. While early approaches to has been formulated and implemented in combination solve the realistic many-body problem in solids focused with realistic electronic structure calculations such as the on perturbative treatments of the interactions [3], over the LDA þ GA approach [23,28], which has been applied last two decades several nonperturbative methodologies successfully to many systems [29–36]. A third important have emerged. Dynamical mean field theory (DMFT) was many-body technique is the slave boson approach (SB) combined with realistic electronic structure methods, for [37,38], which is, in principle, an exact reformulation of example, in the LDA þ DMFT approach [4,5]. This the quantum many-body problem for model Hamiltonians, methodology can be thought of as a spectral density and it reproduces the results of the GA at the saddle-point functional [6] and is nowadays widely used to study 3d, level [24,39]. This technique has recently been extended to 4d, 5d, 4f, and 5f systems. For reviews see, e.g., treat full rotationally invariant interactions [38,40], and it Refs. [7–11].LDAþ DMFT has been implemented in has also been combined with LDA for the study of real different basis sets, such as the linearized augmented plane materials, either in the form of impurity solvers for the wave (LAPW) [12,13], plane-wave pseudopotentials [14], resulting LDAþDMFT impurity models (LDAþDMFTþ SB) [6,41] or directly on the lattice [42]—which are *Corresponding author. equivalent approaches, as we will show. [email protected] On the methodological side, here we show that the † Corresponding author. three above-mentioned methods are closely connected and [email protected] largely complementary of each other. We use the connection between the GA and the SB methods to introduce a Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- functional formulation, which can be used not only at zero bution of this work must maintain attribution to the author(s) and temperature but also at finite temperatures. This functional the published article’s title, journal citation, and DOI. is a first step toward deriving formulas for the forces [43]

2160-3308=15=5(1)=011008(28) 011008-1 Published by the American Physical Society LANATÀ et al. PHYS. REV. X 5, 011008 (2015) and the phonon spectra in the LDA þ GA and LDA þ SB temperature-induced structure transitions are accompa- methods. Our functional formulation of the LDA þ GA nied by significant changes of density. Here, we perform method [23] has a mathematical structure similar to LDA þ LDA þ GA calculations of all of the six phases of Pu and DMFT [7]. This parallelism suggests possible synergistic study how the total energy and the f-electron correlations combinations between the two methods—such as using depend on the volume and the lattice structure. These LDA þ GA for structural relaxation while using the results provide a complete bird’s eye view of this material exact impurity solver in the LDA þ DMFT iteration to and, in particular, indicate that the most important effect determine the spectral properties. Furthermore, it enables originating the above-mentioned large differences us to pattern the LAPW interface [44] between LDA and between the equilibrium volumes of the phases of Pu is the GA after the LDA þ DMFT work of Ref. [12]. These the competition between the Peierls effect and the connections result in a new algorithm for solving the Madelung interaction and not the dependence of the LDA þ GA equations, which is faster and more precise electron correlations on the lattice structure, which we than earlier methods and sheds light on the physical find to be a negligible effect. We point out that the interpretation of the SB amplitudes—which are central quantities both in the GA and in the SB approach. In fact, explanation of this phenomenon is of great interest both we display a connection between the SB amplitudes and physically and from a metallurgic standpoint. the coefficients of the Schmidt decomposition [45]-which The outline of the paper is as follows. In Sec. II, the was also recently used to derive the density matrix formulation and the implementation of the GA/SB devel- embedding theory [46–49]. Our algorithm consists in oped in Refs. [22,24,25,27] are substantially improved. recursively calculating the ground state of a series of In Sec. III, the connection between the GA/SB and DMFT Anderson impurity Hamiltonians (one for each inequivalent is discussed. In Sec. IV, our functional formulation of the impurity within the lattice unit cell), whose baths have LDA þ GA method is derived. Finally, in Secs. V and VI, the same dimension as the corresponding impurities. This our calculations of Pr and Pu are illustrated. enables us to derive accurate equations of state for materials currently far beyond the reach of LDA þ DMFT. The technical advances obtained in this work result in a II. THE GUTZWILLER APPROXIMATION new understanding of the volume-collapse transition in f FOR THE HUBBARD MODEL systems. In particular, we use our all-electron (LAPW) Let us consider the Hubbard model (HM) implementation of the LDA þ GA method to study two X X X X f ˆ αβ † ˆ loc † prototypical systems with partially delocalized electrons: H ¼ ϵk;ijcˆkiαcˆkjβ þ HRi ½fcˆiαg; fcˆiαg; elemental Pr and Pu. k ij αβ Ri Pr is a rare earth like Ce, and it is the next element in ð1Þ the periodic table. An interesting property of Pr is that, similarly to many other rare-earth compounds, it undergoes where k is the momentum conjugate to the unit-cell label R, a volume-collapse structure transition under pressure, the atoms within the unit cell are labeled by i; j, and the which is accompanied by an abrupt delocalization of the spin orbitals are labeled by α; β. We assume that the first f electrons [50]. Here, we compute its pressure-volume term is nonlocal, i.e., that phase diagram, finding very good agreement with the X experiments. In particular, we show that the method is ϵk;ii ¼ 0 ∀ i; ð2Þ able to capture the pressure-induced volume-collapse k structure transition toward the low-symmetry α-U phase, and that the GA correction to the total energy is crucial to and that Hˆ loc includes both the one-body and the two-body correctly determine the stable lattice structure of Pr. Finally, part of the Hamiltonian. Note that no specific assumption we investigate the relation between the f delocalization and needs to be made on the structure of the local interaction. the volume-collapse structure transition—which is one of In particular, in this work we have used the rotationally the most important puzzles in condensed matter physics. invariant Slater-Condon parametrization of the on-site Our main conclusion is that, contrarily to Ce [35],inPr interaction [51]. there would not be any volume-collapse transition without We define the temperature T and the corresponding taking into account the change of lattice structure (at least at fermionic Matsubara frequencies as ω ¼ð2m þ 1ÞπT . low temperatures). As shown in Appendix A, the SB mean-field theory The stable allotrope of Pu at ambient conditions is [37,38] can be formulated in terms of the following α-Pu, and five different crystalline phases (named Lagrange function, which gives the free energy when it β; γ; δ; δ0; ϵ) can be stabilized at higher temperatures. is evaluated at its saddle point and reduces to the GA One of the most intriguing properties of Pu is that these at T ¼ 0:

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c † † c 0 LN½ϕ;E ; R; R ; λ; η; μ; D; D ; λ ; n T X 1 eiω0þ ¼ N Tr log iω − Rϵ R† − λ − η μ k;ω k þ X X X X † loc † † c † † c † þ Tr ϕiϕi Hi þ ð½Diaαϕi FiαϕiFia þ H:c:Þþ ½λi abϕi ϕiFiaFib þ Ei ð1 − Tr½ϕi ϕiÞ i aα ab i XX X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c 0 0 0 − ð½λiab þ½λi abÞ½ni ab þ ð½Diaα½Riaα þ c:c:Þ ½ni aað1 − ½ni aaÞ þ μN; ð3Þ i ab aα

c where the dimension Mi of the matrices Ri; Di; λi; λi ; ηi is and in the last step of Eq. (7), we used that the renorm- the number of spin orbitals within the ðR; iÞ space; R, λ and alization factors in the quasiparticle Green’s function give η are block matrices whose blocks are Ri, λi and ηi, only a frequency-independent term, which does not con- respectively; μ is the chemical potential; N is the total tribute because of the Matsubara summation. Note that number of electrons (normalized to the number of k points since R, λ, and η are block matrices, ΣðzÞ is also block loc N ); the matrices Fia and Hi represent the local operators diagonal, with blocks ˆ ˆ loc fRia and HRi in a given (arbitrary) basis set of local multiplets jΓ;Rii, I − R†R 1 1 I − R†R Σ z ≡−z i i λ η − μ i i : ið Þ † þ R ð i þ iÞ † † Ri Ri i Ri Ri Ri ˆ 0 ½FiaΓΓ0 ≡ hΓ;RijfRiajΓ ;Rii; ð4Þ ð9Þ

Hloc ≡ Γ;RiHˆ loc cˆ† ; cˆ Γ0;Ri ; ½ i ΓΓ0 h j Ri ½f iαg f iαgj i ð5Þ Once the GA solution is determined by imposing the saddle-point conditions of LN with respect to all of its M and ϕi are the SB amplitudes, which are 2 i -dimensional arguments, the expectation value of any observable can be matrices such that readily computed. In particular (see Appendix A), it can be proven that the expectation value of any local operator X † within the site ðR; iÞ is given by ϕi; FiαFiα ¼ 0 ∀ i; ð6Þ α ˆ † † hARi½fcîαg; fcîαgiT ¼ Tr½ϕiϕi Ai; ð10Þ i.e., that couple only local multiplets with the same number of electrons (which amounts to restricting ourselves to ˆ where Ai is the representation of ARi in the same basis used nonsuperconducting phases). in Eqs. (4) and (5), Note that the above finite-temperature extension has been recently used in Ref. [36] to study the thermody- ˆ † 0 ½AiΓΓ0 ≡ hΓ;RijARi½fcîαg; fcîαgjΓ ;Rii: ð11Þ namical properties of Ce in relation to its α-γ isostructural volume-collapse transition. For later convenience, we observe that the first term of Eq. (3) can be equivalently rewritten as A. Physical interpretation of the parameters ϕi T X 1 based on the Schmidt decomposition eiω0þ Tr log † In this section, we show that the space of matrices ϕi N iω − RϵkR − λ − η þ μ k;ω [see the second line of Eq. (3)] can be conveniently mapped X T 1 1 1 into the Hilbert space of states Φi of an impurity system eiω0þ j i ¼ N Tr log R† iω − ϵ μ − Σ iω R composed of the i impurity and an uncorrelated bath with k;ω k þ ð Þ X the same dimension, which is determined self-consistently T 1 ˆ loc eiω0þ ; in order to describe the entanglement between Hi and the ¼ N Tr log iω − ϵ μ − Σ iω ð7Þ k;ω k þ ð Þ rest of the system [see Eq. (1)]. Let us consider the impurity local many-body space Vi where generated by the same Fock fermionic basis that has been used in Eqs. (4) and (5), I − R†R 1 1 I − R†R Σ z ≡−z λ η − μ ; † νΓ † νΓ ð Þ † þ ð þ Þ † † ð8Þ Γ;i cˆ 1 … cˆ Mi 0 ; R R R R R R j i¼½ i1 ½ iMi j i ð12Þ

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Γ where να are the occupation numbers of jΓ;ii, so that the where total number of electrons corresponding to this state is X X ˆ tot ˆ † ˆ † X Ni ≡ fiafia þ cîαcîα ð22Þ Γ NΓ ≡ να: ð13Þ a α α¼1;…;Mi is the total number operator in the embedding system Ei, W V We define a copy of i of i generated by another set of and Mi is the number of spin orbitals in the R; i space. In n; i Fock states j i represented as fact [see Eq. (16)], jΦii is a linear combination of product NΓ Mi − Nn NΓ Nn † νn † νn states with þ electrons, with ¼ . n; i fˆ 1 … fˆ Mi 0 : ˆ j i¼½ i1 ½ iMi j i ð14Þ It can be readily verified that, for any operator A acting within the space generated by the states jΓ;ii, Note that, since the f ladder operators act only on the Wi degrees of freedom, they anticommute with all of the c ˆ † † n hΦijA½fcîαg; fcîαgjΦii¼Tr½ϕiϕi Ai; ð23Þ ladder operators. We define να as the occupation numbers n; i of j i, so that the total number of electrons corresponding ϕ to this state is where i is the matrix of coefficients that appears in Eq. (16) and X n Nn ≡ νa: ð15Þ A ≡ Γ;iAˆ cˆ† ; cˆ Γ0;i : a¼1;…;Mi ½ iΓΓ0 h j i½f iαg f iαgj i ð24Þ

Let us now consider a generic pure state within the Furthermore, it can be shown that tensor-product space Ei ≡ Vi ⊗ Wi, and represent it as X † ˆ † † hΦijcˆiαfiajΦii¼Tr½ϕi FiαϕiFia; ð25Þ Φ ≡ eiðπ=2ÞNnðNn−1Þ ϕ U Γ;i n; i ; j ii ½ iΓn PHj ij i ð16Þ Γn ˆ ˆ † † † hΦijfibfiajΦii¼Tr½ϕi ϕiFiaFib; ð26Þ U where PH is the particle-hole (PH) transformation satisfying the following identities: † hΦijΦii¼Tr½ϕi ϕi; ð27Þ U† fˆ †U fˆ ; PH i PH ¼ i ð17Þ where the matrix elements of Fiα are defined by U† fˆ U fˆ †; PH i PH ¼ i ð18Þ 0 ½FiαΓΓ0 ≡ hΓ;ijcˆiαjΓ ;iið28Þ U† cˆ†U cˆ†; PH i PH ¼ i ð19Þ and can be equivalently calculated as U† cˆ U cˆ ; PH i PH ¼ i ð20Þ ˆ 0 ½Fiαnn0 ≡ hn; ijfiαjn ;ii: ð29Þ i.e., acting only on the f degrees of freedom. For later convenience, we assume that the condition Note, in fact, that Eqs. (28) and (29) represent the matrix [Eq. (6)] is respected by the matrix ϕi appearing in Eq. (16) elements of ladder operators in their own Fock basis [see so that it couples only states with NΓ ¼ Nn. Note that this the definitions (12) and (14)]. The explicit derivation of condition amounts to assuming that Eqs. (23)–(27) is given in Appendix B. Thanks to Eqs. (23)–(27), the GA Lagrange function [see ˆ tot Ni jΦii¼MijΦii; ð21Þ Eq. (3)] can be rewritten as follows:

c † † c 0 LN½Φ;E ; R; R ; λ; η; μ; D; D ; λ ; n T X 1 X eiω0þ Φ Hˆ emb D ; D†; λc Φ Ec 1 − Φ Φ ¼ N Tr log iω − Rϵ R† − λ − η μ þ ½h ij i ½ i i i j iiþ i ð h ij iiÞ k;ω k þ i XX X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c 0 0 0 − ð½λiab þ½λi abÞ½ni ab þ ð½Diaα½Riaα þ c:c:Þ ½ni aað1 − ½ni aaÞ þ μN; ð30Þ i ab aα

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where 1 1 X Π Rϵ R†f Rϵ R† λ η − μ Π N R k i k ð k þ þ Þ i ˆ emb † c ˆ loc † i αa Hi ½Di; Di ; λi ≡ Hi ½fcîαg; fcîαg qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 0 0 † ˆ ¼½Diaα ½ni aað1 − ½ni aaÞ; ð35Þ þ ð½Diaαcîαfia þ H:c:Þ aα X 0 1 X c ˆ ˆ † ½ni aa − 2 þ ½λi abfibfia ð31Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D R : : δ 0 0 ½ iaα½ iaα þ c c ab ab ½ni aað1 − ½ni aaÞ α c − ½λi þ λi ab ¼ 0; ð36Þ and jΦii belongs (by construction) to the subspace Nˆ tot defined by Eq. (21); i.e., it is an eigenstate of i with ˆ emb † c c Hi ½Di; Di ; λi jΦii¼Ei jΦii; ð37Þ eigenvalue Mi. Since the GA solution is stationary with respect to jΦii ð1Þ † ˆ Ec Hˆ emb ½F i αa ≡ hΦijcîαfiajΦii and i , i can be interpreted as an impurity Hamiltonian qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi whose bath has the same dimension of the impurity and is 0 0 − ½Riaα ½n ð1 − ½n Þ ¼ 0; ð38Þ determined by the GA procedure in order to describe the i aa i aa entanglement between the impurity and the rest of the ð2Þ ˆ ˆ † 0 system. ½F i ab ≡ hΦijfibfiajΦii − ½ni ab ¼ 0: ð39Þ We point out that, in principle, a Hamiltonian whose bath has the same dimension of the impurity is sufficient to Note that the projectors Πi appear in Eqs. (32)–(35) represent exactly the ground-state local properties of the because derivatives are taken with respect to the matrix Hubbard model, as it can be readily shown by making use elements of the block matrices η, λi, and Ri. of the Schmidt decomposition [45,46]. What we have A possible way to compute the Gutzwiller solution is the shown in this section is that the GA consists in assuming following [27]: (i) Given ðR; λÞ, use Eqs. (32) and (33) to that the form of this Hamiltonian is limited to an Anderson compute the Lagrange multipliers μ and η and the corre- c Ψ n0 D impurity model [see Eq. (31)], where the coefficients Di; λi sponding j 0i, which determines i through Eq. (34), i c are determined by the stationarity of the Lagrange function through Eq. (35), and λi through Eq. (36). (ii) Thereafter, ˆ emb [Eq. (30)]. This insight makes it clear that taking into build the embedding Hamiltonians Hi and compute jΦii account all of the components of ϕi is crucial to accurately [see Eq. (37)], which determine the left sides of Eqs. (38) describe the local physics of the system—including the off- and (39). Equations (38) and (39) are verified if and only if ˆ loc R; λ diagonal matrix elements in a basis that diagonalizes Hi . ð Þ is the correct set of variational parameters. In conclusion, we have formulated the solution of the Gutzwiller equations as a root problem for ðR; λÞ, which B. Numerical solution of the GA can be formally written as Lagrange equations ð1Þ ð2Þ In Eq. (7) and the text below, we have introduced the ðF i ½R; λ; F i ½R; λÞ ¼ 0 ∀ i ð40Þ block matrices R, λ, and η appearing in the functional [Eq. (30)], whose respective blocks (one for each i within and solved numerically. Note that the vector functions F i the unit cell) are Ri, λi, and ηi. For later convenience, we [see Eqs. (38) and (39)] can be evaluated independently define Πi as the projectors onto the above-mentioned i local through the numerical steps outlined above. subspaces. The symbol f will indicate the Fermi function. It can be readily shown that the saddle-point condition C. Summary of the main results of Sec. II L of N with respect to all of its arguments provides the In this section, we have expressed the GA Lagrange following system of Lagrange equations: function derived in Ref. [27] [see Eq. (3)] in the convenient 1 X form of Eq. (30), from which follows an exceptionally Π f Rϵ R† λ η − μ Π 0 ∀ a ≠ b; efficient numerical scheme [see Eqs. (32)–(39) and text N k i ð k þ þ Þ i ¼ ba below]. ð32Þ We have shown that our algorithm consists in iteratively solving a series of Anderson impurity Hamiltonians X 1 X whose bath has the same dimension as the impurity [see Π f Rϵ R† λ η − μ Π N; N k i ð k þ þ Þ i ¼ ð33Þ Eq. (31)]. This finding provides a useful interpretation of ia aa the Gutzwiller variational parameters based on the Schmidt decomposition and opens up the possibility to exploit 1 X † 0 techniques such as those developed in quantum chemistry ΠifðRϵkR þ λ þ η − μÞΠi ¼½ni ab; ð34Þ N k ˆ emb ba to solve Hi , in order to further speed up our algorithm.

011008-5 LANATÀ et al. PHYS. REV. X 5, 011008 (2015) We point out that in our numerical scheme, the treatment X X X I − R†R T −G iω Σ iω − −iω i i of the correlation effects scales linearly with the number of ð ið ÞÞba ið Þ † i ω ab Ri Ri correlated atoms per unit cell (as in LDA þ DMFT). Since 1 1 I − R†R linear-scaling DFT methods are also available [52], the λ η − μ i i ; linear-scaling property of our solver opens up the possibil- þ R ð i þ iÞ † † ð43Þ i Ri Ri Ri ab ity of studying correlated systems with extremely large supercells [53–55]—even containing several hundreds of where GiðiωÞ are, at the present stage, the Lagrange correlated atoms. multipliers used to enforce the GA definition of ΣiðiωÞ In Appendix C, alternative expressions for the GA [see Eq. (9)]. However, by deriving the so-obtained Lagrange equations (32)–(35) are derived using the Lagrange function with respect to ΣiðiωÞ, one obtains ’ Green s function formalism [see Eqs. (C8) and (C9)], the Dyson equation for the i local Green’s function, which can be preferable if the unit cell of the system contains many atoms and/or if only a few orbitals are 1 X 1 correlated [56]. In Appendix D, the numerical strategy GiðiωÞ¼Πi Πi; ð44Þ N iω − ϵk þ μ − ΣðiωÞ discussed in this section is generalized to Anderson k impurity models. where Πi is the projector onto the i local subspace. The above formal manipulations enable us to express the III. THE GUTZWILLER-BAYM-KADANOFF GA in terms of the following Lagrange function: FUNCTIONAL In this section, we formulate the GA of the Hubbard L00½G; Σ; Φ;Ec; R; R†; λ; η; μ; D; D†; λc; n0 model [see Eq. (1)] as the saddle-point of a functional of the T X 1 coherent part of the local Green’s function, and we show ≡ N Tr log iω − ϵ μ − Σ iω that the mathematical structure of this functional resembles k;ω k þ ð Þ X X the Baym-Kadanoff (BK) theory [57] on top of the DMFT − T Tr½ΣiðiωÞGiðiωÞ approximation. i ω Let us rewrite Eq. (30) as follows: X ~ c † † c 0 Φ ˆ loc Gi; μ; Φi;E ; Ri; R ; λi; ηi; Di; D ; λ ; n ; þ Hi ;N½ i i i i i L0½Φ;Ec; R; R†; λ; η; μ; D; D†; λc; n0 i ð45Þ T X 1 μN ¼ N Tr log iω − ϵ μ − Σ iω þ where k;ω k þ ð Þ X c † † c 0 † † þ ΘHˆ loc ½Φi;Ei ; Ri; Ri ; λi; Di; Di ; λi ; ni ; ð41Þ ~ c c 0 i ΦHˆ loc;N½Gi; μ; Φi;Ei ; Ri; Ri ; λi; ηi; Di; Di ; λi ; ni i i X † I − Ri Ri ≡ T Tr GiðiωÞ −iω where we have implicitly assumed that the regularization R†R factor eiω0þ is present in the Matsubara summation, ω i i † 1 1 I − Ri Ri † † λ η − μ μN c c 0 þ ð i þ iÞ † † þ ΘHˆ loc ½Φi;Ei ; Ri; Ri ; λi; Di; Di ; λi ; ni R i i Ri Ri Ri ≡ Φ Hˆ emb D ; D†; λc Φ Ec 1 − Φ Φ c † † c 0 h ij i ½ i i i j iiþ i ð h ij iiÞ þ ΘHˆ loc ½Φi;Ei ; Ri; Ri ; λi; Di; Di ; λi ; ni : ð46Þ X i c 0 − ð½λi þ½λ Þ½n ab i ab i ab ~ ab We observe that the functional ΦHˆ loc;N can be viewed X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 0 0 as a Lagrange function on its own, which depends þ ð½Diaα½Riaα þ c:c:Þ ½ni aað1 − ½ni aaÞ; aα parametrically on Gi and μ, and explicitly on c † † c 0 ð42Þ Xi ≡ ðΦi;Ei ; Ri; Ri ; λi; ηi; Di; Di ; λi ; ni Þ—which appear ~ ~ only in Φ ˆ loc . The stationary solution of Φ ˆ loc for the ˆ loc Hi ;N Hi ;N and ΘHˆ loc depends on Hi through Eq. (31). Note that, i variables Xi can be formally expressed as a function of Gi thanks to the formal manipulations of Eq. (7), the quasi- and μ itself. This mathematical construction enables us to particle parameters R; λ; η affect the first term of L0 only define the following functional of Gi and μ only: through the Gutzwiller self-energy Σ, which was defined in Eq. (8). ΦGA G ; μ ≡ Φ~ G ; μ; X G ; μ ; ˆ loc ½ i Hˆ loc;N½ i i½ i ð47Þ It is useful to promote the self-energy to an independent Hi ;N i variable by introducing the following additional Lagrange- Legendre term in Eq. (41): which can be substituted back in Eq. (45).

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In conclusion, we have demonstrated that the GA [51],LDAþ DMFT [7], and LDA þ GA [23,28],have solution is the saddle-point with respect to G and Σ of been developed. the functional, In this section, we discuss our implementation of the LDA þ GA method. T X 1 ΩGA G; Σ; μ ≡ ½ N Tr log iω − ϵ μ − Σ iω k;ω k þ ð Þ A. Correlated orbitals X ΦGA G ; μ The application of any LDA þ X method requires the þ Hˆ loc;N½ i i i identification of a proper subset of “correlated” orbitals X X (e.g., d or f), which we indicate with the symbol P. − T Tr½ΣiðiωÞGiðiωÞ; ð48Þ i ω The correlated orbitals are determined on a physical basis. Usually, they are constructed using Wannier function which resembles the BK theory on top of the DMFT methods, e.g., maximally localized Wannier functions [60], approximation. projected Wannier functions [16,61], or quasi-atomic min- ΦGA imal basis-set orbitals [62,63]. In particular, in this work, We point out that, remarkably, the functional Hˆ loc;N i we refer to the construction of Haule et al. (see Ref. [12]), depends on the nonlocal dispersion ϵk only through the ’ G which is exploited in our numerical implementation. coherent part of the local Green s functions i. In other Given a set of P orbitals, we introduce an orthonormal Hˆ loc N words, it is determined onlyP by i and , and is formally subset of “uncorrelated” orbitals Q spanning the orthogonal ΦL T G ϵloc μN ΦL analogous to ˆ int þ ω Tr½ i i þ , where ˆ int is P Hi Hi complement to the corresponding linear space so that P Q the impurity Luttinger functionalP corresponding to the i on- and are a complete basis for the physical system loc site interaction, and T ω Tr½Giϵi is the additional term considered. arising from having included the on-site quadratic part ϵloc For later convenience, we expand the field operator as of the Hamiltonian and the chemical potential μ within the X X Ξˆ r; σ ξ r χ σ cˆ Ξˆ r; σ definition of the self-energy. Note that, since the on-site ð Þ¼ ki;πð Þ σ0 ð Þ kiπσ þ Qð Þ 0 ki;π∈P quadratic operators have to be treated together with the Xσ ˆ ˆ interaction within the GA, the Gutzwiller approximation ≡ ΞPiðr; σÞþΞQðr; σÞ; ð49Þ ΦL for ˆ int alone cannot be defined in general. i Hi

where ξki;πðrÞ represents the πth correlated orbital within A. Summary of the main results of Sec. III the i local space (in the Bloch representation), χσ0 ðσÞ In this section, we have derived a functional formu- represents the eigenstate of the third component of the 0 ˆ lation of the GA which has essentially the same formal spin with eigenvalue σ , and ΞQ is the component of the structure of the Baym-Kadanoff theory on top of the field operator that corresponds to the Q orbitals. DMFT approximation [see Eq. (48)], making an exception Equation (49) provides a prescription to express any for the following technical differences. (1) The one-body operator A in second quantization as Gutzwiller-Baym-Kadanoff functional depends only on Z the coherent part of the Green’s function [see Eq. (44) and X X Aˆ drΞˆ † r; σ AΞˆ r; σ ≡ Aˆ loc Aˆ hop; Eqs. (C5)–(C7)]. (2) Within the GA, it is necessary to ¼ ð Þ ð Þ i þ ð50Þ σ i treat the quadratic part of the local Hamiltonian together with the interaction [see Eq. (2) and text below Eq. (48)]. where This result clarifies the connection between GA/SB and Z DMFT and proves the equivalence between DMFT þ SB X Aˆ loc ≡ drΞˆ † r; σ AΞˆ r; σ ; and SB. i Pið Þ Pið Þ ð51Þ σ X IV. FUNCTIONAL FORMULATION Aˆ hop ≡ Aˆ − Aˆ loc: LDA GA i ð52Þ OF þ i Approximations to DFT [58] represent the state of the Let us define VP as the ith single-particle subspace art of materials simulations. DFT calculations based on i Π LDA [59] enable us to theoretically attack a wide class of and i as the corresponding orthogonal projector. For later materials, but they are generally not satisfactory for the so- convenience, we also define “ ” called strongly correlated systems, such as, e.g., the high- loc Gi ðiωÞ ≡ ΠiGðiωÞΠi; ð53Þ Tc superconductors, transition metal oxides, and rare-earth compounds. In order to study this important class of Σloc iω ≡ Π Σ iω Π ; materials, several “hybrid” techniques, such as LDA þ U i ð Þ i ð Þ i ð54Þ

011008-7 LANATÀ et al. PHYS. REV. X 5, 011008 (2015) X G iω ’ loc loc where ð Þ is the Green s function of the system within Ni ≡ T Tr½Gi ðiωÞ ð58Þ the whole single-particle space, and ΣðiωÞ is the corre- ω sponding self-energy, which is block diagonal in i, within both DMFT and the GA. is the local population of the i correlated electrons. The total number of electrons N (correlated and uncor- LDA GA related) is predetermined by the charge-neutrality condition B. Functional formulation of þ of the system. The purpose of this section is to derive a functional For later convenience, in the rest of this subsection, we formulation of the LDA þ GA method [23] with the same assume a linear double-counting functional, represented as mathematical structure of LDA þ DMFT [7]. X dc loc loc dc loc dc loc Φ G Φ dc G ≡ V T Tr G iω ≡ V N ; ½ i ¼ Vi ½ i i ½ i ð Þ i i ω 1. The LDA þ DMFT functional ð59Þ In Ref. [7], it was shown that the solution of the Vdc LDA þ DMFT method can be formulated as the saddle where i is a given real number. The generalization to the point of the following functional [64]: problem of nonlinear double-counting functionals—such as Eq. (57)—will be obtained in Sec. IV D by reducing it to the simpler case of linear double counting. Ω ρ r ; J r ; Gloc iω ; Σloc iω ; μ N½ ð Þ ð Þ ð Þ ð Þ Note that, under the assumption [Eq. (59)], the LDA þ X T 1 DMFT functional can be written as ¼ Tr log N ˆ ˆ loc ω iω þ Δ − J þ μ − Σ ðiωÞ loc loc X X ΩVdc;N½ρ; J ; G ; Σ ; μ loc loc Z − T Tr½Σi ðiωÞGi ðiωÞ i ω ≡ ΩKSH J ; Gloc; Σloc; μ − drJ r ρ r X Vdc;N½ ð Þ ð Þ L loc dc loc þ ½Φi ½Gi − Φ ½Gi þ μN LDA i þ E ½ρþE ½ρþE ; ð60Þ Z Hxc ion ion-ion ELDA ρ E ρ E − drJ r ρ r ; þ Hxc ½ þ ion½ þ ion-ion ð Þ ð Þ where

ð55Þ ΩKSH J ; Gloc; Σloc; μ Vdc;N½ X μ ρ r T 1 where is the chemical potential, ð Þ is the electron ¼ Tr log N ˆ ˆ loc Δˆ J r ω iω þ Δ − J þ μ − Σ ðiωÞ density, is the Laplacian, ð Þ is the corresponding X X constraining field, and loc loc − T Tr½Σi ðiωÞGi ðiωÞ i ω X Z X ˆ † L loc loc J ≡ drΞˆ r; σ J r Ξˆ r; σ Φ G − Φ dc G μN 61 ð Þ ð Þ ð Þð56Þ þ ½ i ½ i Vi ½ i þ ð Þ σ i

is the DMFT approximation to the BK functional for the ΦL is the corresponding operator. The functional i is the Kohn-Sham-Hubbard (KSH) Hamiltonian i -impurity Luttinger functional associated with a local Z Hˆ int Φdc X interaction operator i , and is an appropriate dou- Hˆ KSH J ≡−Δˆ drΞˆ † r; σ J r Ξˆ r; σ Vdc ½ þ ð Þ ð Þ ð Þ ble-counting correction. In particular, in this work, we σ ˆ int X X assume that Hi is the general (rotationally invariant) Hˆ int − VdcNˆ loc; Slater-Condon parametrization of the on-site interaction, þ i i i ð62Þ i i which is identified by the (atom-dependent) interaction- U ’ J ˆ loc strength parameters i and Hund s coupling constant i, where Ni is the number operator for the correlated and we employ the following standard form for the double- electrons at the site i. counting functional [51]: 2. The LDA þ GA functional dc loc loc Φ ½Gi ¼E ½Ni dc In the previous section, we have shown that, in the case U J Nloc i Nloc Nloc − 1 − i Nloc i − 1 ; of linear double counting [see Eq. (59)], the LDA þ DMFT ¼ i ð i Þ i ð57Þ ΩKSH 2 2 2 functional can be rewritten as in Eq. (60), where Vdc;N is the DMFT functional [Eq. (61)] of the Hubbard model Hˆ KSH J where Vdc ½ [see Eq. (62)]. This point of view suggests a

011008-8 PHASE DIAGRAM AND ELECTRONIC STRUCTURE OF … PHYS. REV. X 5, 011008 (2015) natural method to derive the LDA þ GA functional. In fact, our derivation of the LDA þ GA functional consists ΩKSH Hˆ KSH J in replacing in Eq. (60) the DMFT functional Vdc;N of the Hubbard model Vdc ½ [see Eq. (62)] with the corresponding GA functional. Note that the GA functional for a generic Hubbard model was already derived in Sec. II [see Eq. (30)]. Specializing Eq. (30) to Eq. (62) gives

ΩKSH ½J ; Φ;Ec; R; R†; λ; η; μ; D; D†; λc;n0 Vdc;N T X 1 ¼ Tr log N ˆ hop ˆ hop † ω iω þ R½Δ − J R − λ − η þ μ X X ˆ loc † ˆ loc † ˆ int † dc † þ hΦij − Δi ½fcîαg; fcîαg þ J i ½fcîαg; fcîαg þ Hi ½fcîαg; fcîαg − Vi cîαcîα i α X X † ˆ c ˆ ˆ † c þ ð½Diaαcîαfia þ H:c:Þþ ½λi abfibfiajΦiiþEi ð1 − hΦijΦiiÞ aα ab XX X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c 0 0 0 − ð½λiab þ½λi abÞ½ni ab þ ð½Diaα½Riaα þ c:c:Þ ½ni aað1 − ½ni aaÞ þ μN: ð63Þ i ab aα

Note that the operators Δˆ and Jˆ , which appear in the δELDA δE ½ρ J ðrÞ¼ Hxc þ ion ; ð65Þ Hubbard Hamiltonian [Eq. (62)], have been split into their δρðrÞ δρðrÞ local and nonlocal components consistently with the definitions (51) and (52). This is because the local and J r nonlocal operators have to be treated differently within the while the stationarity condition with respect to ð Þ is GA [see, e.g., point (2) in Sec. III A]. More precisely, the i X local parts of Δˆ and Jˆ have to be treated together with the T 1 ρGAðrÞ¼ Tr Hˆ int N ˆ hop ˆ hop † interaction i , while the nonlocal parts are accounted for ω iω þ R½Δ − J R − λ − η þ μ within the first term of Eq. (63) [see Eq. (2) and text below]. X † Note also that, since the Q states are uncorrelated, R is (by × R Ξˆ ðr; σÞΞˆ ðr; σÞ construction) a block matrix acting as the identity within σ the space Q and with blocks Ri within the corresponding i X X ˆ † ˆ † correlated spaces. The matrices λ and η are instead zero − ΞPiðr; σÞΞPiðr; σÞ R Q λ η σ i within the space and with blocks i and i, respectively, X X within the corresponding i correlated spaces. ˆ † ˆ þ hΦijΞPiðr; σÞΞPiðr; σÞjΦii In summary, the LDA þ GA functional for linear double σ i counting is given by X ≡ Ξˆ † r; σ Ξˆ r; σ ; σ ð Þ ð Þ ð66Þ c † † c 0 T ΩVdc;N½ρ;J ;Φ;E ;R;R ;λ;η;μ;D;D ;λ ;n ≡ΩKSH J ;Φ;Ec;R;R†;λ;η;μ;D;D†;λc;n0 Vdc;N½ where the GA expectation value of the local part of the Z density operator is not computed from the quasiparticle − drJ r ρ r ELDA ρ E ρ E ; ð Þ ð Þþ Hxc ½ þ ion½ þ ion-ion ð64Þ Green’s function but is computed using jΦii according to Eqs. (10) and (23). Finally, the stationarity condition with c † † respect to the variables Φi , E , Ri, R , λi, ηi, μ, Di, D , where ΩKSH is given by Eq. (63). j i i i i Vdc;N λc, n0 amounts to solving within the GA the KSH N i i As in LDA þ DMFT, the total number of electrons Hamiltonian [Eq. (62)] following the procedure described (correlated and uncorrelated) is predetermined by the in Sec. II B. charge-neutrality condition of the system. In conclusion, under the assumption (59),bothLDAþ GA and LDA þ DMFT can be solved numerically as follows C. Charge self-consistency and KSH Hamiltonian (see Fig. 1): (1) Given an initial electron density ρ0ðrÞ (e.g., ˆ KSH In this section, we discuss the general structure of our the LDA electron density), we construct H using ˆ KSH implementation of the charge self-consistent LDA þ GA Eqs. (62) and (65);(2)wesolveH within the GA (see method in the case of linear double counting. Sec. II B) and compute the corresponding electron density The stationarity condition of Eq. (64) with respect to ρGAðrÞ according to Eq. (66). The procedure is iterated until ρðrÞ is the charge self-consistency condition is satisfied.

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loc the correlated electrons Ni , as in Eq. (57). In this case, Eq. (55) reduces to

loc loc ΩN½ρðrÞ; J ðrÞ; G ðiωÞ; Σ ðiωÞ; μ T X 1 ¼ Tr log N ˆ ˆ loc ω iω þ Δ − J þ μ − Σ ðiωÞ X X loc loc − T Tr½Σi ðiωÞGi ðiωÞ i ω X X L loc dc loc þ Φi ½Gi − Ei T Tr½Gi ðiωÞ þ μN i ω Z ELDA ρ E ρ E − drJ r ρ r : þ Hxc ½ þ ion½ þ ion-ion ð Þ ð Þ

FIG. 1. Schematic flow chart of the LDA þ GA charge self- ð67Þ consistent procedure for linear double counting. The solution of For later convenience, we promote the average local Hˆ KSH is calculated as discussed in Sec. II B. occupations X loc loc The generalization of the above numerical procedure to a Ni ≡ T Tr½Gi ðiωÞ ð68Þ more general class of double-counting functionals is given ω in the next section. to independent variables by adding to Eq. (67) the follow- LDA GA ing Lagrange-Legendre term: D. The þ for general double counting X X dc loc loc It is useful to generalize the procedure described above Vi T Tr½Gi ðiωÞ − Ni : ð69Þ to the case when the double-counting functional is a i ω generic nonlinear function of the on-site occupations of This step enables us to rewrite Eq. (67) as follows:

loc dc loc loc ΩN½N ;V ; ρðrÞ; J ðrÞ; G ðiωÞ; Σ ðiωÞ; μ T X 1 X X X ¼ Tr log − T Tr½ΣlocðiωÞGlocðiωÞ þ ½ΦL½Gloc − EdcðNlocÞ N ˆ ˆ loc i i i i i i ω iω þ Δ − J þ μ − Σ ðiωÞ i ω i Z X X μN ELDA ρ E ρ E − drJ r ρ r Vdc T Gloc iω − Nloc þ þ Hxc ½ þ ion½ þ ion-ion ð Þ ð Þþ i Tr½ i ð Þ i X i ω loc loc dc loc dc loc ≡ ΩVdc;N½ρ; J ; G ; Σ ; μþ ½Ei ½Ni − Vi Ni ; ð70Þ i dc loc where Ei is now a function of the new variable Ni and ΩVdc;N is the LDA þ DMFT functional valid for the special case of linear double counting [see Eq. (60)]. Consequently, using Eq. (64), we obtain that the LDA þ GA functional is represented as loc dc c † † c 0 ΩEdc;N½N ;V ; ρ; J ; Φ;E ; R; R ; λ; η; μ; D; D ; λ ;n X c † † c 0 dc loc dc loc ¼ ΩVdc;N½ρ; J ; Φ;E ; R; R ; λ; η; μ; D; D ; λ ;n þ ½Ei ½Ni − Vi Ni i T X 1 X ˆ loc † ˆ loc † ¼ Tr log þ hΦij − Δi ½fcˆ g; fcîαg þ J i ½fcˆ g; fcîαg N ˆ hop ˆ hop † iα iα ω iω þ R½Δ − J R − λ − η þ μ i X X X ˆ int † dc † † ˆ c ˆ ˆ † c þ Hi ½fcîαg; fcîαg − Vi cîαcîα þ ð½Diaαcîαfia þ H:c:Þþ ½λi abfibfiajΦiiþEi ð1 − hΦijΦiiÞ α aα ab XX X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c 0 0 0 − ð½λiab þ½λi abÞ½ni ab þ ð½Diaα½Riaα þ c:c:Þ ½ni aað1 − ½ni aaÞ þ μN i ab aα Z X − drJ r ρ r ELDA ρ E ρ E Edc Nloc − VdcNloc ; ð Þ ð Þþ Hxc ½ þ ion½ þ ion-ion þ ½ i ½ i i i ð71Þ i where ΩVdc;N is the LDA þ GA functional previously derived for the case of linear double counting [see Eq. (64)].

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iteration afterwards, in order to also have access to the spectral properties of the system of interest.

F. Summary of the main results of Sec. IV We have derived a functional formulation of the LDA þ GA method [23] with the same mathematical structure of LDA þ DMFT [7] [see Eq. (71)]. This parallelism has enabled us to use the same LAPW interface between DMFT/GA and the LDA code, and it suggests possible synergistic combinations between the two methods. We have derived a very stable and numerically efficient implementation of the LDA þ GA method, whose structure dc FIG. 2. Schematic flow chart of the LDA þ GA outer loop that is as follows. (1) The double-counting potentials Vi are Vdc determines the double-counting potentials i . The step con- determined by the outer loop represented in Fig. 2. (2) Each Vdc cerning the solution of the problem at fixed i is implemented as iteration of the outer loop requires us to calculate the in Fig. 1 (see Sec. IV C). dc LDA þ GA solution at fixed Vi —i.e., the saddle point of the functional [Eq. (63)]—which is computed numerically Equation (71) enables us to reduce the LDA þ GA using the charge self-consistent procedure represented in problem for generic double counting to the problem solved Fig. 1. Each iteration of the charge self-consistency loop in Sec. IV C. In fact, the saddle-point condition with respect consists in solving the Kohn-Sham-Hubbard Hamiltonian c † to the variables ρðrÞ, J ðrÞ, jΦii, Ei , Ri, Ri , λi, ηi, μ, Di, determined by the input electron-density according to † c 0 loc dc Di , λi , ni —at fixed Ni and Vi —of the so-modified Eq. (62) and computing the corresponding output electron Eq. (63) can be solved numerically following the procedure density according to Eq. (66) until convergence. (3) The of Sec. IV C (see Fig. 1). Kohn-Sham-Hubbard Hamiltonian is solved using the loc dc The stationarity conditions with respect to Ni and Vi procedure given in Sec. II B. give the additional equations V. THE PRESSURE-VOLUME PHASE DIAGRAM dEdc Vdc i ; OF PRASEODYMIUM i ¼ loc P ð72Þ dNi Nloc Φ cˆ† cˆ Φ i ¼h ij α iα iαj ii In this section, we apply our LDA þ GA implementation to the elemental praseodymium. As in Refs. [35,36],we which determine the self-consistent Vdc and can be solved i employ the “standard” prescription for the double-counting numerically, e.g., as shown in Fig. 2. functional and the general Slater-Condon parametrization Note that the structure of the algorithm discussed above of the on-site interaction [51], assuming that the Hund’s and represented in Figs. 1 and 2 is applicable to both coupling constant is J ¼ 0.7 eV and that the value of the LDA þ GA and LDA þ DMFT. interaction strength is U ¼ 6 eV, which is consistent with previous constrained LDA calculations [50]. E. A possible synergistic combination of LDA þ GA U LDA DMFT However, since the value of is generally difficult to and þ predict exactly, here we also perform calculations for In the previous subsection we have shown that the U ¼ 5 eV and U ¼ 7 eV. LDA þ GA and the LDA þ DMFT methods require, in The lattice structure of the elemental praseodymium is dc order to determine the double-counting potentials Vi and dhcp at ambient conditions, and it undergoes the following the charge density ρðrÞ, solving iteratively the correlated sequence of transformations under pressure: dhcp → fcc → Hubbard Hamiltonian [Eq. (62)]. distorted − fcc → α-U [50,67]. While no appreciable vol- dc We observe that, remarkably, the calculation of Vi and ume collapse is associated with the transitions between ρðrÞ, as well as the total energy, does not require us to the three lower-pressure phases—which are all character- compute the spectral properties of the system but only the ized by relatively high symmetry and/or good packing ground state. ratios [68]—the distorted − fcc → α-U transition is accom- Since the GA ground-state properties are generally in panied by a sizable volume collapse (about 10% at room very good agreement with DMFT for strongly correlated temperature). metals [23,27,35,65,66]—even though the GA is much less It is widely believed that the low-symmetry α-U lattice computationally demanding—this observation opens up structure in the elemental praseodymium is stabilized at the possibility to use the GA for structural relaxation and to high pressures by the delocalized f electrons, in accordance dc determine Vi and ρðrÞ, and to perform a single DMFT with a general argument based on the Peierls theorem

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theoretically in the fcc and α-U lattice structures (see Fig. 3). The theoretical LDA and LDA þ GA phase diagrams are shown in the lower panels of Fig. 4, in comparison with the experimental data. The pressure was obtained as P ¼ −dE=dV from the corresponding total-energy curves, which are shown in the upper panels. Note that the calculations at U ¼ 6 eV are shown in the main panels, while the calculations at U ¼ 5 eV and U ¼ 7 eV are FIG. 3. Representation of the fcc and α-U crystal structures of Pr. shown in the insets. The LDA þ DMFT calculations of fcc Pr of Ref. [72], which were performed at T ¼ 632 K, are also reported. Note that in these LDA þ DMFT calcu- (see Refs. [69,70]). The deformation from a high-symmetry lations the charge self-consistency was not carried out, and structure to a low-symmetry structure can lower the band’s part of the Slater integrals and the non-density-density terms in the local f-electron interaction were neglected. The energy by opening a Peierls gap between a “bonding” band theoretical and experimental values of equilibrium volume, (below the Fermi level) and an “antibonding” band (above bulk modulus, and critical pressure are reported in Table I. the Fermi level). This effect competes with the electrostatic The agreement between the LDA GA theoretical Madelung interaction, which favors the high-symmetry þ results and the experimental data is very good. In particular, lattice structures [71] such as the fcc. Nevertheless, there U ¼ 7 eV gives an overall better agreement with the are a couple of important aspects of the physics underlying experimental phase diagram. Remarkably, the GA correc- the volume-collapse transition of Pr that are still not fully tion to the LDA total energy is very important for Pr, understood and that require further investigation. (1) What is especially at low pressures. In particular, the LDA equilib- the role of the electron correlations for the volume-collapse VLDA ≃ 21 3= transition? (2) Would Pr display a volume-collapse transition rium point is about eq Å atom, while the exp 3 even without taking into account the change of structure? experimental value is approximately Veq ≃34.5Å =atom, In order to further investigate the physics underlying the which is better reproduced by the LDA þ GA calculations, VLDAþGA ≃ 32 3= volume-collapse transition of Pr, in this work, we study it which give eq Å atom. Furthermore, while

FIG. 4. Theoretical total energy as a function of the volume (upper panels) at zero temperature for the fcc and the α-U phases. The corresponding pressure-volume phase diagrams (lower panels) are shown in comparison with the experimental data of Ref. [50] (colored markers) and Ref. [67] (gray markers), which refer to measurements at room temperature. The LDA calculations are reported in the left panels, and the LDA þ GA calculations are reported in the right panels. The LDA þ DMFT calculations of fcc Pr at T ¼ 632 Kof Ref. [72] are also shown (see the dotted red lines in the right panels).

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V TABLE I. Theoretical equilibrium volume eq, bulk modulus K, and critical pressure Pc of Pr in comparison with the experiments. The LDA results are also given for α-U Pr, which has a lower energy with respect to fcc Pr (within LDA). V 3 K Gp P Gp Pr eqðÅ Þ ð Þ cð Þ Experiments 34.54 (dhcp) 26–37 (dhcp) 20 LDA 20.9 (fcc) 61 (fcc) <0 LDA 20.0 (α-U) 84 (α-U) <0 LDA þ GA (U ¼ 6 eV) 32.3 (fcc) 33 (fcc) 13 LDA þ GA (U ¼ 5 eV) 31.9 (fcc) 30 (fcc) 8 LDA þ GA (U ¼ 7 eV) 32.6 (fcc) 38 (fcc) 18 LDA þ DMFT (Ref. [72]) 34.5 (fcc) 31 (fcc) — the LDA predicts that the α-U structure becomes less stable than the fcc phase only at negative pressures, the LDA þ GA method predicts correctly that the volume collapse occurs at positive pressures (see the common- tangent construction in the upper-right panel of Fig. 4). Our LDA þ GA calculations are also consistent with the

LDA þ DMFT calculations of fcc Pr of Ref. [72]. FIG. 5. Theoretical on-site f occupation probabilities Wf Our theoretical results indicate that in Pr, the correlation for the fcc phase (left panels) and for the α-U phase (right panels). effects energetically favor the fcc lattice structure with The LDA results are reported in the upper panels, and the respect to the α-U, stabilizing it at the equilibrium point and LDA þ GA results at U ¼ 6 eV are reported in the lower for a wide range of positive pressures. This fact is also panels. For the fcc phase, the LDA þ DMFT results of clearly illustrated in the inset of the upper-left panel of Ref. [73] are also reported (dots). Fig. 4, which represents the energy difference between the fcc and α-U structures as a function of the volume, both in LDA and in LDA þ GA. We point out that, contrarily to Ce [35], the fcc phase of Pr would not display any isostructural volume-collapse transition by applying pressure. In fact, the second derivative of the fcc energy-volume curve is positive within the entire range of volumes and for all of the values of U considered. Let us examine how the correlation effects taken into account by the GA correction influence the on-site f occupation probabilities Wf and the f quasiparticle renormalization weights, which are determined as Z ¼ R†R according to Eq. (8). In the upper panels of Fig. 5, the on-site f occupation probabilities are illustrated for the fcc and for the α-U phases as a function of the volume. While the LDA probability distribution is very “broad” at all pressures, the LDA þ GA probability distribution is relatively narrow, especially at large volumes, where we find that the majority of the f electrons lie within the f2 space, in agreement with recent experiments [69]. The averaged f quasiparticle renormalization weights are shown in the upper panels of Fig. 6. Note that, because of the spin-orbit coupling, the f quasiparticle weights are split into two groups with total angular momentum J ¼ 5=2 and 7=2, respectively. While at small volumes both of the Z’s FIG. 6. Evolution as a function of the volume of the averaged LDAþGA (U ¼ 6 eV) quasiparticle renormalization weights Z decrease as a function of the volume, at larger volumes 5=2 7=2 f Z (upper panels) of the averaged and electrons and they develop a qualitatively different behavior: 5=2 corresponding orbital occupations (lower panels) for the fcc becomes significantly smaller than 1—indicating that the phase (left panels) and for the α-U phase (right panels) of Pr. For system is very correlated in this regime—and Z7=2 the fcc phase, the LDAþDMFT results of Ref. [72] are also increases. This behavior is a consequence of the spin-orbit reported (dots).

011008-13 LANATÀ et al. PHYS. REV. X 5, 011008 (2015) effect, which occurs also in Ce [35,36]. In particular, Z7=2 is accompanied by an abrupt delocalization of the f grows because the 7=2 electrons disappear at larger electrons, and it is not even surprising that a correlation volumes, as indicated in the upper panels of Fig. 6. between these two phenomena is found experimentally in Note that the difference for Wf and the Z’s is very small several other rare-earth materials [50]. On the other hand, between the two lattice structures. based on our calculations, it does not seem appropriate to As shown in the lower panels of Fig. 4, the GA regard the above-mentioned correlation as a general cause- correction to the pressure-volume phase diagram becomes effect relation. In fact, Pr would not display the transition more substantial at large volumes, where the quasiparticle without taking into account the change of structure (at least renormalization weights are considerably smaller than 1. at low temperatures). Note also that other f systems, such This is not surprising, as decreasing the distance between as americium [74], display volume-collapse structure the atoms increases the bandwidth of the f electrons, which transitions maintaining essentially a constant f valence, reduces the relative importance of the GA correction to the indicating that f localization is not a crucial prerequisite for LDA functional—i.e., of the interaction and the double- the volume-collapse transitions in f systems. counting terms [see Eq. (63)]. Nevertheless, it is important to observe that the above-mentioned correction to the VI. PHASE DIAGRAM OF PLUTONIUM pressure-volume phase diagram is essentially identical Plutonium is the most exotic and mysterious element in for the two lattice structures considered. Note that this the periodic table. Its stable structure at ambient conditions consideration is consistent with the fact that, according to is α-Pu, which has a low-symmetry monoclinic structure our calculations, the strength of the electron correlations is α with 16 atoms within the unit cell grouped in eight very similar for the fcc and the -U lattices. nonequivalent types. At higher temperatures (see Fig. 7), Let us now analyze the role of the electron correlations in Pu can assume the following distinct lattice structures: β the determination of the more stable structure as a function of (monoclinic, with 34 atoms within the unit cell grouped in the volume. As we have anticipated, the Peierls mechanism, γ δ δ0 α seven inequivalent types), (orthorhombic), (fcc), (bct), which stabilizes the -U phase, relates to the itinerant and ϵ (bcc). One of the most intriguing properties of Pu f character of the electrons, and it is consequently less is that these temperature-induced structure transitions are f effective at large volumes, where the bandwidth is smaller. accompanied by significant changes of density. In particular, Note that this effect is qualitatively well captured already by the equilibrium volumes of the δ and δ0 phases are very large the LDA. In fact, at very large volumes (negative pressures), with respect to the other allotropes. Another interesting the LDA total energy of the fcc lattice structure becomes puzzle is that δ- and α-Pu have negative thermal-expansion α lower with respect to the -U (see the upper-left panel of coefficients within their respective range of stability, unlike Fig. 4). As shown in the inset of the upper-right panel of Fig. 4, the effect of the GA correction on the total energy difference between the two lattice structures is of the same order of magnitude for all volumes. Consequently, we attribute the consequent improved quantitative agreement with the experiments for the transition volume to the overall more realistic evaluation of the total energy in LDA þ GA with respect to LDA. In other words, we argue that the behavior of the energy difference between the fcc and the α-U structures is essentially already qualitatively captured by the LDA, and it is not directly related to the f-electron localization—which is substantial only at large volumes, as indicated by the f quasiparticle weights shown in the lower panels of Fig. 6. In conclusion, we have observed that the behavior of the quasiparticle weights and of the f configuration probabilities as a function of the volume is essentially equal for the fcc and the α-U lattice structures. In particular, at small volumes we find that Z ≃ 1, while at large volumes we find that Z ≪ 1. Consistently with the Peierls mechanism, while at small volumes the α-U structure has a lower energy, at large volumes the fcc lattice configuration becomes more stable. Since both the energy difference between the phases FIG. 7. Experimental volume-temperature phase diagram of and the quasiparticle weights are controlled by the volume, Pu. The dotted lines indicate the zero-temperature equilibrium it is not surprising that the volume-collapse transition of Pr volumes extrapolated by linear interpolation.

011008-14 PHASE DIAGRAM AND ELECTRONIC STRUCTURE OF … PHYS. REV. X 5, 011008 (2015) the vast majority of materials. These facts have stimulated In this section, we provide a bird’s eye view of Pu by extensive theoretical and experimental studies. However, a studying all of its crystalline phases at zero temperature convincing explanation of the metallurgic properties of Pu using our implementation of the LDA þ GA method, whose based on fundamental principles is still lacking, and none of description of the ground-state properties is generally in the previous theories has been able to describe simulta- very good agreement with LDA þ DMFT but is consid- neously the energetics and the f electronic structure of all of erably less computationally demanding. In particular, we the phases of Pu on the same footing. employ the general Slater-Condon parametrization of the Previous state-of-the-art DFT calculations [75–78] were on-site interaction with parameters U ¼ 4.5 eV and J ¼ able to reproduce reasonably well the equilibrium volumes 0.36 eV—as we find that LDA þ GA calculations per- of the phases of Pu, but in order to describe all of them on formed using these values give a better overall agreement the same footing, it was necessary to introduce artificial [79] with the experiments with respect to U ¼ 4.5 eV and J ¼ spin and/or orbital polarizations—thus compromising the 0.51 eV (which are the values previously assumed in description of the electronic structure. In fact, without spin Ref. [84]). Calculations of α-Pu with different values of and orbital polarization, these techniques predict that the U and J are shown in Ref. [85]. Very interestingly, our equilibrium volumes of all of the phases are essentially study indicates that the electron correlations are only identical, in contrast with the experiments (see Fig. 7). weakly dependent on the lattice structure, while the most Calculations within the framework of DFT in combination important element originating the differentiation between with dynamical mean field theory (DFT þ DMFT) have the equilibrium densities of the phases of Pu is the com- explained several aspects of the electronic structure of Pu petition between the Peierls effect and the Madelung (see, e.g., Refs. [80–84]). Nevertheless, the computational interaction [70,71]. complexity of this approach made it impossible to calculate In the upper panels of Fig. 8, we show the LDA (left) and the pressure-volume phase diagram of all of the phases of Pu. LDA þ GA (right) evolutions of the total energies E as a

FIG. 8. Theoretical total energies for the crystalline phases of Pu as a function of the volume (upper panels) and corresponding pressure-volume curves (lower panels) in comparison with the experimental data of α-Pu from Refs. [88] (black circles), [89] (blue squares), and [86] (red diamonds). Our results are shown both in LDA (left panels) and in LDA þ GA (right panels). The right insets are zooms of the curves in the corresponding panels. In the upper-left inset, we show the correlation energies, while the corresponding contributions to the pressure are shown in the lower-left inset. The vertical lines indicate the minima of the energy curves. The horizontal lines of the legend indicate the estimated zero-temperature equilibrium volumes, which are assumed to lie between the zero-temperature values extrapolated by linear interpolation and the experimental values at the temperatures in which the allotropes are stable (see Fig. 7).

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TABLE II. Zero-temperature theoretical equilibrium volumes, bulk modulus, and total energies of the crystalline phases of Pu in comparison with the experiments [78,90]. Consistently with Fig. 8, the zero-temperature equilibrium volumes are assumed to lie between the zero-temperature values extrapolated by linear interpolation and the experimental values at the temperatures in which the allotropes are stable (see Fig. 7).

Pu αβγδδ0 ϵ V 3 – – – – – – expðÅ Þ 19.5 20.4 21.5 22.6 21.9 23.5 25.0 25.5 24.7 27.6 22.3 24.5 3 Vth 0.5ðÅ Þ 21.1 22.0 21.3 25.5 25.0 21.2 EexpðKÞ 0 470 550 620 710 1130 Eth 100ðKÞ 0 460 110 310 330 870 KexpðGPaÞ 70.2 38 K GPa – – – – – – thð Þ 50 70 40 55 45 70 15 35 10 25 35 55 function of the volume V for all of the crystalline phases of the LDA þ GA zero-temperature energies of the allo- of Pu. In the lower panels, we show the corresponding tropes of Pu is clearly inherited by the LDA energy-volume evolutions of the pressure P ¼ −dE=dV in comparison curves in the region highlighted in the upper-left panel of with the experimental data of α-Pu. In the left insets, Fig. 8, which transform into the region highlighted in the we show the correlation energies—here defined as the upper-right panel of Fig. 8 when the correlation energies differences between the LDA þ GA and LDA total ener- are taken into account. The same considerations apply to gies—and the respective contributions to the pressure. Note the evolutions of the pressure, as indicated by the gray that in our calculations, we have not performed structure circles in the lower panels of Fig. 8. relaxation, but we have assumed a uniform rescaling of the The above observations explain from a simple perspec- experimental lattice parameters (see Refs. [86,87]). tive how the electron correlations determine the unusual In Table II, the theoretical zero-temperature equilibrium energetics of Pu. In fact, the energy crossings between the volumes are shown in comparison with the zero- LDA energy curves of the high-symmetry structures (δ- and temperature experimental volumes, which we assume to α-Pu) and the other phases can be simply understood in be in between the thermal-equilibrium volumes and the terms of the competition between the Peierls effect and the zero-temperature values extrapolated by linear interpola- Madelung interaction—which is known to energetically tion in Fig. 7. The bulk modulus and energies (referred to as favor the low-symmetry structures at small volumes and the the ground-state energy of α-Pu) are shown in comparison high-symmetry structures at large volumes [69–71]. The with the experimental data of Refs. [78,90]. Remarkably, most important effect of the correlation energies is to shift while the theoretical equilibrium volumes of all phases the position of the equilibrium volumes to larger values, of Pu are very similar in LDA, they are very different in near where the above-mentioned energy crossings take LDA þ GA, and in good quantitative agreement with the place and the LDA energy differences are relatively small. zero-temperature experimental values. Furthermore, while Interestingly, a similar interplay between correlation effects LDA predicts very large equilibrium energy differences and bands structure is displayed in Pr (see Sec. V) and between the phases of Pu, these differences are very small might emerge in even greater generality. in LDA þ GA, in agreement with the experiments. Note In the upper panels of Fig. 9, we show the occupations of also that the LDA þ GA ground-state energies increase the f electrons. The total number of f electrons in δ-Pu is monotonically from each phase to the next-higher- nf ≃ 5.2, which is consistent with previous LDA þ DMFT temperature phase, consistently with the experiments calculations [81]. Here, we find that nf ≃ 5.2 also for γ-, 0 (see Fig. 7 and Table II). The only exception is β-Pu, δ -, and ϵ-Pu. In the monoclinic structures, nf is different whose theoretical equilibrium energy is larger than γ-, δ-, for the inequivalent atoms within the unit cell; it runs and δ0-Pu. between 5.21 and 5.32 in α-Pu, while it runs between 5.17 In order to understand how the electron correlations so and 5.21 in β-Pu. In the middle panels of Fig. 9 we show the drastically affect the energetics of Pu, it is enlightening to averaged orbital populations with total angular momentum look at the behavior of the correlation energies (see the left J ¼ 7=2 and J ¼ 5=2. Note that for all of the phases of Pu, insets in Fig. 8). In fact, the evolution of the correlation the number of 7=2 f electrons decreases as a function of energies as a function of the volume is essentially struc- the volume, while the number of 5=2 f electrons increases. tureless and identical for all of the phases (except for a This behavior simply indicates that the spin-orbit effect is uniform structure-dependent energy shift whose main more effective at larger volumes, as expected. Finally, in the effect is to slightly increase the energy of α-Pu with respect lower panels, we show the behavior of the branching ratio to the other phases). As a result of this correction, the LDA B, which is a measure of the strength of the spin-orbit total energies are transformed as indicated by the gray coupling interaction in the f shell and is calculated from the circles in the upper panels of Fig. 8. The relative behavior orbital populations using the following equation:

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The α-Pu theoretical value of nf at equilibrium is in good agreement with the values extrapolated from the x-ray absorption near-edge structure (XANES) measurements of Ref. [93]. On the other hand, while our calculations indicate that nf is slightly smaller in δ-Pu than in α-Pu, according to the extrapolations of Ref. [93], the 1.9%-Ga δ-Pu alloy has a larger nf with respect to α-Pu. Also, the theoretical values of B are in good agreement with values extrapolated in Refs. [94,95] from electron energy-loss spectroscopy (EELS) and x-ray absorption spectroscopy (XAS) [91,96–98]. Let us study the behavior of the many-body reduced density matrix ρˆf of the f electrons, which is obtained from the full many-body density matrix of the system by tracing out all of the degrees of freedom with the exception of the f local many-body configurations of one of the Pu atoms. We define FIG. 9. Upper panels: Evolution as a function of the volume of ˆ ˆ the averaged orbital populations of the 5=2 and 7=2 f electrons. F ≡−ln ρf þ k; ð74Þ Middle panels: Total orbital occupations in comparison with the values extrapolated in Ref. [93] from XANES measurements at where k is an arbitrary constant that we determine so that ambient conditions of α-Pu (black stars) and the 1.9%-Ga δ-Pu the lowest eigenvalue of Fˆ is zero by definition. Within alloy. Lower panels: Theoretical branching ratios in comparison −Fˆ ˆ this definition, ρˆf ∝ e ; i.e., F represents an effective with the values extrapolated in Refs. [94,95] from XAS (black local Hamiltonian of the f electrons that depends on the cross) and EELS (blue crosses) experiments of α-Pu and the δ volume and that is renormalized with respect to the atomic 0.6%-Ga -Pu alloy. The colors in the first and second panels f from the left correspond to the inequivalent atoms of α-Pu and Hamiltonian because of the entanglement with the rest of β-Pu. The vertical dotted lines indicate the LDA þ GA equilib- the system (see Ref. [99]). rium volumes for the respective phases. Figure 10 shows the eigenvalues Pn of ρˆf as a function f Fˆ of the eigenvalues n of for all of the allotropes of Pu 3 4 1 3 (that are computed at their respective theoretical zero- B ¼ − n7=2 − 2n5=2 ð73Þ temperature equilibrium volumes). Consistently with 5 15 14 − n5=2 − n7=2 2 Ref. [81], we find that for δ-Pu there are two dominant groups of multiplets: one with N ¼ 5 and J ¼ 5=2 (that is 6 (see Refs. [91,92]). Consistently with the behavior of the times degenerate) and one with N ¼ 6 and J ¼ 0 (that is orbital populations, B increases as a function of the volume. nondegenerate). Interestingly, our results show that this Note that the behavior of nf and B is very similar for all of conclusion also applies to all of the other phases. The f the phases of Pu. probability distribution of δ- and δ0-Pu is slightly less broad

−Fˆ −Fˆ FIG. 10. Configuration probabilities of the eigenstates of the reduced density matrix ρˆf ≡ e =Tr½e of the f electrons as a function ˆ of the eigenvalues fn of F. Each configuration probability is weighted by the degeneracy dn ¼ 2Jn þ 1 of the respective eigenvalue fn, where Jn is the total angular momentum.

011008-17 LANATÀ et al. PHYS. REV. X 5, 011008 (2015) with respect to the other phases. This is to be expected, as VII. CONCLUSIONS δ- and δ0-Pu are stable at larger volumes, such that the local We have developed an exceptionally efficient algorithm f degrees of freedom are less entangled with the rest of the to implement the GA. Furthermore, we have derived a system with respect to the other phases. The f probability functional formulation of LDA þ GA that has the same distributions of α- and β-Pu are considerably different for mathematical structure of LDA þ DMFT [7]. This insight inequivalent atoms, as they depend on the number and has enabled us to pattern the LAPW interface [44] between relative distances of the nearest-neighbor atomic positions, LDA and GA after the LDA DMFT work of Ref. [12]. consistently with Ref. [84] (see also Ref. [85]). Note that in þ Using our LDA GA code, we have performed first- β-Pu the atom dependency of the f probability distribution þ principles calculations of Pr and Pu under pressure, which is less pronounced than in α-Pu. are prototypical systems with partially localized f We point out that nf ≃ 5.2 reveals that the f electrons of Pu are in a pronounced mixed-valence state [100]. Indeed, electrons. Our calculations of Pr indicate that its volume-collapse the probability of the N ¼ 6, J ¼ 0 multiplet is very large, transition is not driven only by the concomitant delocal- as indicated by the fact that it has the lowest Fˆ eigenvalue. ization of the f electrons. In fact, contrarily to Ce [35],Pr The reason why nf is closer to 5 than to 6 is that the N ¼ 6, J 0 N 5 J would not display any volume-collapse transition without ¼ multiplet is nondegenerate, while the ¼ , ¼ taking into account the change of lattice structure (at least at 5=2 Fˆ — eigenvalue is 6 times degenerate so its contribution low temperatures). This suggests that there is no reason to n 6 2 5=2 1 to f is weighted by a factor ¼ × þ . The exclude the possibility that, in other f materials, a volume- f observation that the electrons have a significant collapse transition may occur without any concomitant f mixed-valence character indicates that the local degrees substantial f delocalization—as indicated, for instance, by of freedom are highly entangled with the rest of the system recent experiments on the elemental Am [74]. Note that the [99]. This observation is consistent with the fact that the understanding of the connection between f delocalization δ Pauli susceptibility of the -Pu Ga alloy is Pauli-like at low and volume-collapse transitions in f systems is one of the temperatures [79]. Furthermore, it is consistent with the most important puzzles in condensed matter theory. statement of Ref. [90] that Pu is an ordinary quasiharmonic Our calculations of Pu constitute the first theoretical crystal in all of its crystalline phases; i.e., already at description of all of the crystalline phases of this material T ≳ 200 K, the electronic entropy is very small with giving good agreement with all of the experiments on the respect to the quasiharmonic contributions. same footing—including both the thermodynamical prop- In conclusion, we have calculated from first principles erties and the f electronic structure. A particularly impor- the zero-temperature energetics of Pu, finding good agree- tant conclusion of our study is that the most important ment with the experiments. Our analysis has clarified how effect originating the differentiation between the equilib- the electron correlations determine the unusual energetics rium densities of the phases of Pu is the competition of Pu, including the fact that the different allotropes have between the Peierls effect and the Madelung interaction very large equilibrium-volume differences while they are and not the dependence of the electron correlations on the very close in energy. Remarkably, in our calculations, we lattice structure, which is a negligible effect. Note that the did not introduce any artificial spin and/or orbital polar- explanation of this phenomenon is of great interest both izations [79], while this was necessary in previous state-of- physically and from a metallurgic standpoint. – the-art DFT calculations [75 78]. This advancement has From a technical point of view, our calculations clearly f also enabled us to describe the electronic structure of Pu demonstrate the exceptional capabilities of the computa- — on the same footing. Our calculations indicate that tional scheme presented in this work. Indeed, our method — f similarly to Pr the ground-state electronic structure is enables us to rapidly perform accurate first-principles f similar for all of the phases of Pu and that the -electron calculations of strongly correlated materials even for atomic probabilities display a significant mixed-valence systems so complex that other state-of-the-art methods character. Our zero-temperature calculations of Pu also are too time-consuming to be practically applicable. constitute an important step toward the theoretical understanding of its peculiar temperature-dependent properties, ACKNOWLEDGMENTS e.g., the negative thermal expansion of δ- and α-Pu. In fact, above room temperature, the contributions to the free We thank XiaoYu Deng, Kristjan Haule, Xi Dai, Hugo energy of the nonadiabatic effects and of the thermal U. R. Strand, and Michele Fabrizio for useful discussions. excitations of the electrons from their ground state are N. L. and G. K. were supported by the U.S. Department expected to be negligible in Pu [90]. Consequently, the total of Energy (DOE), Office of Basic Energy Sciences under energy could be used to investigate this problem either Grant No. DE-FG02-99ER45761. Research at Ames within direct Monte Carlo simulations or using molecular Laboratory was supported by the U.S. DOE, Office of dynamics [101,102] with atomistic potentials extrapolated Basic Energy Sciences, Division of Materials Sciences and from this work. Engineering. Ames Laboratory is operated for the U.S.

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ˆ † ˆ Department of Energy by Iowa State University under hΨ0jPRiPRijΨ0i¼1; ðA7Þ Contract No. DE-AC02-07CH11358. ˆ † ˆ † † hΨ0jPRiPRicˆRiαcˆRiβjΨ0i¼hΨ0jcˆRiαcˆRiβjΨ0i; ðA8Þ

APPENDIX A: SUMMARY OF THE STANDARD which are commonly named “Gutzwiller constraints.” FORMULATION OF THE GUTZWILLER (2) The so-called GA is assumed, which is an approxima- VARIATIONAL METHOD tion scheme that, like DMFT [9], becomes exact in the limit Let us write the Hubbard Hamiltonian in an infinite- of infinite coordination lattices (see, e.g., Ref. [22]). coordination lattice [see Eq. (1)] in the form The derivation of the Lagrange formulation of the GA — — X X X [Eq. (3)] that we employed in Sec. II as a starting point ˆ αβ † loc 0 is summarized in this appendix. The material presented H ¼ ϵk;ijcˆkiαcˆkjβ þ ½Hi ΓΓ0 jΓ;RiihΓ ;Rij; kij;αβ Ri ΓΓ0 in this appendix makes large use of ideas developed in previous works by several authors [22–25,27,103]. ðA1Þ where here, in order to derive our formalism, we conven- 1. Reformulation of the Gutzwiller problem iently assume that jΓ;Rii are local Fock states, In this section, we briefly summarize the reformulation of the Gutzwiller problem derived in Refs. [24,25]. † n1 † nM jΓ;Rii¼½cRi1 …½cRiM j0i: ðA2Þ a. The mixed-basis representation From now on, we will name the above c basis set the “original” basis. Let us consider the matrix The GA consists in variationally determining a projected Ψ cˆ† cˆ Ψ ≡ ρ0 : wave function represented as h 0j Riα Riβj 0i ½ i αβ ðA9Þ Y 0 ˆ ˆ Since ρi are Hermitian, there always exists a unitary jΨGi¼PGjΨ0i ≡ PRijΨ0i; ðA3Þ Ri transformation Ui that diagonalizes it, i.e., such that X ˆ ˆ † † where jΨ0i is a Slater determinant and PRi is a general fRia ¼ ½UiαacˆRiα; ðA10Þ operator acting on the local configurations at the site ðR; iÞ, α which we represent in the original basis as follows: Ψ fˆ † fˆ Ψ ≡ n0 δ n0 : X h 0j Ria Ribj 0i ½ i ab ¼ ab½ i bb ðA11Þ ˆ ¯ 0 PRi ¼ ½ΛiΓΓ0 jΓ;RiihΓ ;Rij: ðA4Þ f ΓΓ0 The so-obtained ladder operators Rib are named natural- basis [22] operators. Here, we assume that jΨ0i is an eigenstate of the number Note that the connection between the natural basis f and ˆ c Ψ Ψ operator and that the local operators PRi satisfy the the original basis depends only on j 0i.Atgivenj 0i, ¯ commutation rule the coefficients Λ that determine the Gutzwiller projector [see Eq. (A4)] are free variational parameters. X Pˆ ; cˆ† cˆ 0 ∀ R; i; Instead of expressing the Gutzwiller projector in terms of Ri Riα Riα ¼ ðA5Þ the original basis as in Eq. (A4), it is convenient to adopt α the following mixed original-natural [24] basis form which implies that jΨGi is also an eigenstate of the number X Pˆ Λ Γ;Ri n; Ri ; operator. Note that these assumptions are no longer valid Ri ¼ ½ iΓnj ih j ðA12Þ when the method is generalized to study superconductivity Γn (see, e.g., Refs. [22,25]), but this subject will not be where jΓ;Rii are Fock states in the original basis [see addressed in the present work. Eq. (A2)], while jn; Rii are Fock states in the natural basis, In order to analytically evaluate the expectation value of ˆ represented as H with respect to jΨGi, † † n; Ri fˆ n1 … fˆ nM 0 : 1 j i¼½ Ri1 ½ RiM j i ðA13Þ E Ψ ; Λ¯ Ψ Pˆ † Hˆ Pˆ Ψ ; ½ 0 ¼N h 0j G Gj 0i ðA6Þ For later convenience, we adopt the convention that the order of the Γ;Ri and the n; Ri states is the same. For the following two additional approximations are done. j i j i Γ cˆ† cˆ 0 (1) The manifold of variational wave functions is further instance, if the second vector in Eq. (A12) is 1↑ 2↓j i, ˆ † ˆ † restricted by the following conditions [22]: then the second n vector is f1↑f2↓j0i.

011008-19 LANATÀ et al. PHYS. REV. X 5, 011008 (2015) X † ˆ † As we will see, the mixed-basis parametrization of the cˆRiα → ½RiaαfRia: ðA22Þ Gutzwiller projector enables us to gauge away from the a formalism the unitary matrix U that relates the original basis and the natural basis [24] [see Eq. (A10)], which is a Within the definitions given in this section, it can be shown great simplification. that the renormalization matrices Ri can be expressed as

0 † † 0 b. Gutzwiller expectation values ½Riaα ¼ Tr½Pi Λi FiαΛiFia=½ni aa: ðA23Þ In an infinite-coordination lattice, the expectation value of any observable can be computed analytically. Note that, thanks to the mixed-basis representation of the Let us define the uncorrelated occupation-probability Gutzwiller projector, the unitary transformation that relates 0 the natural-basis operators f to the original ones c need not matrices Pi with elements [22] be known explicitly, which is a great simplification. 0 0 ½Pi nn0 ≡ hΨ0jjn ;Riihn; RijjΨ0i YM c. The ϕ matrix and the GA total energy δ n0 na 1 − n0 1−na : ¼ nn0 ð½ i aaÞ ð ½ i aaÞ ðA14Þ The formalism can be further simplified by defining the a¼1 following matrix [24]: We also introduce the matrix representations of the oper- qffiffiffiffiffiffi ˆ 0 ators fRib and cˆRiβ, ϕi ¼ Λi Pi : ðA24Þ

F n; Ri fˆ n0;Ri : ½ ibnn0 ¼h j Ribj i ðA15Þ Note that Eq. (A5) translates into Eq. (6) for ϕi. Within this definition, the expectation value of any local Note that, since we have assumed that the order of the observable [see Eq. (A17)] reduces to jΓ;Rii and the jn; Rii states is the same, the matrix F elements of ib can be equivalently computed as ˆ † ˆ ˆ † hΨ0jPGARiPGjΨ0i¼Tr½ϕiϕi Ai; ðA25Þ F Γ;Ricˆ Γ0;Ri : ½ iβΓΓ0 ¼h j Riβj i ðA16Þ the Gutzwiller constraints [see Eqs. (A19) and (A20)] With the above definitions, it can be readily verified simplify as follows: that the expectation value of any local observable can be † calculated as Tr½ϕi ϕi¼1; ðA26Þ ˆ † ˆ ˆ 0 † Ψ0 P ARiPG Ψ0 Tr P Λ AiΛi ; A17 h j G j i¼ ½ i i ð Þ † † ˆ † ˆ Tr½ϕi ϕiFiaFib¼hΨ0jfRiafRibjΨ0i where 0 ≡ δab½ni bb; ðA27Þ ˆ 0 ½AiΓΓ0 ¼hΓ;RijARijΓ ;Rii; ðA18Þ and Eq. (A23) for the renormalization factors reduces to and that the Gutzwiller constraints can be written as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi † † 0 0 0 † ½Riaα ¼ Tr½ϕi FiαϕiFia= ½ni aað1 − ½ni aaÞ: ðA28Þ Tr½Pi Λi Λi¼1; ðA19Þ

0 † † ˆ † ˆ Tr½Pi Λi ΛiFiaFib¼hΨ0jfRiafRibjΨ0i: ðA20Þ In conclusion, the variational energy [Eq. (A6)] is given by [24] The average of the intersite density matrix reduces to 1 X X [24,26] E R ϵ R† Ψ fˆ † fˆ Ψ ¼ N ½ i k;ij j abh 0j kia kjbj 0i ˆ † † ˆ k;ij ab hΨ0jPGcˆRiαcˆR0jβPGjΨ0i X X † loc ˆ † ˆ þ Tr½ϕiϕi Hi ; ðA29Þ ¼ hΨ0jð½RiaαfRiaÞð½Rj fR0jbÞjΨ0i: ðA21Þ bβ i ab

In other words, the intersite single-particle density matrix and it has to be minimized by fulfilling Eqs. (A26)-(A28). averaged on jΨGi is computed by averaging over jΨ0i a Following Ref. [27], we take into account the constraints renormalized density matrix with natural fermionic oper- by applying the theorem of the Lagrange multipliers as ators, replacing the physical ones according to the rule follows:

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c † † c 0 L½Ψ0;E; ϕ;E ; R; R ; λ; η; μ; D; D ; λ ; n 1 X X X X R ϵ R† Ψ fˆ † fˆ Ψ ϕ ϕ†Hloc E 1 − Ψ Ψ Ec 1 − ϕ†ϕ ¼ N ½ i k;ij j abh 0j kia kjbj 0iþ Tr½ i i i þ ð h 0j 0iÞ þ i ð Tr½ i iÞ k;ij ab i i X ϕ†F† ϕ F X D~ pTrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½ i iα i ia − R : : λc ϕ†ϕ F† F − n0 þ ½ iaα 0 0 ½ iaα þ c c þ ½ i abðTr½ i i ia ib ½ i abÞ i;aα ½ni aað1 − ½ni aaÞ i;ab 1 X 1 X λ Ψ fˆ † fˆ Ψ − n0 η Ψ fˆ † fˆ Ψ : þ N ½ iabðh 0j Ria Ribj 0i ½ i abÞþN ½ iabh 0j Ria Ribj 0i ðA30Þ Ri;ab Ri;a≠b

0 Note that ni and Ri have been conveniently promoted to independent variables [104]. Within this formulation, the numerical problem arising from the stationarity condition of L is particularly easy to solve numerically (see Sec. II B). It is convenient to rewrite Eq. (A30) as follows:

c † † c 0 L½Ψ0;E; ϕ;E ;R; R ; λ; η; D;D ; λ ; n 1 Ψ Hˆ qp R; R†; λ η Ψ E 1 − Ψ Ψ ¼ N h 0j G ½ þ j 0iþ ð h 0j 0iÞ X X X † loc † † c † † þ Tr ϕiϕi Hi þ ð½Diaαϕi FiαϕiFia þ H:c:Þþ ½λi abϕi ϕiFiaFib i aα ab X XX X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c † c 0 0 0 þ Ei ð1 − Tr½ϕi ϕiÞ − ð½λiab þ½λi abÞ½ni ab þ ð½Diaα½Riaα þ c:c:Þ ½ni aað1 − ½ni aaÞ ; ðA31Þ i i ab aα where X X X X Hˆ qp R; R†; λ η ≡ R ϵ R† fˆ † fˆ λ fˆ † fˆ η fˆ † fˆ G ½ þ ½ i k;ij j ab kia kjb þ ½ iab Ria Rib þ ½ iab Ria Rib ðA32Þ k;ij ab Ri;ab Ri;a≠b is the GA quasiparticle Hamiltonian of the system [105].

d. GA Lagrange functional in the canonical ensemble If the Hubbard Hamiltonian [see Eq. (A1)] has to be solved in the canonical ensemble, i.e., at a fixed number of particles per site N, the functional [Eq. (A31)] becomes

c † † c 0 LN½Ψ0;E; ϕ;E ; R; R ; λ; η; μ; D; D ; λ ; n 1 Ψ Hˆ qp R; R†; λ η; μ Ψ E 1 − Ψ Ψ ¼ N h 0j G ½ þ j 0iþ ð h 0j 0iÞ X X X X † loc † † c † † c † þ Tr ϕiϕi Hi þ ð½Diaαϕi FiαϕiFia þ H:c:Þþ ½λi abϕi ϕiFiaFib þ Ei ð1 − Tr½ϕi ϕiÞ i aα ab i XX X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c 0 0 0 − ð½λiab þ½λi abÞ½ni ab þ ð½Diaα½Riaα þ c:c:Þ ½ni aað1 − ½ni aaÞ þ μN; ðA33Þ i ab aα where X X X X X Hˆ qp R; R†; λ η; μ ≡ R ϵ R† fˆ † fˆ λ fˆ † fˆ η fˆ † fˆ − μfˆ † fˆ : G ½ þ ½ i k;ij j ab kia kjb þ ½ iab Ria Rib þ ½ iab Ria Rib Ria Ria ðA34Þ k;ij ab Ri;ab Ri;a≠b Ri;a

Note that the constraint on the number of particles has been imposed on jΨ0i instead of jΨGi. This is licit because ˆ ˆ we have assumed that the Gutzwiller projector PG commutes with the number operator N.

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† νΓ † νΓ † νn † νn 2. Finite-temperature extension Γ n ≡ cˆ 1 … cˆ Mi fˆ 1 … fˆ Mi 0 ; j ij i ½ i1 ½ iMi ½ i1 ½ iMi j i ðB7Þ We observe that the terms involving jΨ0i in Eq. (A33) qp Hˆ νn n νΓ Γ can be substituted with the ground-state energy of G , 0 0 ˆ M ˆ ν1 M ν hΓ jhn j ≡ h0j½fiM i …½fi1 ½cˆiM i …½cˆi1 1 ; ðB8Þ which can be written as i i U T X 1 and PH is the particle-hole (PH) transformation satisfying eiω0þ ; N Tr log iω − Rϵ R† − λ − η μ ðA35Þ the following identities: k;ω k þ † ˆ † ˆ U faU ¼ fa; ðB9Þ where the summation over the fermionic Matsubara PH PH frequencies ω is continuous, μ is the chemical potential, U† fˆ U fˆ †; and R, λ, and η are block matrices, whose blocks are Ri, λi, PH a PH ¼ a ðB10Þ and ηi, respectively. Within the SB mean field theory [37,38], discretizing the U† cˆ†U cˆ†; PH α PH ¼ α ðB11Þ Matsubara summation by assuming that ω ¼ð2m þ 1ÞπT (with m integer) actually amounts to performing calcula- U† cˆ U cˆ : tions at the finite temperature T . In fact, the so-obtained PH α PH ¼ α ðB12Þ functional is equivalent to Eq. (51) of Ref. [38] written in the natural-basis gauge—once R and n0 are promoted to Let us finally assume that independent variables, as we did in Eq. (A33). Nˆ totjΦi¼MjΦi; ðB13Þ

APPENDIX B: DERIVATION OF EQS. (23)–(27) where In order to facilitate the reading, we summarize the main X X ˆ tot ˆ † ˆ † equations from the main text, which are necessary for the N ≡ fafa þ cˆαcˆα: ðB14Þ proofs, and drop the impurity label i. a α We consider the impurity local many-body space V generated by fermionic Fock states

† Γ † Γ ν1 νM jΓi¼½cˆ1 …½cˆM j0i; ðB1Þ 1. Proof of Eqs. (23) and (27) Let us consider a generic (bosonic) operator acting only with the number of electrons ˆ † on the c degrees of freedom A½fcˆαg; fcˆαg and define its X Γ Γ matrix elements in the j i basis as follows: NΓ ≡ να: ðB2Þ α 1;…;M ¼ ˆ † 0 ½AΓΓ0 ≡ hΓjA½fcˆαg; fcˆαgjΓ i: ðB15Þ We define a copy W of V (bath) generated by By using the definitions [Eqs. (B5) and (B6)], we obtain ˆ † νn ˆ † νn jni¼½f 1 …½fM M j0i; ðB3Þ 1 ˆ † hΦjA½fcˆαg; fcˆαgjΦi X X with the number of electrons −i π=2 N N −1 i π=2 N N −1 ¼ e ð Þ n0 ð n0 Þe ð Þ nð n Þ X n Γn Γ0n0 Nn ≡ νa; ðB4Þ † 0 0 † ˆ a 1;…;M ϕΓnϕ 0 0 hΓ jhn jU AU jΓijni ¼ nXΓ X PH PH ϕ ϕ† Γ0 Aˆ Γ δ where the f ladder operators anticommute with the c ladder ¼ Γn n0Γ0 h j j i nn0 Γn Γ0n0 operators. Finally, we define the tensor-product space E ≡ † V ⊗ W and represent the most general state in E as ¼ Tr½ϕϕ A: ðB16Þ X iðπ=2ÞNnðNn−1Þ jΦi ≡ e ½ϕΓnU jΓijni; ðB5Þ Equation (27) is obtained as a special case of Eq. (B16) PH ˆ Γn by assuming that A is the identity operator. X −i π=2 N N −1 † † Φ ≡ e ð Þ n0 ð n0 Þ ϕ Γ0 n0 U ; h j ½ n0Γ0 h jh j PH ðB6Þ 2. Proof of Eq. (25) Γ0n0 Let us write the expectation value of a quadratic where impurity-bath operator explicitly:

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† P P N ˆ −iðπ=2ÞN 0 ðN 0 −1Þ iðπ=2ÞNnðNn−1Þ n hΦjcˆαfajΦi¼ e n n e basis. The phase ð−1Þ appearing in the first step of Γn Γ0n0 Eq. (B19) is due to the fact that the c and f ladder operators † 0 0 † † ˆ ϕΓnϕ 0 0 hΓ jhn jU cˆαfaU jΓijni anticommute, so a minus sign is generated whenever nXΓ X PH PH † † they are transposed. Note that in the last step we used that Nn 0 0 ˆ † ¼ ð−1Þ ϕΓnϕ 0 0 hΓ jhn jcˆαfajΓijni; ðB17Þ † n Γ Fa is real (its matrix elements can only assume the values 0, Γn Γ0n0 1, and −1), so doing the Hermitian conjugate amounts to transposing the labels. where we used Nn0 ¼ Nn þ 1,so By substituting Eq. (B19) in Eq. (B17), we obtain −i π=2 N N −1 i π=2 N N −1 −iπN N e ð Þ n0 ð n0 Þe ð Þ nð n Þ ¼ e n ¼ð−1Þ n : ðB18Þ X X Φ cˆ†fˆ Φ ϕ ϕ† F† F It can be readily verified that h j α aj i¼ Γn n0Γ0 ½ αΓ0Γ½ ann0 Γn Γ0n0 † † 0 0 ˆ † Nn 0 0 ˆ † hΓ jhn jcˆαfajΓijni¼ð−1Þ hΓ jcˆαjΓihn jfajni † † ¼ Tr½ϕ FαϕFa: ðB20Þ Nn † † ≡ ð−1Þ ½FαΓ0Γ½Fan0n † ≡ −1 Nn F F ; ð Þ ½ αΓ0Γ½ ann0 ðB19Þ 3. Proof of Eq. (26) † where we introduced the matrix elements of Fα, which is the Let us write the expectation value of a quadratic bath representation of the ladder operators in their own Fock operator explicitly:

X X ˆ ˆ † −iðπ=2ÞN 0 ðN 0 −1Þ iðπ=2ÞNnðNn−1Þ hΦjfbfajΦi¼ e n n e XΓn XΓ0n0 † † † −i π=2 N N −1 i π=2 N N −1 ϕ ϕ Γ0 n0 U fˆ fˆ U Γ n e ð Þ n0 ð n0 Þe ð Þ nð n Þ Γn n0Γ0 h jh j PH b a PHj ij i¼ Γn 0 0 X XΓ n ϕ ϕ† Γ0 n0 fˆ †fˆ Γ n ϕ ϕ† n0 fˆ †fˆ n δ Γn n0Γ0 h jh j b aj ij i¼ Γn n0Γ0 h j b aj i ΓΓ0 XΓn XΓ0n0 ϕ ϕ† n fˆ †fˆ n0 ϕ†ϕF†F : ¼ Γn n0Γh j a bj i¼Tr½ a b ðB21Þ Γn n0

1 X 1 Gqp z Π Π i ð Þ¼ i N z − Rϵ R† − λ − η μ i ðC2Þ APPENDIX C: EXPECTATION VALUES k k þ IN TERMS OF THE LOCAL GREEN’S FUNCTIONS as follows: 1 X In this section, we derive useful alternative expressions Π f Rϵ R† λ η − μ Π for Eqs. (32)–(35). N k i ð k þ þ Þ i ba It is well known that the Fermi function can be expressed 1 X as a partial fraction decomposition (PFD), T Gqp z ; ¼ 2 þ n i ð nÞ ðC3Þ 1 X 1 ba f ϵ T ; ð Þ¼2 þ z − ϵ ðC1Þ 1 1 X n n Π Rϵ R†f Rϵ R† λ η − μ Π k i k ð k þ þ Þ i N Ri αa where several alternative choices are possible for zn.For 1 X instance, the well-known Matsubara expansion is given T z − λ − η μ Gqp z − Π : ¼ n½ð n i i þ Þ i ð nÞ i in terms of the Matsubara frequencies zn ¼ iωn, which Ri αa are purely imaginary. However, the convergence of the ðC4Þ Matsubara series is very slow. A numerically more convenient alternative to the Matsubara expansion was pro- qp We observe that Gi can be rewritten as posed in Ref. [106], which converges faster than z 1 1 X 1 1 exponentially, and where the poles n are complex, with Gqp z Π Π i ð Þ¼ † i N z − ϵ μ − Σ z i R finite real and imaginary components. Ri k k þ ð Þ i Equation [106] enables us to express the left members of 1 1 Eqs. (32)–(35) in terms of the local quasiparticle Green’s ≡ GiðzÞ ; ðC5Þ R† R functions i i

011008-23 LANATÀ et al. PHYS. REV. X 5, 011008 (2015) where ΣðzÞ, which represents the GA approximation for the where Hˆ loc is the Hamiltonian of the impurity, which self-energy, is a block-matrix [see Eq. (8)] whose blocks are includes both the interaction and the one-body component. given by The purpose of this section is to solve hˆ within the GA in the grand-canonical ensemble. I − R†R 1 1 I − R†R Σ z ≡−z i i λ η − μ i i ; We observe that any IAM can be viewed as a Hubbard ið Þ † þ R ð i þ iÞ † † Ri Ri i Ri Ri Ri model, with no translational invariance and where only the impurity site is correlated. In this special case, the GA C6 ð Þ equations derived in the previous section reduce to and X 1 1 T G z 0 ∀ a ≠ b; X † ð nÞ R ¼ ðD2Þ 1 1 n R ba GiðzÞ ≡ Πi Πi ðC7Þ N z − ϵk þ μ − ΣðzÞ k 1 X 1 1 T G z n0 ; þ n † ð nÞ ¼ ab ðD3Þ is the GA approximation for the coherent part of the local 2 R R ba Green’s function. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 1 Within the above definitions, it can be readily shown that T Δ z G z D n0 1 − n0 ; n ð nÞ ð nÞ ¼ aα aað aaÞ ðD4Þ Eqs. (C3) and (C4) can also be represented as R αa 1 X n0 − 1 X Π f Rϵ R† λ η − μ Π aa 2 c k i ð k þ þ Þ i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DaαRaα þ c:c: δab − ½λ þ λ ab ¼ 0; N ba 0 0 naað1 − naaÞ α 1 1 X 1 T G z ; ðD5Þ ¼ 2 þ † n ið nÞ R ðC8Þ Ri i ba Hˆ emb D; D†; λc Φ Ec Φ ; ½ j i¼ j i ðD6Þ 1 1 X Π Rϵ R†f Rϵ R† λ η − μ Π qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k i k ð k þ þ Þ i † N Ri αa F ð1Þ ≡ Φ cˆ fˆ Φ − R n0 1 − n0 0; ½ αa h j α aj i ½ aα aað aaÞ ¼ ðD7Þ X 1 ¼ T ½ðzn þ μ − ΣiðznÞÞGiðznÞ − Πi : ðC9Þ n R ð2Þ ˆ ˆ † 0 i αa ½F ab ≡ hΦjfbfajΦi − nab ¼ 0; ðD8Þ Note that the evaluation of Eqs. (C8) and (C9) only requires where knowledge of the local Green’s functions Gi of the X correlated sites [see Eq. (C7)]. As we will show, the ˆ emb † c ˆ loc † ˆ H D; D ; λ ≡ H D cˆαfa H:c: operation to compute Gi at given Σi relates to the embed- ½ þ ð½ aα þ Þ aα ding procedure of DMFT [9]. X c ˆ † ˆ We observe that the computational time necessary to þ ½λ abfbfa; ðD9Þ calculate Eq. (C7) scales as the square of the number of ab correlated atoms. On the contrary, Eqs. (32)–(35) require ϵ X V V† 1 the diagonalization of k, whose computational time scales Δ z αl lβ as the cube of the number of total atoms (correlated and αβð Þ¼ n z − e ðD10Þ l l not). Consequently, Eqs. (C8) and (C9) can be preferable if the unit cell contains many atoms and/or if only a few is the hybridization function, orbitals are correlated. 1 GðzÞ¼ ðD11Þ APPENDIX D: APPLICATION TO z − ΔðzÞ − ΣðzÞ IMPURITY MODELS In the previous section, we discussed our method to is the coherent part of the impurity Green’s function, and solve the GA equations for a generic Hubbard model. I − R†R 1 1 Let us now consider a generic impurity Anderson model ΣðzÞ ≡−z þ ðλ þ ηÞ ðD12Þ (IAM), R†R R R† X X V is the Gutzwiller self-energy of the impurity (that also hˆ e a†a alffiffiffi b†a : : Hˆ loc b† ; b ; ¼ l l l þ pn a l þ H c þ ½f ag f ag includes the on-site energies in our notation). l la Note that the impurity quasiparticle Green’s function that ðD1Þ corresponds to Eq. (D11) is given by

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