Dependence of DFT+DMFT Results on the Construction of the Correlated Orbitals
DDependence of DFT+DMFT Results on the Construction of the Correlated Orbitals
Jonathan Karp, ∗ Alexander Hampel, and Andrew J. Millis
2, 3 Department of Applied Physics and Applied Math,Columbia University, New York, NY 10027, USA Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA Department of Physics, Columbia University, New York, NY 10027, USA (Dated: February 18, 2021)The sensitivity of Density Functional Theory plus Dynamical Mean Field Theory calculationsto different constructions of the correlated orbitals is investigated via a detailed comparison ofresults obtained for the quantum material NdNiO using different Wannier and projector methodsto define the correlation problem. Using the same interaction parameters we find that the differentmethods produce different results for the orbital and band basis mass enhancements and for theorbital occupancies, with differing implications regarding the importance of multiorbital effects andcharge transfer physics. Using interaction parameters derived from cRPA enhances the difference inresults. For the isostructural cuprate CaCuO , the different methods give quantitatively differentmass enhancements but still result in the same qualitative physics. I. INTRODUCTION
The quantum many body problem is both complexand difficult. “Difficulty” refers to the combinationof the exponential growth of Hilbert space with sys-tem size and quantum entanglement (in particular theFermion sign problem) which render standard methodsfor dealing with large interacting problems ineffective.“Complexity” refers to the issues involved in formulat-ing the many-body problem, in particular defining andcomputing the large number of basis functions and in-teraction parameters required to capture the chemicaland structural effects that distinguish e.g. aluminum,a low transition temperature superconductor well de-scribed by conventional Migdal-Eliashberg theory, frome.g. La − x Sr x CuO , a high transition temperature su-perconductor believed to have properties inconsistentwith conventional Migdal-Eliashberg theory.Some methods, such as Density Functional Theory,map the correlation problem onto a one body problemwith a self-consistently determined potential. While inprinciple exact, these methods fail in practice for materi-als with strong electronic correlations. Other approaches,such as the coupled cluster theories of quantum chem-istry [1], treat the full complexity of molecular systemsbut work well only for relatively small, relatively weaklycorrelated systems where the level of quantum mechan-ical difficulty is not large. Conversely, Bethe ansatz ap-proaches [2] provide an exact solution for model systems,thus fully taking into account the quantum mechanicaldifficulty while omitting the complexity needed to de-scribe real materials.A complete treatment of the full quantum many-bodyproblem in all its difficulty and complexity is not cur-rently feasible. For most systems of interest, progress hascome from the combination of reducing the complexity by ∗ [email protected] “downfolding” the full problem to a much smaller andtherefore more tractable one and managing the difficultyvia an approximate solution of the quantum many bodyproblem as defined in the downfolded subspace. Down-folding typically involves the identification of a subset ofsingle particle states which are then used to construct themany-particle Fock space in which the quantum manybody physics is to be solved and the projection of theHamiltonian into this basis. A widely studied exampleof downfolding is the single band Hubbard model [3] inwhich the single-electron electronic structure is approx-imated as a one band tight binding model defined on asimple lattice and only the onsite term in the electron-electron interaction is retained.Interest in downfolding has been renewed by the re-cent discovery [4] of superconductivity in infinite layerrare earth nickelates such as hole doped NdNiO . Thisfamily of materials has been of long-standing interest asa potential analog of the layered copper oxide (cuprate)superconductors [5–8] and is currently the subject of in-tense theoretical and experimental research. Electronicstructure calculations [8–25] performed on NdNiO re-veal some similarities to the cuprates, including a similarnominal d valence and a band of transition metal d x − y character crossing the Fermi level, but also some differ-ences including the importance of Nd-derived bands, arather different charge transfer energy, and potential rel-evance of other d -multiplet states. The similarities anddifferences raise the question: can one use essentially thesame downfolded model to study superconductivity andother properties of the two compounds?The combination of Density Functional Theory andDynamical Mean Field Theory (DFT+DMFT) [26–29]has emerged as a powerful and widely used method forstudying quantum materials, materials whose propertiesare determined by quantum many-body effects, becauseit combines a downfolding based on density functionaltheory that produces a reasonably realistic descriptionof particular “correlated” orbitals in the structural andchemical environment defined by the rest of the material a r X i v : . [ c ond - m a t . s t r- e l ] F e b with a many-body method that focuses on the solutionof a local correlation problem.The DFT+DMFT methodology has been broadly suc-cessful in describing the physics of many quantum ma-terials [27–30]. Motivated by this success, many authorshave performed DFT+DMFT calculations on the infinitelayer nickelates [9, 19, 20, 24, 25, 31–39], but with dif-ferent and sometimes conflicting results. Some papersfind that multiorbital physics is crucial to the correlationeffects in NdNiO [32–35, 38, 40], while others [9, 25]claim that the important correlation physics lies in a sin-gle self-doped band. Some papers state that NdNiO is inthe Mott-Hubbard regime with little influence of chargetransfer effects [25, 31], while others claim that, as in thecuprates, charge transfer physics is important [9]. Someof the differences arise from different choices of Coulombinteraction parameters, but it now appears that some ofthe differences arise from choices made in the downfold-ing procedure.In the DFT+DMFT context, discussion of downfold-ing issues has centered on topics related to the appropri-ate treatment of interactions. Important questions haveincluded the limits of applicability of the single-site dy-namical mean field approximation, the question of whichCoulomb matrix elements are treated dynamically andwhich via a mean field or one-loop theory [41–44] andthe “double counting” problem of how the portion of theinteractions included in the underlying density functionaltheory are accounted for [45–50]. The issue of abstract-ing a one electron basis for the correlated subspace out ofa more chemically realistic background electronic struc-ture has been assumed to be less problematic.In this paper we show, using NdNiO as an example,that the aspect of downfolding involving the choice of ba-sis set requires more attention than has heretofore beenassumed. Different methods of downfolding, all of whichreproduce the underlying band structure, are shown tolead to markedly different results for many body prop-erties of interest. We trace the origin of the differencesback to different partitioning of the band theory elec-tronic states into correlated and uncorrelated orbitals.The results suggest that the accuracy of different down-folding approximations should be revisited. Our specificresults are derived in the context of the layered nickelatesbut the conclusions should be more generally valid.The rest of this paper is organized as follows. In sec-tion II we review the DFT+DMFT method, describethe different downfolding methods, and provide detailson the calculations performed in this paper. In sectionIII we compare the physical content of the localized or-bitals constructed in the different downfolding methods.Section IV presents our results from DFT+DMFT calcu-lations with the different downfolding methods. Finally,we offer further analysis and concluding thoughts in sec-tion V. II. METHODSA. Theoretical Overview
In DFT+DMFT the main object of interest is the oneelectron Green’s functionˆ G ( r, r (cid:48) ; ω ) = (cid:16) ω − ˆ H ref − ˆΣ( r, r (cid:48) ; ω ) (cid:17) − (1)with non-interacting reference Hamiltonian ˆ H ref takento be the Kohn-Sham Hamiltonian resulting from the so-lution of the equations of density functional theory andthe self energy ˆΣ constructed by identifying (on physicalgrounds) particular correlated orbitals with wave func-tions φ mα ( r − R α ) corresponding to orbitals m and local-ized near sites R α . The Green’s function is calculatedby making the single site DMFT approximation in whichonly the site-local matrix elements of the self energy be-tween correlated orbitals are retained. For example, ina transition metal oxide the φ mα might be chosen to rep-resent the orbitals in the 3 d shell of the transition metalion with nucleus at site R α .The site-local self energy ˆΣ αQI is obtained from the so-lution of a quantum impurity model, a 0 space +1 timedimensional quantum field theory describing m corre-lated orbitals coupled to a non-interacting bath. Theinteractions of the quantum impurity model are chosento represent the matrix elements of the screened Coulombinteraction among the correlated orbitals φ mα . The one-electron parameters of the quantum impurity model aredetermined from a self-consistency equation relating theGreen’s function of the quantum impurity model ˆ G QI to the projection onto site α of the full lattice Green’sfunction.The specification of the correlated orbitals and of theircoupling to the other degrees of freedom in the solid isthus fundamental to the DFT+DMFT method. Twoclosely related methods, referred to in the literature as“projector” and “Wannier” methods, are widely used forthis purpose.In the projector methodology [51, 52], outlined in [53],one predefines a set of atomic-like correlated orbitals | ˜ φ αm (cid:105) , typically chosen to be centered on positions R α of particular atoms of interest with the symmetry appro-priate to the correlated orbital of interest (e.g. transi-tion metal d ), and vanishing for | r − R α | greater thansome pre-set value. One then represents Σ in the “Kohn-Sham” basis of eigenstates ψ νk ( r ) of H ref . All practicalcalculations retain only a finite set of bands within awindow W (which may depend on k ) so the orbitals aredefined as | ˜ φ mα (cid:105) = (cid:88) ν,k ∈W ( k ) ˜ P α,mν,k | ψ ν,k (cid:105) (2)with ˜ P α,mν,k = (cid:104) ˜ φ mα | ψ ν,k (cid:105) (3)The | ˜ φ mα (cid:105) defined in Eq. 2 are a sum over an incompletebasis and must be orthonormalized. The result after or-thonomalization is a set of states | φ mα (cid:105) that deviate tosome degree from the originally defined atomic like states | ˜ φ αm (cid:105) , and in particular have tails that extend outsidethe originally defined state radius. This considerationsuggests that it is often advantageous to formulate theproblem in as wide an energy range as feasible, to makethe | φ αm (cid:105) as similar as possible to the | ˜ φ αm (cid:105) .With the | φ mα (cid:105) in hand one “upfolds” the self energyto the Kohn Sham basis viaΣ νν (cid:48) ( k, ω ) = (cid:88) αmm (cid:48) P αkνm (cid:2) Σ α QI ( ω ) (cid:3) mm (cid:48) (cid:0) P αkm (cid:48) ν (cid:48) (cid:1) (cid:63) (4)where the P are the coefficients in an expansion of | φ mα (cid:105) in the | ψ ν,k (cid:105) : P αkνm = (cid:104) ψ ν,k | φ mα (cid:105) (5)so that full Green’s function, Eq. 1 is written (cid:2) G latt ( k, ω ) (cid:3) νν (cid:48) = (cid:104) ω − ˆ H ref ( k ) − ˆΣ( k, ω ) (cid:105) − νν (cid:48) (6)The same basis transformation may be used to “down-fold” the lattice Green’s function to the basis of corre-lated orbitals, yielding the self-consistency equation re-lating the quantum impurity model Green’s function G QI to the downfolded lattice Green’s function: G mm (cid:48) QI ; α ( ω ) = (cid:88) ν,ν (cid:48) ,k (cid:0) P αkmν (cid:1) (cid:63) G ν,ν (cid:48) ( k, ω ) P αkν (cid:48) m (cid:48) (7)Eq. 7 can be rearranged to determine the one-electronparameters of the quantum impurity model. It is im-portant to note that the equation is formulated directlyin terms of the projection of the lattice Green’s functiononto the pre-specified correlated orbitals and the upfold-ing of the impurity model self energy to the Kohn-Shambasis. While it is possible to define the projection ofthe Kohn-Sham Hamiltonian onto the correlated orbitals,this Hamiltonian is not used in the formalism; in partic-ular it is not what enters the impurity model.In the Wannier methodology one first identifies a set of N Kohn Sham bands that are not significantly entangledwith other energy bands. If the N bands are containedwithin an energy window that does not contain any otherbands, the identification is straightforward. In the morecommon case in which there is no energy window thatfully isolates a relevant set of bands, a disentanglementprocedure [54] is performed in which an energy window W containing more than N bands is defined and then ateach k point N optimized bands are constructed as linearcombinations of bands inside the entanglement windowvia | ψ opt µk (cid:105) = (cid:88) ν ∈W T dis( k ) µν | ψ νk (cid:105) (8) Following Ref. [54] the disentangling transformation T dis( k ) µν (which may be non-unitary) is chosen to minimizea spread function that ensures k -point connectivity, or“global smoothness of connection” [54] in the optimizedstates and also includes an orthonormalization step.Finally, N Wannier states φ Ia ( r ) localized at positions R Ia in unit cell I are defined as [54]. φ Ia ( r ) = (cid:88) k,ν U aνk | ψ opt ν,k ( r ) (cid:105) e − ik ( r − R Ia ) (9)and a number N c ≤ N of these are designated as corre-lated orbitals.The unitary operators U mνk are chosen to optimizesome desired property of the Wannier functions, typicallylocalization about the Wannier centers R a = (cid:104) φ a | r | φ a (cid:105) .In the Maximally Localized Wannier Function (MLWF)procedure [54, 55] one minimizes the average over allWannier functions of the mean square positional uncer-tainty (cid:80) a δR a ≡ (cid:80) a (cid:104) φ a ( r ) | ( r − R a ) | φ a (cid:105) (the unit cellindex is suppressed here since the localization is the samefor each cell). In the Selectively Localized Wannier Func-tion (SLWF) procedure [56] one minimizes the spreadonly of the designated correlated orbitals, and also opti-mizes the center position and symmetry of these states,but the procedure is otherwise the same.The projection of the Kohn-Sham Hamiltonian ontothe Wannier orbitals defines an N orbital tight bindingmodel with Hamiltonian H abIJ = (cid:104) φ Ia | H KS | φ Jb (cid:105) . An im-portant test of the Wannierization procedure is that theeigenvalues of H abIJ reproduce the Kohn-Sham bands withhigh precision: failure to reproduce the DFT band struc-ture means that the Kohn-Sham Hamiltonian has matrixelements between the | φ a (cid:105) and Bloch functions | ψ nk (cid:105) notincluded in Eq. 9, so that the Wannier basis is not acomplete expression of the single particle physics in therelevant energy range.In the Wannier method the dynamical mean field selfconsistency is expressed in terms of the Wannier Green’sfunction G abIJ ( ω ) = (cid:104) ω − ˆ H − ˆΣ( ω ) δ IJ (cid:105) − IJab (10)where the self energy matrix has nonzero elements onlyin the N c × N c correlated orbital subspace, with thesematrix elements being precisely equal to the quantumimpurity model self energy. The Wannier analog of Eq.7 is then given by equating the quantum impurity modelGreen’s function to the sub-block of the onsite G : G ˜ a ˜ bQI ( ω ) = G ˜ a ∈ N c ˜ b ∈ N c II ( ω ) (11)In the Wannier method the self consistency is thus car-ried out directly in the Wannier basis, with “upfolding”to the Kohn Sham basis only required for reasons outsidethe scope of this paper such as computing the charge den-sity required for the “full charge self consistency” step ofthe DFT+DMFT procedure.The Wannier procedure requires construction of H abIJ ,which makes it in a sense less elegant than the projectormethod, but as will be seen, the form of H abIJ providesphysical insight, and the orbitals and energies permit theuse of cRPA methods for computing the interactions.Advantages of the projector method include the abilityto specify in an intuitively or chemical reasonable mannerthe shape and location of the correlated orbitals (subjectto the orthonormalization issues discussed above) and theavoidance of the multiparameter optimization requiredto construct the Wannier functions. Advantages of theWannier procedure include a flexibility in determiningthe correlated orbital wave function (which the Wanniermethod will adapt, for example, to changes in lattice con-stant). Additionally, analysis of the intermediate tight-binding model can provide physical insight and permitsthe use of cRPA methods for computing the interactions.Both the Wannier and the projector methods involve aspecification of the correlated orbital wave function, im-posed a priori in the projector method and computed aspart of the process in the Wannier approaches. The im-portance of the specification for the correlation physicsmay be seen by consideration of a simple two-site modelof a d orbital of energy ε d , a ligand (“ p ”) orbital ofenergy ε p , a hybridization t pd , and a correlation term U d †↑ d ↑ d †↓ d ↓ . The strength of the correlation effects de-pends on both U and ( ε d − ε p ) /t pd . On the noninter-acting ( U = 0) level, the model has two levels with en-ergy difference ∆ E = (cid:113) ( ε d − ε p ) + 4 t pd . We see thata range of t pd and ε d − ε p can fit the same energy dif-ference; pinning down the parameters requires additionalinformation such as the d content of the states. In closeanalogy, in the solid state case the Kohn-Sham eigenval-ues and eigenfunctions do not by themselves determinethe energies of the correlated orbitals or their relation tothe uncorrelated orbitals. The values of the analogs of ε p , ε d , and t pd can be read off directly from the WannierHamiltonian, and they can be inferred from the projec-tor results. The crucial finding of our paper is that dif-ferent projector and Wannier methods which have iden-tical bands on a DFT level produce different results forthese parameters, leading to strikingly different correla-tion physics. B. Computational details
We perform one-shot DFT+DMFT calculations forNdNiO using different combinations of downfolding ap-proach and energy window. The four cases we considerare:1. MLWF: We construct 13 Wannier functions, corre-sponding to the Ni-3 d , O-2 p , Nd-5 d z and Nd-5 d xy orbitals. We use a disentanglement energy win-dow of − . . − . . d Wannier functions and ignorethe spread of the rest.3. Projectors in an energy window from −
10 eV to10 eV around the Fermi energy. This guaranteesthat all relevant low energy states are included inthe window, and projectors are quite localized.4. Projectors in an energy window from −
10 eV to3 eV around the Fermi energy, going high enoughin energy to include the self-doping band but notthe tail of the Nd-5 d z and Nd-5 d xy densities ofstates.For comparison we also used the two Wannier functionmethodologies to perform calculations for the “infinitelayer” high-T c cuprate CaCuO , retaining in this caseonly 11 bands because the Ca- d states are far above theFermi level.For the projector cases, we perform DFT calculationsusing WIEN2k [57] and the standard PBE GGA func-tional [58]. We use the experimental crystal structurewith a = b = 3 . (cid:6) A and c = 3 . (cid:6) A [4] (NdNiO ) and a = b = 3 . (cid:6) A and c = 3 . (cid:6) A (CaCuO ). We treat theNd-4 f bands as core states. The DFT calculations areconverged with an RK max = 7 and with a k -point gridof 40 × ×
40. We use the dmftproj software [59] tocreate the projectors.For the MLWF and SLWF cases we perform the DFTcalculations with Quantum Espresso [60]. We use thesame structure parameters and PBE functional as withWien2k. We use PAW pseudopotentials with the the Nd- f states in the core. We use a k -point mesh of 16 × × we con-sider two cases: a “two orbital” theory treating the dy-namical correlations among the two Ni- e g orbitals and a“one orbital” theory treating only the d x − y orbital ascorrelated; for CaCuO we perform a “two orbital” cal-culation. The calculations are one-shot in the sense thatthe DFT density is not further updated. For the pro-jector cases we also run calculations with full charge selfconsistency, finding that imposing full charge self consis-tency does not alter the results significantly.For the impurity problem, we choose the interactionsto be of the Kanamori form [64]: H = U (cid:88) m n m ↑ n m ↓ + (cid:88) m 9, with an energy cutoff of 500 eV,and ∼ 300 empty bands (plus 21 occupied bands). To ex-tract symmetrized interaction parameters we average thefull four index interaction tensor assuming cubic symme-try, obtaining the parameters for the Hubbard-KanamoriHamiltonian used for the Ni- e g orbitals [69]. III. ORBITAL CONTENT In this section, we examine the physical content of theWannier and projector representations of the band the-ory. The Wannier methods produce an explicit repre-sentation of the Kohn-Sham bands and eigenfunctionswithin a given energy window and the correlated orbitalsare defined as particular linear combinations of thesestates, permitting a straightforward analysis. Figure 1shows the bands obtained from the MLWF and SLWFHamiltonians along a high symmetry path in k space, X M Z R A Z86420246 EE F ( e V ) a) MLWF X M Z R A Z b) SLWF other d x y FIG. 1. Energy bands obtained from diagonalizing the Wan-nier H ab ( k ) in the MLWF and SLWF methods. The color ofthe bands at each energy eigenvalue represents the amount ofthe d x − y Wannier function in the corresponding eigenvec-tors. along with the d x − y content, indicated in pseudocolor.The energy dispersions produced in the two methods areessentially identical and are indistinguishable from theKohn-Sham bands (not shown), but the orbital contentof the bands is different in the different Wannierizationschemes. The most relevant bands are the one crossingthe Fermi level between Γ − X − M and Z − R − A , andthe weakly dispersing band at ∼ − d x − y content is more concentrated in theband that crosses the Fermi level, with less weight in the − d x − y content of relevant bands at the Brillouin zone M point. We compare the orbital content obtained fromthe MLWF and SLWF methods to that provided by theQuantum Espresso and Wien2k codes, which use a pro-jector method. We see that the different methods, whileexactly reproducing the energy dispersions, lead to quitedifferent orbital contents. The difference arises because(as qualitatively seen in the two site model discussed inthe previous section), the same dispersion may be fit bydifferent tight binding parameters, which in turn leadto different orbital content of bands. Table II presentsthe p - d energy difference (obtained from the orbital andsite-diagonal terms of the Wannier Hamiltonian) and hy-bridization (the first neighbor p - d hopping term in theWannier Hamiltonian). We see as expected that the M-point d x − y content Band at ∼ ∼ − d x − y content of the two bands with significant d x − y content at the M point. In the MLWF and SLWFcases the orbital content is the modulus squared of the over-lap of the band basis state with the d x − y Wannier function.In the Wien2k case, the orbital content is obtained from theprojection of the band on the 3 d x − y basis state inside theNi muffin tin. In the Quantum Espresso (QE) case, the pro-jection is onto orthogonalized atomic wavefunctions. ε d − ε p t pd d x − y d z d xz/yz d xy totalMLWF 4.32 1.28 1.19 1.83 1.94 1.97 8.89SLWF 3.38 1.41 1.32 1.88 1.96 1.98 9.11Proj -10 to 10 2.66 - 1.19 1.58 1.89 1.95 8.50Proj -10 to 3 4.18 - 1.21 1.71 1.95 1.99 8.82TABLE II. Left: Difference between the onsite energies of theNi- d x − y and O- p σ Wannier functions ( ε d − ε p ) and hoppingbetween them ( t pd ). In the Wannier cases the parameters areread off directly from the appropiate entries in the real spaceWannier Hamiltonian H abIJ ). In the projector cases, ε d − ε p isdetermined by downfolding the Kohn-Sham Hamiltonian, but t pd is not defined. Right: Orbital occupancies of Ni- d definedas the square of the projection of the occupied k-states asobtained from DFT onto the local orbitals summed over spin. SLWF and MLWF methods trade off the values of ε d − ε p and t pd to obtain comparable fits to the band structure.Figure 2 shows the orbitally projected density of statesobtained using Wannier and projector methods. The up-per left panel shows the projection onto the d x − y or-bital. Two peaks are observed, reflecting the strong hy-bridization of Ni- d x − y and O- p σ states which dividesthe d density of states into bonding (low energy) andantibonding (near Fermi energy) portions. The differentmethods predict different d weights in the bonding (lowenergy) region.Fig. 2(b) shows the d z density of states. Around ∼ − . d z densityof states. Here the hybridization is not with the oxygenbut with the Nd- d states, as there are no oxygen statesat this energy (see Fig. 2(d)). We also see clearly thatat higher energies the d z orbital is hybridized with or-bitals of Nd character (not shown here) lying above the D O S ( / e V ) a) d x y b) d z E E F (eV)0.00.20.40.60.81.0 D O S ( / e V ) c) d x y (zoomed in) MLWFSLWFproj -10 to 10proj -10 to 3 8 6 4 2 0 2 E E F (eV) d) O- p (per O atom) FIG. 2. Uncorrelated DOS (per spin), obtained from theimaginary part of the local Green’s function in the Wannierbasis without self energy. Fermi level. The different methods treat the hybridiza-tion to the higher lying states differently, and this affectsthe final results.Fig. 2 (c) shows the d x − y DOS in a narrow fre-quency range around the Fermi level. In this energyrange the MLWF d x − y DOS is slightly greater becauseit has more content from the Ni-derived band that crossesthe Fermi level. However, the differences are minimal, in-dicating that in this case the choice of downfolding is im-portant primarily in affecting the character of the statesfarther from the Fermi level.The differences are quantified in Table II, which showsthe occupancy of the d orbitals as obtained from eachmethod. In all cases, the t g orbitals are almost full, jus-tifying the use of a two-orbital model that neglects corre-lations in these orbitals. All methods produce a greaterthan half filled d x − y orbital due to charge transfer fromthe ligand orbitals. However, the different methods leadto quantitatively different results. The d x − y filling isroughly the same in the MLWF and projector cases, butit is significantly greater in the SLWF case. The d z or-bital is less filled in the projector than in the Wanniercases, especially the case with the larger energy window,reflecting the difference in the feature at ∼ − . t pd , but the physics is revealed by acomparison of the orbitally projected density of statesshown in Figure 2 to the bare hybridization function,shown in Figure 3 and defined as ˆ∆ ( ω ) = ω − ˆ ε − ˆ G − loc, ( ω ), where ˆ G loc, ( ω ) is the uncorrelated site lo-cal Green’s function projected onto the basis of corre-lated orbitals and ˆ ε is the onsite energy obtained from lim ω →∞ (cid:16) ω − ˆ G − loc, ( ω ) (cid:17) . In the simple two level modelconsidered above, the bare hybridization function wouldbe t pd / ( ω − ε p ). In the general case ˆ∆ ( ω ) has poles at theenergies of the levels with which the correlated orbitalsare hybridized, while the integrated weight ( (cid:82) Im ∆ ( ω ))gives the total hybridization strength.Examination of the hybridization function reveals pro- - I m () ( e V ) a) d x y b) d z MLWFSLWFprojector -10 to 10projector -10 to 3 FIG. 3. Negative imaginary part of the real frequency hy-bridization of the a) d x − y and b) d z orbitals. The inset ofa) shows the d x − y hybridization zoomed in on the windowof − − p σ . nounced differences between the methods. In particular,Fig. 3(a), inset shows that the SLWF method yields anintrinsically less dispersive but more strongly hybridizedO- p σ state at a noticeably lower (less negative) energythan the other methods. For these reasons the SLWFmethod has a substantially larger Ni- d -admixture in theDOS in the bonding energy range, reflecting the larger t pd and smaller ε d − ε p found in this method. Conversely,looking at the O- p σ DOS (d), we see that the MLWF O- p σ DOS is smaller at the Fermi level but larger in the − d x − y Wannier function. The projec-tor method produces results in between the SLWF andMLWF methods, although closer to MLWF, indicating asmaller effective t pd and larger ε d − ε p than in the SLWFmethod, but not quite as small (large) as in the MLWFmethod. DOS, − Im∆ ( ω )MLWF 0.204 16.86SLWF 0.288 21.41Proj -10 to 10 0.213 17.92Proj -10 to 3 0.218 17.98TABLE III. Integral of the DOS over the energy range − − d x − y over regionsof significant overlap with oxygen. The differences in hybridization strength are quanti-fied in Table III. The integral of the d x − y DOS overthe energy region with oxygen-derived bands from − ε d − ε p and larger effective t pd in the SLWF case, in agreement with the values in ta-ble II. Likewise, the integral of − Im∆ ( ω ) in the regionof dominant peaks from − − t pd . Forboth of these quantities, the values in the projector casesare in between those for the MLWF and SLWF cases butcloser to the MLWF case, indicating an effective ε d − ε p slightly smaller than the MLWF case and an effective t pd slight larger than the MLWF case. Thus in summary wesee that the selectively localized Wannier function leadsto the smallest p - d energy difference and strongest p - d hybridization; the MLWF leads to the largest p - d hy-bridization, with the projector method intermediate, butcloser to the MLWF method.The differences in mapping from orbital to band basisin the different downfolding methods also imply a differ-ence in interaction strengths. In the DMFT approach,the interaction parameters are chosen to represent theon-site terms in the screened Coulomb interaction. Theyare sometimes chosen phenomenologically or to obtainagreement with experiment (for the case of perovskitenickel oxides see [65]), and in most of the calculationsreported in this paper these phenomenologically deter-mined parameters are used. IV. DMFT RESULTS This section investigates the ways in which the dif-ferent downfoldings lead to different results in the in-teracting theory. Unless otherwise specified this sectionpresents DMFT calculations for a ”two orbital” model ofNdNiO in which the d x − y and d z orbitals are consid-ered to be dynamically correlated and the phenomenolog-ically determined U = 7 eV and J = 0 . d x − y orbital is considered to be cor-related and a brief discussion of the analogous cupratematerials, where a one band description is more widelyaccepted. A. Self Energy and Mass Enhancement Figure 4 compares the imaginary part of the Mat-subara self energy of the different models for both the d x − y and d z orbitals. The difference in self en-ergy corresponds to a difference in predicted correla-tion strength. We quantify the strength of electroniccorrelations by the inverse quasiparticle renormalization Z − = 1 − ∂Re Σ( ω → /∂ω related, in the single-site DMFT approximation, to the quasiparticle mass en-hancement as m (cid:63) /m = Z − . At low T in a Fermi liquidregime, Z − can be expressed in terms of the Matsubaraself energy as Z − = 1 − ∂ ImΣ( iω n → /∂ω n . We esti-mate the derivative by fitting a 4th order polynomial tothe first 6 Matsubara points and taking the linear term,following [70, 71]. The resulting mass enhancements areshown in Table IV.Figure 4 and Table IV show that for the d x − y or-bital the self energy at all frequencies as well as the massenhancement is much larger in the MLWF case than theSLWF case, with projector cases being intermediate. The d z orbital mass enhancement is small in all cases. Weattribute the differences in d x − y self energy and massenhancement to the differences in p - d energy splittingand p - d hybridization strength discussed in the previ- n (eV)2.01.51.00.50.0 I m ( i n ) ( e V ) a) d x y n (eV) b) d z MLWFSLWFproj -10 to 10proj -10 to 3 FIG. 4. Imaginary part of the Matsubara self energy obtainedfrom a DMFT solution for a two orbital Ni- e g model with aKanamori Hamiltonian with U = 7 eV and J = 0 . m (cid:63) m d x − y m (cid:63) m d z m (cid:63) m X U eff d x − y MLWF 7.6 1.2 6.7 3.6SLWF 3.9 1.3 3.3 2.8Proj -10 to 10 4.6 1.3 3.8 2.5Proj -10 to 3 5.6 1.3 4.7 2.5TABLE IV. Left: Orbital basis mass enhancements for the d x − y and d z orbitals, obtained by fitting a 4th order poly-nomial to the first 6 Matsubara points of the imaginary partof the self energy along with the band basis mass enhance-ment obtained from the quasiparticle band nearest the Fermilevel at the X point of the band structure. Right: Effective U values for the d x − y , defined as the distance between theHubbard peaks of the d x − y momentum integrated spectralfunction. X M Z R A Z EE F ( e V ) a) MLWF X M Z R A Z b) SLWF X M Z R A Z EE F ( e V ) c) Proj -10 to 10 X M Z R A Z d) Proj -10 to 3 FIG. 5. Pseudocolor plot of the quasiparticle mass enhance-ments in the band basis along a high symmetry k path. Inthe Wannier cases the bands plotted are the same in Figure 1and in the projector cases they are the Wien2k DFT bands.The color corresponds to the mass enhancement of the DFTband determined from upfolding the self energy to the bandbasis. The self energy is obtained from a DMFT solution fora two orbital Ni- e g model with a Kanamori Hamiltonian with U = 7 eV and J = 0 . ous section, consistent with previous literature on thecharge-transfer to Mott insulator crossover [72, 73]. Note,however, that in contrast to the situations considered inprevious literature, where only hybridization to oxygenbands is relevant, for NdNiO the mass enhancementsalso depend on the hybridized d z /Nd bands, which de-pend on the projection window.One important caveat is that that the values reportedin Table II are “orbital basis” mass enhancements, deter-mined from the diagonal elements of the projection of theself energy operator onto the correlated orbitals. A quan-tity of more direct relevance to the low energy physics isthe “band basis” mass enhancement, which is propor-tional to the admixture of the uncorrelated orbitals inthe band of interest and gives the renormalization of thequasiparticle bands with respect to the DFT bands. Forthe MLWF and SLWF methods, we obtain the band ba- sis mass enhancement by transforming the self energy tothe band basis using the eigenvectors of the uncorrelatedWannier Hamiltonian H ( k ). For the projector methods,we use the projectors to upfold the self energy back tothe Kohn-Sham basis (Eq. 4). Figure 5 shows the bandbasis mass enhancement along the same high symmetrypath on which the bands are plotted in Figure 1, andTable II gives the value of the band basis mass enhance-ment for the near Fermi surface state at the X point. Inthe band basis, the difference between the MLWF andSLWF cases is even greater than in the orbital basis, forthe same reason– in the MLWF case there is less admix-ture of oxygen in the near Fermi surface band so the d self energy has a greater effect on the dispersion.Another potential caveat is that different choices fordownfolding may lead to different interaction parame-ters. To investigate the basis dependence of the interac-tion parameters we have used the “constrained RandomPhase Approximation” (cRPA) approach to estimate theCoulomb parameters corresponding to the two Wannierdownfoldings. This approximation is believed to under-estimate the true interactions, but gives trends correctly.Symmetrizing our compouted screened Coulomb tensorover the two active orbitals gives parameters U ≈ . J ≈ . U ≈ . J ≈ . ε d − ε p found in the SLWF method. Use of the cRPA in-teraction parameters (smaller for SLWF and for MLWF)yields a Ni- d x − y orbital mass enhancement of 2 . . m (cid:63) /m − 1. Thusthe difference in interaction parameters arsising from dif-ferences in downfold amplify, rather than decrease, thedownfolding-induced differences in self energy. B. Spectral function A () ( / e V ) a) d x y b) d z MLWFSLWFproj -10 to 10proj -10 to 3 FIG. 6. Momentum integrated spectral functions per spinobtained from DMFT solutions for a two orbital Ni- e g modelwith a Kanamori Hamiltonian with U = 7 eV and J = 0 . Figure 6 shows the orbitally resolved momentum in-tegrated spectral function ˆ A ( ω ) = i (cid:104) ˆ G ( ω ) − ˆ G ( ω ) † (cid:105) / π for the d x − y and d z orbitals for a range of energiesnot too far from the chemical potential. The d x − y or-bital (left panel) exhibits a three peak structure similarto that found in the single-band Hubbard model at mod-erate correlation strength. Interpreting the structure interms of a low energy effective model, we identify theelectron removal peak at ω ≈ − . ∼ ω = 0 withthe quasiparticle band. The energy separation betweenthe lower and upper “Hubbard peaks”, shown in TableIV then provides an estimate for the effective interaction U eff characterizing an effective low energy model. Whileall methods provide qualitatively similar spectral func-tions, in the MLWF case the Hubbard peaks are furtheraway from each other, indicating a greater effective Hub-bard repulsion due to the greater ε d − ε p and smaller t pd and consistent with the larger mass enhancement foundin the previous subsection.The spectral function for the d z orbital shows a weaktail at energies above the chemical potential, a sharp peakat ∼ − ∼ − ∼− FIG. 7. Pseudocolor plots of the momentum resolved spectralfunctions A ( k, ω ) obtained from DMFT solutions for a twoorbital Ni- e g model with a Kanamori Hamiltonian with U =7 eV and J = 0 . Figure 7 shows the momentum resolved spectral func-tion A ( k, ω ) = − Tr (cid:104) Im ˆ G ( k, ω ) (cid:105) /π along a high symme-try path in the Brillouin zone. The bands with signifi-cant correlation effects appear more diffuse because of thelarger imaginary part of the self energy. Comparison toFig 6 shows that the correlated band crossing the Fermi d x − y d z LS N=2 HS N=2 N=3 N=4MLWF 1.13 1.91 0.04 0.05 0.78 0.13SLWF 1.27 1.93 0.03 0.02 0.69 0.26Proj -10 to 10 1.14 1.65 0.11 0.15 0.64 0.09Proj -10 to 3 1.15 1.81 0.07 0.08 0.72 0.12TABLE V. Left: Orbital occupancies of the correlated or-bitals, obtained from the Matsubara Green’s function. Right:Occurrence probabilities of multiplet configurations obtainedfrom the impurity density matrix. LS stands for low spin( s = 0) and HS stands for high spin ( s = 1). level (e.g. between X and M) arises from the d x − y orbital. Corresponding to the different mass enhance-ments, this band is renormalized the most in the MLWFcase (see e.g. the distance below the Fermi level of theband at the Z point), then the projector cases, and thenthe SLWF case. The sharp peak visible in the d z densityof states in Fig. 6 arises from the almost dispersionlesscorrelated band visible from Z to A, while the features inthe − − C. Orbital Occupancies The occupancy of the different orbitals has been viewedas an important diagnostic of correlation physics. Forexample, in a model with a single relevant orbital, theorbital is more Mott-Hubbard like as it gets closer tohalf filling, while an occupancy noticeably greater thanhalf filling implies important charge transfer effects. Con-versely, a significant probability of occupation (by holes)of more than one orbital is a necessary condition forHund’s metal physics. Here we consider how the cal-culated orbital occupancies depend on the downfoldingmethodology.The left side of Table V shows the orbital occupan-cies, obtained directly from the impurity Green’s func-tion ˆ G QI ( iω n ) without analytic continuation. Compari-son to Table II shows that in all methods the main ef-fect of adding correlations is to drive the d x − y orbitalcloser to half filling while the d z orbital gets more full.However, the d z orbital is significantly less full in theprojector cases than in the Wannier cases, with the fill-ing depending on the energy window employed and beingsmallest for the wider window extending to 10 eV. Thisdemonstrates the importance of the hybridization to theNd orbitals at positive energy.The right side of Table V shows the probabilities ofdifferent multiplet configurations of the correlated statesobtained from the impurity density matrices determinedfrom the CTHYB solver. Important quantitative differ-ences are evident. In all cases the N = 3 configuration is0dominant, but in the Wannier cases the fluctuation into N = 4 (fully occupied e g , spin singlet) are larger than thefluctuations into N = 2 (2 holes in the e g ; potential highspin state), whereas in the projector methods the situ-ation is opposite. These differences have been used toargue for and against the relative importance of Hund’sand charge-transfer physics. [9, 31, 33, 35, 38, 39] D. One vs. Two Orbital Results One way to approach the physics of a complicated ma-terial such as the multilayer nickelates is to attempt todefine a “minimal model” of the correlation effects. Themuch larger value of the d x − y self energy than the d z self energy suggests that a minimal model might involveonly one correlated orbital. Insight into this possibilitymay be obtained by comparing results obtained from amodel with multiple correlated orbitals to those obtainedfrom a model with only one correlated orbital. n I m ( i n ) MLWF 1 orbitalMLWFSLWF 1 orbitalSLWFproj -10 to 10 1 orbitalproj -10 to 10 2 orbitalsproj -10 to 3 1 orbitalproj -10 to 3 2 orbitals FIG. 8. Comparison of imaginary part of the Matsubaraself energy of the d x − y orbital obtained from two orbitalDMFT calculations (solid lines) to self energy obtained fromone orbital DMFT calculation (dashed lines) with downfold-ing methods unchanged. We use exactly the same downfolding, definition of cor-related orbitals, and interaction U as in our previoustwo orbital DMFT calculations to perform “one orbital”DMFT calculations in which only the d x − y orbital istreated as correlated. Figure 8 compares the resulting d x − y self energies, and Table VI compares the mass en-hancements. In the SLWF case, the d x − y self energyand mass enhancement are not changed considerably byincluding the d z orbital, presumably because the d z orbital is almost completely full. In the MLWF and pro-jector cases, there is a significant difference in the self en-ergies and mass enhancements between the one and twoorbital results. Referring to Tables V and VI we at-tribute the differences to a combination of the differencefrom half filling of the d x − y orbital (with the larger oc-cupancy in the SLWF case indicating a greater relevanceof charge transfer physics) and (especially in the pro-jector cases) greater number of holes in the d z orbital. The SLWF results imply that correlation physics relatedto the d z orbital may be neglected without adverselyaffecting the accuracy of the results; the other methodswould suggest that this is not the case. E. Cuprate results The layered d nickelates of primary interest in this pa-per have a potentially rich physics associated with mul-tiple bands at the Fermi surface and important ligandstates lying both above and below the strongly corre-lated d states. In this subsection we examine the ex-tent to which our qualitative considerations apply alsoto the electronically simpler cuprate system, where onlythe d x − y is correlated (the d z orbital is to good ap-proximation completely full) but charge transfer to oxy-gen is also relevant. In this examination, a difficultyimmediately arises. A straightforward application ofthe DFT+DMFT methodology outlined above predictsa rather weakly correlated system, essentially because ε d − ε p is so small in magnitude that the d - d interaction U is irrelevant. Previous work has argued that straight-forward application of the DFT+DMFT method is notappropriate, essentially because the DFT approximationpredicts that the oxygen levels are about 1 eV closer tothe Fermi level than they are in practice. Adjusting the p -level energy “by hand” to match photoemission exper-iments [74] provides cuprate correlation physics in betteragreement with experiment and we follow this route here.Table VII shows that the phenomenon found in the-ories of the nickelate materials occurs also in theories ofcuprates: the MLWF method yields substantially larger d x − y mass enhancements than does the SLWF method.Examination of the Wannier fits (not shown) revealsthat origin is the same–the MLWF parametrization cor-responds to a larger ε d − ε p and smaller t pd and hence tostronger correlations.The occupancy analysis shown in Table VII confirmsthis conclusion, revealing a larger covalence (more N = 4weight) for SLWF than for MLWF. It is important tonote, however, that in contrast to the nickelate case,where the different methods point towards different un-derlying physics, in the cuprate case both the MLWF andthe SLWF methods paint the same picture of a chargetransfer material, different only in quantitative aspects. m (cid:63) m m (cid:63) m n x − y n x − y d x − y orbital in the one and two orbital DMFT calcula-tions on NdNiO . m (cid:63) m d x − y n d x − y N=3 N=4MLWF 1.9 1.41 0.56 0.42SLWF 1.5 1.52 0.45 0.54TABLE VII. Results for the d x − y orbital from a two orbitalcalculation on CaCuO where ε p is reduced by 1 eV. Left:Mass enhancement Middle: Orbital occupancy. Right: Oc-currence probabilities of multiplet configurations. V. DISCUSSION Quantum embedding methods approach the correlatedelectron problem by defining a subset of “correlated or-bitals” whose contributions to the physics are determinedby the use of a high level many body method and are self-consistently embedded into a more complex electronicstructure specified by an inexpensive, lower-level method.In the DFT+DMFT approach the correlated orbitals areidentified as partly filled, atomic-like orbitals relativelytightly localized to particular ions. Implementation ofthis appealing idea encounters the difficulty that the intu-itively clear idea of a “transition metal d orbital” cannotbe defined unambiguously because an atomic-like stateis not an eigenstate of any reasonable single particles ap-proximation to the electronic Hamiltonian. What mustbe done, in effect, is to define a single particle basis thatincludes atomic-like states with the desired spatial struc-ture and enough other states so that the projection ofthe Kohn-Sham Hamiltonian onto this basis reproducesthe DFT band structure. Different methods have beenused to define the correlated orbitals and select the addi-tional states; see Sec. II for a discussion of the principaltechniques. While the issue seems not to have been ex-tensively investigated (see [49] for an exception), the con-sensus in the field has been that all methods that producecorrelated states with approximately the desired spatialstructure and reproduce the DFT bands accurately areapproximately equally good. This paper investigates theissue carefully and finds that this is not at all the case,because the different methods in effect lead to differentpartitioning of the band states into correlated and uncor-related components, and these differences in partition-ing have a substantial effect on the computed correlationphysics.Focusing on one system of intense current interest, thelayered nickelate NdNiO , this paper performs a compar-ative study of the implications for the many body physicsof the methodology used to construct the correlated or-bitals. In the layered nickelates the important correlationphysics is believed to relate to states arising from theNi d x − y and perhaps also from d z orbitals. Straight-forward quantum chemical considerations suggest thatthe primary valence configuration of the Ni is d withone hole in the d x − y orbital, but other configurationsmay also be important. Currently debated questions in-clude the relevance of “Hund’s metal” physics arisingfrom high-spin d (two holes, one in d x − y and one in d z ), the importance of correlation physics in the Ni/Nd hybrid bands crossing the Fermi level, and the relevanceof valence fluctuations involving the O- p . In addressingthese questions, the definition of the Ni- d states and theirhybridization to other orbitals is evidently crucial.We use different variants of the two most widely usedtechniques, the projector and Wannier methods discussedin detail in Sec. II, to compute various physical quanti-ties, while keeping everything else the same. In Sec. IV Awe show that different methods lead to almost factor-of-two differences in the predicted renormalization factor(“mass enhancement”). In Sec. IV C we show that thedifferent methods also give quite different predictions forthe relevance of multiorbital (high spin, “Hund’s metal”)physics, and this conclusion is reinforced in Sec. IV Dwhich shows that the different methods give also verydifferent results for the changes in many body propertiesbetween models with two and one correlated orbitals.The issues are not specific to the NdNiO : Sec. IV Eshows that similar results are obtained in a model of thecopper-oxide superconductor CaCuO , where only onecorrelated orbital is relevant and charge transfer physicsplays a larger role.Before proceeding to the discussion of origin and im-plications of the results, we dispose of two side issues.First, the results mentioned in the previous paragraphall pertain to properties of the correlated orbitals as de-fined in the different methodologies. The correlated or-bitals themselves are only auxiliary quantities used inintermediate stages of computations of experimental ob-servables. Physically meaningful results are experimen-tal observables such as the mass enhancement, relativeto the underlying DFT mass, of the theoretically de-rived quasiparticle bands (see Fig. 7), which are measur-able in angle-resolved photoemission. Table IV showsthat the differences between these “band basis” mass en-hancements are actually greater than the orbital basisself energy renormalizations. One may similarly considerthe many-body density of states (local spectral function)measurable in angle-integrated photoemission and ana-lyzed in Sec. IV B. We find that large differences be-tween methods also appear in the local spectral function.Of special interest are the upper Hubbard feature around2 eV and the strong shift of the as Ni- d z characterizedband between the different methods (see Fig. 6).Second, most of our calculations investigate variationsat fixed values of the interaction parameters, while dif-ferent specifications of the correlated orbitals also im-ply differences in the interaction parameters governingthe physics of these orbitals, which might compensate tosome degree for the differences in orbital specification.We examine this issue in Sec. IV A, which presents re-sults of cRPA calculations of effective interaction param-eters for different downfolding schemes. We find that thedifferences in interaction parameters arising from differ-ences in specification of correlated orbitals are such as toenhance the differences between methods. In summary,different prescriptions for defining correlated orbitals leadto different results for physically measurable quantities.2We now discuss the interpretation and implications ofour results. Sec. III shows that the different parameter-izations lead to quite different fillings of the correlatedorbitals, already on the DFT level. Orbital filling is animportant determinant of correlation physics; for exam-ple, Hund’s metal physics requires at least two partiallyfilled orbitals while Mott physics is most pronounced ifthere is one nearly half filled correlated orbital. We findthat the MLWF Ni- d x − y occupation is much closer tohalf-filling than the SLWF Ni- d x − y occupation, and isthus more likely to be found more correlated. Both pro-jector approaches have very similar Ni- d x − y occupationcompared to MLWF. However, the Ni- d z content is lowerfor projectors, depending on the energy window, whichcould be an indicator that projectors fail to capture inter-stitial contributions in strongly hybridized systems. Thisis observed both with Wien2k and VASP projectors.The differences arise because different constructionsof the correlated orbitals correspond to different embed-dings of the correlated orbital in the underlying band the-ory, or in other words to different overlaps of correlatedorbitals with Kohn-Sham eigenfunctions. To be explicit,for transition metal oxides such as NdNiO , importantparameters include the energy level difference betweenoxygen p and transition metal d orbitals ε d − ε p , andthe p - d hybridization t pd (see Tab. II) and (see Tab. III).Different constructions of the correlated orbitals lead toequally accurate parametrizations of the calculated bandstructures but with drastically different values of ε d − ε p and t pd . Furthermore, we find that the hybridization tothe Nd- d orbitals right above the Fermi level is stronglymethod dependent (see Fig. 3). We emphasize that thisbehavior is found both for VASP and Wien2K band the-ory codes.Section IV C shows that the different methods leadto differences in the multiplet occurrence probabilities inthe interacting theory. The multiplet occurrence proba-bilities are often used to gauge the nature of correlations.In a typical one orbital Mott-Hubbard system we expecta dominance of N = 3 with roughly equal amounts of N = 4 and low spin N = 2. In a charge transfer mate-rial, we expect more N = 4 than N = 2. This is seenin the MLWF case and to a much greater extent in theSLWF case, but not in the projector cases. Conversely,a large amount of high spin N = 2, seen in the projectorcases but not the Wannier cases, points to an importanceof Hund’s correlations. Using these results to classify thematerial will therefore lead to different conclusions basedon the downfolding method employed, and may explainthe differences in classification found in the literature.Likewise, our results in Section IV D show that compar-ing one and two orbital calculations using the differentmethods leads to different results on the importance ofmultiorbital effects.The results presented in this paper, along with the ex-isting discussions in the literature around the issues ofvalue and frequency dependence of interaction parame-ters and the double counting correction, underscore the fact that the DFT+DMFT methodology requires choicesat various points in the calculation. As noted in othercontexts [65], experiment can to some degree help toguide the required choices. Experimental probes involv-ing form factors that can distinguish between d and p orbitals can help pin down orbital content of differentbands. Further, differences in e.g. the position of the d z orbital relative to the chemical potential (see Fig. 6)and broadening of the − RuO wherethe correlated bands are to good approximation disen-tangled from the other bands and a low energy theoryinvolving only the correlated bands may be constructed,the physics is independent of the method used to con-struct the correlated orbitals.The success of DFT+DMFT in many contexts moti-vates further research to determine the optimal down-folding approach for different physical contexts. Identi-fication of experimental observables that will distinguishdifferent downfoldings will also be valuable, as would bethe determination of quantities that are robust with re-spect to choice of downfolding. VI. ACKNOWLEDGEMENTS J.K. and A.J.M. acknowledge funding from the Ma-terials Sciences and Engineering Division, Basic EnergySciences, Office of Science, US DOE. We thank F. Lecher-mann, S. Beck, M. Zingl, A. Botana, and A. Georges forvery helpful discussions. The Flatiron Institute is a divi-sion of the Simons Foundation.3 Appendix A: t g DOS and Hybridization Figure 9 show the density of states and Figure 10 showthe imaginary part of the real frequency hybridizationfor the Ni- t g orbitals, corresponding to the plots for the e g orbitals in Figures 2 and 3. The plots show somedifferences between methods, but less drastic than the e g orbitals. E E F (eV)0.00.20.40.60.81.01.21.41.6 D O S ( / e V ) a) d xy E E F (eV) b) d xz / yz MLWFSLWFproj -10 to 10proj -10 to 3 FIG. 9. Uncorrelated density of states (per spin) of the t g orbitals with the different methods. - I m () ( e V ) a) d xz / yz MLWFSLWFproj -10 to 10proj -10 to 3 b) d xy FIG. 10. Negative imaginary part of the real frequency hy-bridization of the t g orbitals. Appendix B: Doping Dependence To simulate the effects of hole doping, we use Wien2kwith the virtual crystal approximation, adjusting theatomic numbers of the Nd ions to fractional values andcorrespondingly change the number of electrons.We assess the behavior of each of the downfoldingmethods upon doping by running the same calculationswith a hole doping of 0 . 2. We achieve this doping usingthe virutal crystal approximation, where we artificiallychange the Nd atomic number to 59 . . . d x − y filling de-creases with hole doping, as expected. The filling de- creases more in the SLWF case than the other cases. Inall cases, the d z filling does not change considerably,but the behavior depends on the method. It increases d x − y d z MLWF -0.048 -0.004SLWF -0.078 -0.003Proj -10 to 10 -0.053 0.004Proj -10 to 3 -0.049 0.015TABLE VIII. Orbital occupanices (summed over spin) at holedoping of 0.20 minus the orbital occupanices at stoichiometry.Positive values indicate that the filling increases with holedoping. the most in the projector from -10 to 3 case because theself doping band moves above 3 eV so more d z weighthas to be given to the main d z -derived band below theFermi level. In the projector from -10 to 10 case it alsoincreases. In the MLWF and SLWF cases it decreases. Appendix C: Full Charge Self Consistency For the case of projectors in the wide window from − 10 to 10 eV, we compare the results of a fully chargeself consistent (FCSC) DFT+DMFT calculation to theone-shot results. In the FCSC case, we find the same d z orbital occupancy as the one shot case and a d x − y occupancy of 1 . 16, very close to the one shot results of1 . 14. Likewise, the self energies are not so different inthe one shot and FCSC case, as shown in Figure 11. 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