Many-body Electronic Structure of NdNiO$_2$ and CaCuO$_2$

The demonstration of superconductivity in nickelate analogues of high $T_c$ cuprates provides new perspectives on the physics of correlated electron materials. The degree to which the nickelate electronic structure is similar to that of cuprates is an important open question. This paper presents results of a comparative study of the many-body electronic structure and theoretical phase diagram of the isostructural materials CaCuO$_2$ and NdNiO$_2$. Important differences include the proximity of the oxygen $2p$ bands to the Fermi level, the bandwidth of the transition metal-derived $3d$ bands, and the presence, in NdNiO$_2$, of both Nd-derived $5d$ states crossing the Fermi level and a van Hove singularity that crosses the Fermi level as the out of plane momentum is varied. The low energy physics of NdNiO$_2$ is found to be that of a single Ni-derived correlated band, with additional accompanying weakly correlated bands of Nd-derived states that dope the Ni-derived band. The effective correlation strength of the Ni-derived $d$-band crossing the Fermi level in NdNiO$_2$ is found to be greater than that of the Cu-derived $d$-band in CaCuO$_2$, but the predicted magnetic transition temperature of NdNiO$_2$ is substantially lower than that of CaCuO$_2$ because of the smaller bandwidth.

Introduction: The remarkable physics of layered copper-oxide materials [1][2][3][4], including high transition temperature superconductivity for carrier concentrations not too far from optimal doping, electronic pseudogaps, and various forms of long-and short-range order, has challenged researchers over the more than thirty years since the discovery of superconductivity in La 2−x Ba x CuO 4 [5]. Many cuprate superconductors are known; all share the structural motif of CuO 2 planes weakly coupled in the third dimension and the electronic motif of an approximately d 9 electronic configuration for Cu, but differ in other details. A key question is whether the novel physics of these materials can be essentially understood in a one-band model with strong local correlations [1] or whether other physics is important [6,7].
It was recognized early on that perspective on this issue could be gained from the analysis of materials with similar features but with a different local chemistry. Ni, which is adjacent to Cu on the periodic table, has been of particular interest in this regard [8][9][10][11]. Because Ni has one fewer proton than Cu, the d-electron count for Ni is typically expected to be lower than for Cu, but with appropriate chemistry a configuration close to Ni d 9 , with one hole in the d x 2 −y 2 orbital, might be achieved. Proposals to use artificial superlattices [12] have yet to yield such a configuration, but recently, trilayer systems with a formal valence of Ni 4/3+ (d 8.67 ) and d x 2 −y 2 holes were synthesized [13]. One of these materials (Pr 4 Ni 3 O 8 ) is metallic but not superconducting, likely because the carrier concentration corresponds to a doping far beyond the optimal doping of typical cuprate superconductors. NdNiO 2 is isostructural to the "infinite layer" cuprate CaCuO 2 and has a formal Ni d 9 valence. The recent discovery [14] that NdNiO 2 can be synthesized, hole doped, and made superconducting has created intense excitement. There has been an outpouring of theoretical interest, with DFT [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29], DMFT [24][25][26][27]30], and model-system [18,19,21,[31][32][33][34][35] studies of the material. These papers have come to a variety of conclusions about the important low energy physics of the nickelate materials and its relation to the low energy physics of the cuprates. In this paper we aim to clarify some of the issues.
Because transition metal oxides such as cuprates and nickelates are relatively ionic, the basic electronic structure problem may be posed in terms of corrections to a "formal va- configuration vis a vis d 10 L is not yet determined. Further, Nd 2+ is low enough in energy that there is some charge transfer to Nd, meaning that at least the d 9 L configuration, and perhaps also the d 8 , needs to be considered even in the stoichiometric material. If d 8 is important, one may ask whether the relevant d 8 state is high spin (implying one hole in the d x 2 −y 2 and one in the d 3z 2 −r 2 ), as proposed or implied by some calculations [10,15,24,31,34,35], or low spin (both holes in the d x 2 −y 2 orbitals).
Methods: Our study, in common with most theoretical studies of many-body electronic structure of solids, is formulated in a subspace of the full electronic Hilbert space that is constructed from single particle states obtained from a relatively inexpensive mean fieldtype calculation. We employ DFT (Quantum Espresso [41] with the PBE-GGA exchangecorrelation functional [42]); our DFT results are in agreement with prior literature [15-21, 23, 24, 31]. We then use Wannier90 [43] to construct a basis of maximally-localized Wannier functions [44,45] spanning an energy window of interest. We employ two energy 3 windows: One is a wide energy window that includes the full transition metal d-manifold, the O-p states, and, in NdNiO 2 , the relevant Nd states. The second is a narrow energy window that includes only the "frontier orbitals" that cross the chemical potential. The wide energy window enables a natural connection to formal valence considerations and high energy spectroscopies and helps define the frontier orbital model, while the frontier orbital window is better suited to discussions of the low energy physics. In both cases the resulting Wannier bands reproduce the DFT bands very well. Details, including lattice constants, k- The Wannierization of the Nd-derived 5d states in NdNiO 2 has been the subject of discussion in the literature [16,17,19,23]. We use selective localization methods [46] as implemented in Wannier90 [43] to localize only the Ni-d Wannier functions, which are also constrained to be centered on the Ni sites. The resulting fit yields not only physically reasonable Ni-d Wannier functions but also two Nd-centered orbitals, one of xy and one of 3z 2 − r 2 symmetry.
We define an effective Hamiltonian by projecting the Kohn-Sham states onto the Wannier basis and adding interactions that couple some of the states in the subspace. The interaction terms depend on the energy window and the choice of correlated orbitals, both because the sizes of the relevant orbitals are window-dependent and because the value of U depends on screening which again is affected by the energy window. For our wide energy window calculations we choose the transition metal d x 2 −y 2 and d 3z 2 −r 2 orbitals as the correlated subspace and use and interaction of Kanamori form [47] with U = 7 eV and J = 0.7 eV, representative of nickelates [48] and cuprates [36]. To compensate for the Hartree shifts induced by the added interactions, we include a double counting correction in the form proposed by Held [49] (see Supplementary Material for details). For the frontier orbital calculations we correlate only the d x 2 −y 2 -derived orbitals and consider a range of U , noting that recent cRPA calculations suggest a value of U ≈ 3.1 eV [16]. We then perform singlesite DMFT calculations using the TRIQS software library [50,51] with the continuous-time hybridization expansion solver (CT-HYB) [52]. These calculations are "single shot" in the sense that the DFT density is not further updated. The zero of energy is defined to be the chemical potential.

Results:
The main panels of Fig. 1 show the many-body density of states projected onto the Wannier orbitals computed in the wide energy window for CaCuO 2 (right panel) and NdNiO 2 (left panel). In both materials, the removal spectrum (ω < 0, with the chemical potential defined as the zero of energy) is dominated by O-p and non-d x 2 −y 2 states, and the addition spectrum (ω > 0) by a transition metal-d x 2 −y 2 and O-p hybrid, along with nearly empty Nd-derived bands in NdNiO 2 . The transition metal t 2g and d 3z 2 −r 2 states are essentially filled, although in NdNiO 2 close inspection of Fig. 1 reveals a tail of these states at ω > 0. This tail is a consequence of hybridization with the Nd bands and will be discussed in more detail below. In both materials the near chemical potential transition metal d x 2 −y 2 spectrum exhibits a van Hove peak, which is at the chemical potential in the Ni material and below in the Cu material, a weak maximum at ω ∼ −2 eV, and a gap (due to hybridization with the O-p states) at ∼ 1-2 eV below the chemical potential. The higher lying parts of both the CaCuO 2 and NdNiO 2 d x 2 −y 2 and O-p addition spectra are similar, implying a non-negligible mixing of d x 2 −y 2 and O-p σ in both materials.   While the wide spread in energy of the oxygen-related removal features makes it difficult to precisely define a charge transfer energy, we may conclude that the DMFT data places the Ni material in the same charge transfer insulator class as the cuprates but with less O-p involvement in the near Fermi-level states.
We now consider in more detail the nature of the correlated near chemical potential states in the Ni material. Several authors have investigated the infinite layer nickelates using the spin density functional plus U (sDFT+U) method [9,10,15], which is in effect the Hartree approximation to the wide energy window DFT+DMFT method [54], although the use of a spin density functional rather than the simple density functional method is an important difference [55,56]. For a range of U including U = 0 this method finds that the ground state is antiferromagnetic and metallic with some hole doping on the majority spin d x 2 −y 2 orbital and an empty minority spin d x 2 −y 2 orbital. This state is the Hartree (single determinant) version of the DMFT state reported here. As the magnitude of the on-site interaction is increased beyond a critical value U c ≈ 6 eV, the DFT+U method with the FLL double counting scheme [57] finds [10,15] a first-order transition to a state with a fully filled majority spin d x 2 −y 2 orbital and holes on the d 3z 2 −r 2 orbital in the minority spin channel (the transition is absent if the AMF double counting scheme [58] is used instead).
The full occupancy of the majority spin d x 2 −y 2 state indicates an orbitally selective Mott transition, while the holes in the minority spin channel indicate an admixture of high spin d 8 in the ground state, thus identifying the nature of the first-order transition. This transition is not found in the DFT+DMFT calculations at the U values we have studied, but within a DFT+DMFT perspective at U = 10 eV Lechermann [24] finds an orbitally selective state with insulating d x 2 −y 2 and metallic d 3z 2 −r 2 , implying a small but important admixture of d 8 . Studies of other compounds find that the DFT+U approximation very substantially underestimates the critical U of the low-spin to high-spin transition [55] and that the spinpolarized DFT functionals overestimate the spin splitting of the d states [56]. We therefore believe that this aspect of the DFT+U calculations is not relevant to NdNiO 2 . However, as an exploration of a theoretically interesting model, a DMFT-based investigation of orbitally selective Mott and high spin-low spin transitions in models with two correlated orbitals coupled to one or more uncorrelated bands would be desirable.
In charge transfer systems the effective interaction strength is determined not only by the U and J parameters but by the double counting correction, which is the subject of some discussion [59,60] and is one of the significant uncertainties in the DFT+DMFT and Finally, we quantify the strength of correlations via the d x 2 −y 2 component of the electron self-energy shown in Fig. 2. We see immediately that the self-energy magnitude in the Cu material is much less than in the Ni material. The self-energy corresponding to the calculation with increased charge transfer energy (ε d −ε p ) in CaCuO 2 is still somewhat smaller than the self-energy for the nickelate material. One quantitative metric is the renormalization give U ≈ 3.1 eV for the Ni material [16].
In Fig. 2, we show the frontier-orbital self-energies for both materials obtained for this U value. The frequency dependence of the self-energy is quite different at high frequencies from that found in the wide energy window calculations, as expected from the further truncation of the Hilbert space in the frontier orbital case, but the behavior in the low frequency regime can be compared. We obtain Z = 0.25 for the Ni-derived band, slightly lower than what we found in the band basis (Z ∼ 0.3) for the wide energy window calculation.
The larger bandwidth of the cuprate material leads to less correlation and only Z = 0.42.
That is much smaller than the renormalization we obtained in the wide energy window calculations (Z = 0.83) and still smaller than in the calculation with increased ε d − ε p difference (Z = 0.62).
The resulting many-body Fermi surfaces (trace of the many-body spectral function evaluated at ω = 0) are shown in Fig. 3 as plots in the space of in-plane momenta for different k z . Due to the smaller self-energy, the many-body Fermi surface is quite sharply defined for the cuprate; it overlaps the non-interacting Fermi surface precisely so only the latter quantity is shown. NdNiO 2 displays clear differences in its Fermiology from CaCuO 2 . The Nd d 3z 2 −r 2 orbital gives rise to a band that crosses the chemical potential, leading to an oblate Fermi surface centered on the Γ point (see panels for k z = 0.0 and 0.125), while the Nd d xy orbital gives rise to an oblate Fermi surface pocket centered on A (see panels for k z = 0.375, 0.45, and 0.5), as also found in previous DFT and DFT+U calculations [9,10,15] and in a recent DFT+DMFT study [24]. Interestingly, in DFT calculations for LaNiO 2 the pocket at Γ is noticeably smaller than in NdNiO 2 [9] and it is absent in DFT+DMFT calculations for the La material [27].  The most important difference between the two phase diagrams, however, is that the self-doping provided by the Nd 5d bands means that up to a moderately large U , the Ni material is an antiferromagnetic metal, as found in other DMFT calculations [25]. Only if U is increased beyond a large, temperature-dependent value of about 5 eV is an antiferromagnetic insulator phase found in this calculation. The physics of the transition is that the antiferromagnetic order parameter opens a gap in the Ni spectral function; when this gap becomes large enough so that the top of the lower antiferromagnetic band falls below the bottom of the Nd 5d bands, then all of the charge from the Nd 5d states is transferred to the Ni band, which becomes half-filled such that an insulating state may result. The transition in the paramagnetic phase is similarly driven when the splitting between the upper and lower

The Nd-derived bands centered at Γ and
Hubbard band becomes large enough. This transition depends crucially on the energetics of Ni-Nd charge transfer, which, however, may not be correctly captured by the three-band frontier orbital model used here. It may also be preempted by a high-spin/low-spin transition in which holes are added to the Ni-d 3z 2 −r 2 sector. Both of these effects will tend to push the transition to larger U . Understanding these issues is a theoretically interesting question requiring a fully charge self-consistent calculation and a theoretically justified double counting correction.

Conclusion
We have presented a comparative study of two isostructural materials, NdNiO 2 and CaCuO 2 . Both have a transition metal configuration near d 9 and correlation physics dominated by the transition metal d x 2 −y 2 orbital. A salient difference between the two materials is that the energy splitting between the oxygen 2p and transition metal 3d bands is less in the cuprate than in the nickelate material, leading to relatively weaker correlations in the cuprate case. However, we find that both the nickelate and the cuprate should be regarded as charge transfer materials with a significant admixture of an oxygen hole/d 10 configuration and a negligible role of the d 8 configuration. This conclusion is different from that found in DFT+U calculations at U 6 eV [10,15] and DFT+DMFT calculations at U 10 eV [24].
Other differences between the materials are a modest decrease in the proportion of oxygen holes and, most importantly, that in the nickelate material bands of Nd 5d origin cross the Our conclusions are at variance with others reported in the literature. Several authors argue for the importance of the high-spin d 8 state [10,15,24], which combined with an orbital selective Mott transition leads to a picture of a Nd/Ni hybridized band Kondohybridized to local moments on the Ni sites. Our belief is that this physics is unlikely to be relevant to the 112 nickelates. However, higher-U parameter regimes the models we and others have derived for the Ni materials display very interesting physics, including the Kondo and multi-orbital effects discussed in Ref. [30,33] and the high-spin/low-spin and orbitally selective Mott physics found in DFT+U [10,15] and DFT+DMFT [24], which warrant further investigation.
Most importantly, we find that the low-energy physics of the Ni material is described by a three-band model, with one correlated Ni-d x 2 −y 2 -derived band with a van Hove singularity that crosses the Fermi level as k z is varied and two Nd-derived weakly correlated spectator bands. The effect of the van Hove singularity and the spectator bands on the superconducting physics of the Ni-derived band remains to be determined. For example, one may ask, within the theory as defined here, why the (self) doped Ni compound is not superconducting even in the nominally stoichiometric case [14].

A. DFT Calculations
We use Quantum Espresso [41] to perform the DFT calculations. We use PAW pseudopotentials [66] with the Nd-f states as part of the core. For both compounds we take the ideal tetragonal structure. For NdNiO 2 we use the structure parameters a = 3.95Å and c = 3.37Å and for CaCuO 2 we use a = 3.86Å and c = 3.20Å. We employ the PBE-GGA functional [42], a k-point mesh of 16 × 16 × 16, an energy cutoff of 70 Ry for the wavefunctions, and an energy cutoff of 280 Ry for the density and potential. We get an identical band structure using a k-point mesh of 32 × 32 × 32, a wavefunction energy cutoff of 150 Ry and a density and potential cutoff of 600 Ry. For CaCuO 2 , we compared the band structure to an all electron calculation (WIEN2k [67]) and find excellent agreement.

B. Energy Windows and Wannierization
The first step of the many-body calculations is to define a basis of single particle states, some of which are treated as correlated and some as uncorrelated or "background" states.
The interaction parameters used for the correlated states depend on the basis chosen. In DMFT calculations the single particle basis is typically defined by choosing an energy window and then defining the single particle states in terms of the DFT bands that fall within this window. For transition metal oxides, two choices of energy window are common and both are employed here.
The "wide energy window" is an energy range reaching ∼ 6 to 10 eV below the chemical potential, chosen to incorporate the O-p-derived states. Experience with other compounds indicates that the basis states constructed from the wide energy window are substantially similar to the corresponding free ion states. The "narrow energy window" is an energy range that captures only the near Fermi level "frontier" orbitals, particularly the antibonding bands composed of Cu/Ni d x 2 −y 2 -O p-derived states. These states are physically rather more extended than the rest of the states defined in the wide energy window construction.
In NdNiO 2 it is essential that the narrow energy window calculations also incorporate the Nd-derived bands that cross the Fermi level.
To construct the single particle basis we construct maximally localized Wannier func-  tions [44,45] using Wannier90 [43,68] and a dense 21 × 21 × 21 k-point mesh. The construction of the Nd-derived Wannier states has been the subject of discussion in the literature [16,17]. For NdNiO 2 in both the wide and narrow energy windows we use the "selective localiztion" method of Ref. [46] to localize only the Ni-d orbitals, which are also constrained to be centered on the Ni sites. These restrictions improve convergence and provide a Hamiltonian which is to good accuracy real.

C. DMFT calculations
We perform single-shot single-site DMFT calculations using the TRIQS software library [50][51][52]. In the wide energy window calculations, we only treat the Ni/Cu-e g orbitals as correlated, but we include all of the Wannier functions in the DMFT self-consistency condition. In the narrow energy window calculation, we treat only the Ni/Cu-d x 2 −y 2 orbital as correlated, but in the nickelate case we also include the Nd orbitals in the self-consistency condition. For the DMFT calculations, we interpolate the Wannier function on a denser 40 × 40 × 40 k-point mesh.
For the impurity problem, we use a Kanamori interaction Hamiltonian [47] with U = U − 2J. In the wide energy window calculation, we use an on-site Hubbard interaction of U = 7 eV and a Hund's coupling of J = 0.7 eV for both materials, generally accepted values [36,48], at a temperature of T = 290K. For the narrow energy window, we perform calculations at different values of U and T to construct a phase diagram. We allow for in-plane antiferromagnetism by doubling the unit cell in the c(2 × 2) scheme and forcing the self-energy for neighboring sites and opposite spins to have the same value. To solve the impurity problem, we employ the continuous-time hybridization expansion solver [52]. We typically use ∼ 10 8 measurements, but we use ∼ 10 9 when analytic continuation is needed and ∼ 10 7 for parts of the phase diagram. We employ Held's double counting formula [49]: where D is the number of correlated orbitals and n is the total density, for which we take the density of the correlated orbitals obtained from their local non-interacting Green's function.
We employ the maximum-entropy method [69] to perform analytic continuation on the Green's function and self-energy.   For NdNiO 2 , the paramagnetic to antiferromagnetic transition is found at fixed U or β by varying β or U just above the transition, fitting β or U as a function of magnetization squared to a polynomial, and taking the y-intercept of the resulting fit as the transition point. Fig. 11 shows an example at constant temperature. To find the metal-insulator transition in the nickelate, we make a linear fit to µ(U ) in the region of the transition and For the cuprate where the transitions are sharper, we determine the transition points by bisection. Since the cuprate is at half filling, it is a metal if the Matsubara self-energy goes to 0 as ω n → 0 and an insulator if the Matsubara self-energy diverges as ω n → 0. Fig. 12 shows the spectral function in the paramagnetic phase for different values of U .
The figure shows that the metal insulator transition coincides with the Nd bands emptying out. Fig. 13 shows that in the paramagnetic phase a higher value of U is needed to empty out the Nd bands than in the antiferromagnetic phase, which explains why the metal-insulator transition is at a higher U for the paramagnetic phase than the antiferromagnetic phase.
The Nd occupancy is greater at higher temperatures, so the metallic phase is more stable at higher temperatures.

G. Role of Nd bands
In the nickelate material in the narrow energy window calculation, the Nd orbitals hybridize only weakly with the Ni-d x 2 −y 2 state (see Tab. II), but their main effect is to provide a doping reservoir for the system. We test this claim by doing a one band Wannier fit, keeping just the Ni-d x 2 −y 2 -derived orbital. We fix the electron density to 0.903, the value obtained in the three-band model with U = 3.1 eV, and perform a DMFT calculation in the one-band model with the same U = 3.1 eV. We find that the imaginary part of the Matsbuara self-energy is nearly indistinguishable from the three-band case (see Fig. 14). and Nd (green) states. As U increases, a gap opens in the d x 2 −y 2 DOS above the chemical potential.
At around U = 7 eV the gap reaches the chemical potential when the Nd orbitals empty out.