Normal state of Nd$_{1-x}$Sr$_x$NiO$_2$ from self-consistent $GW$+EDMFT

The recent discovery of superconductivity in hole-doped NdNiO$_2$ thin films has captivated the condensed matter physics community. Such compounds with a formal Ni$^+$ valence have been theoretically proposed as possible analogues of the cuprates, and the exploration of their electronic structure and pairing mechanism may provide important insights into the phenomenon of unconventional superconductivity. At the modeling level, there are however fundamental issues that need to be resolved. While it is generally agreed that the low-energy properties of cuprates can to a large extent be captured by a single-band model, there has been a controversy in the recent literature about the importance of a multi-band description of the nickelates. The origin of this controversy is that studies based entirely on density functional theory (DFT) calculations miss important correlation and multi-orbital effects induced by Hund coupling, while model calculations or simulations based on the combination of DFT and (extended) dynamical mean field theory ((E)DMFT) involve ad-hoc parameters and double counting corrections that substantially affect the results. Here we use a multi-site extension of the recently developed $GW$+EDMFT method, which is free of adjustable parameters, to self-consistently compute the interaction parameters and electronic structure of hole-doped NdNiO$_2$. This full ab-initio simulation demonstrates the importance of a multi-orbital description, even for the undoped compound, and produces results for the resistivity and Hall conductance in qualitative agreement with experiment.

ables an ab-initio simulation of strongly correlated materials without adjustable parameters, to clarify the electronic structure and the importance of multi-orbital physics in undoped and hole-doped NdNiO 2 . For this purpose, we extend the GW +EDMFT method to two coupled interacting low energy models for Nickel and Neodymium containing five and two bands, respectively. This low-energy theory with self-consistently computed dynamically screened interaction parameters is embedded without double counting of interaction energies into an ab-initio bandstructure, as illustrated in Fig. 1. Our computational scheme starts with a DFT calculation in the local density approximation (LDA), where we employ the Virtual Crystal Approximation (VCA) to realistically account for the shifts of the bands induced by hole doping. The downfolding to the low-energy subspace hosting the Nd and Ni bands is achieved with a single-shot G 0 W 0 calculation. In this way, the high-energy degrees of freedom are incorporated via a frequency and momentum dependent self-energy and polarization into the bare propagators and bare interactions of the low-energy model. The latter is solved using self-consistent GW +EDMFT, with separate EDMFT impurity models for Nd and Ni. The GW +EDMFT formalism 20   hole dopings (dark red: undoped, red: close to optimal doping, orange: overdoped). c, Frequency and doping dependence for the Hund coupling between Ni-3d x 2 −y 2 and Ni-3d z 2 .
effects, while EDMFT, the extension of DMFT to systems with nonlocal interactions, 21 In particular, we notice that the J values decrease with hole doping (see also Fig. 2c), while the inter-orbital interactions increase more strongly than the intra-orbital ones.
In a strongly correlated, half-filled single-band model one would expect to find spins with magnitude |S z | close to one half, and fluctuations to states with |S z | = 0. As can be seen from Fig. 2b  population of spin-1 moments with hole doping 5,9,10 only sets in on the overdoped side of the experimental T c dome.
The multi-orbital nature of the undoped and hole-doped nickelate compounds is the consequence of local energy renormalizations induced by the local EDMFT self-energies. In particular we find a strong frequency dependence of the real part ofΣ Ni loc , which is positive and (except for the d x 2 −y 2 orbital) increasing with decreasing energy. As a consequence, local energies which correspond to fully occupied orbitals at high energy, become available for low energy hole-like charge fluctuations (or fast virtual charge excitations), in agreement with the previously discussed configurational statistics. To demonstrate this effect of the self-energy we present in Fig. 3 the local energies obtained using three different approaches.
We first consider the diagonal entries of the real space LDA Hamiltonian at the origin, In agreement with the DFT results reported in the literature, 7,8 this yields a picture compatible with a single-band description. To represent the local energies of the interacting system, we consider the center of mass of the local spectral function, ∞ = ωA(ω)dω. (This includes all the self-energy terms connecting the different tiers, while just adding the real part of the local EDMFT self-energy to LDA would for example miss the GW contributions.) Also in this picture, which may be thought of as the Hartree limit of our result, the Ni states are mostly occupied, except for the 3d x 2 −y 2 orbital.
Finally, to illustrate the effect of the frequency dependence of Σ Ni loc (ω n ), we recall that the fermionic Weiss is defined with respect to an effective local energy E loc determined by the self-consistency equations. In particular, E loc incorporates the modifications of the bandstructure in the downfolding and the k-dependent GW contributions to the self-energy. In the EDMFT impurity calculation, the local self-energy is then added by the solver to this effective level.
We thus plot in the right hand level diagrams the renormalized impurity level positions 0 = E loc − µ + Σ(0). The result indicates a substantial shift of the 3d z 2 and 3d xz,yz orbitals to higher energies, compared to the Hartree limit, while the 3d x 2 −y 2 and 3d xy orbitals shift to lower energies. The arrows in the level diagrams demonstrate that increasing the frequency ω n , where the self-energy is evaluated, to 3 eV (approximate bandwidth) brings the levels closer to their ∞ values. In recent LDA+DMFT studies, Lechermann 12,14 emphasized the behaviour of the Ni-3d z 2 orbital, which for his large value of the onsite interaction (U = 10 eV) empties out and thus enables the system to undergo an orbital selective Mott transition. Even though our selfconsistently computed static onsite interactions are a factor of two smaller, the result in Fig. 3 is reminiscent of this phenomenology. The same tendency can be seen in the Ni-3d z 2 dispersion of the interacting system (see SM Figs. [6][7][8] in which the flat part of the band is shifted up in energy very close to the Fermi level. As a further support of our interpretation, we plot in Fig. 3(b, d, f) the Fourier transforms of the local charge susceptibilitiesχ nn (τ ) = n(τ )n(0) on the real frequency axis. This quantity defines the screening due to local charge fluctuations and the contribution of each orbital is easily identifiable. As expected no contribution is found from Ni-3d xy and Nd-5d xy which are, respectively, completely filled and completely empty, while Nd-5d z 2 contributes only in undoped NdNiO 2 which hosts the hole pocket at the Γ point. Ni-3d xz/yz fluctuates strongly in the undoped compound, in agreement with the energy-dependent shift of the effective level position (red arrow). Overall, the optimally doped compound is least affected by charge fluctuations, which suggests that the latter do not play a role in the pairing mechanism.
In Fig. 4 we show how the tight-binding Fermi surfaces are modified by the interactions.
The interacting result corresponds to the trace of the k-resolved spectral function evaluated at ω = 0. In agreement with the existing literature, 7,8,12,25 our non-interacting reference system contains two hole pockets centered at the A and Γ points, associated with Nd-5d xy and Nd-5d z 2 states, respectively. DFT predicts that this latter band is continuously pushed above the Fermi level as the hole doping is increased. On top of this shift our results, already at the G 0 W 0 level, indicate a substantial flattening of the bands along the Γ-X and Γ-M directions (see SM and sketch of the Brillouin zone in Fig. 1a). Due to the very low carrier concentration there are little additional deformations induced by the local interactions. The difference between the G 0 W 0 and GW +EDMFT treatment concerns mainly an increase in the local energy. The combination of these two effects yields a broadened Γ pocket, which has the highest intensity at k z = 0 in the undoped compound (see Fig. 4a). Moving to the intermediate value of k z = 0.25 (see Fig. 4b), in addition to the dominant Ni-3d x 2 −y 2 contribution, we notice some spectral weight in the M-A direction (at the corners of the Brillouin zone), which has a Ni-3d xz,yz character. This feature is consistent with the charge susceptibility results for undoped NdNiO 2 . Finally, at k z = 0.5, we find significant deviations from the DFT result. The Ni-3d x 2 −y 2 pocket centered at the Z point gives way to a Ni-3d z 2 feature, as can be seen from the lobes emerging in the Z-R direction. In this region, the dispersion is already flat at the DFT level but, as several of our results suggest, the interaction effects significantly increase the local energy. Upon doping, the Γ pocket and the small M-A weight are removed, leaving the typical d x 2 −y 2 shape at k z = 0, 0.25, while the lobes originating from Ni-3d z 2 states persist. The latter indicates an active role of these states at the Fermi level in all the studied setups. A general observation is that the Fermi surface, and especially the Ni-3d x 2 −y 2 contribution, gets closer to the LDA result with increasing hole doping, which is consistent with the system becoming more metallic and less correlated.
In  like peak at low frequency, originating from excitations within the quasiparticle band, and incoherent structures σ inc (ω) at higher energies stemming either from inter-band excitations or, for a correlated band, from excitations to the Hubbard satellites: 28 where τ denotes the relaxation time and D = e 2 n m * is the Drude weight written in terms of the carrier density n and effective mass m * . Our results in Fig. 5a capture both the low and high energy peaks with weight that is transferred from the latter to the former as a function of hole doping. By separately computing the contributions from the Nd and Ni sites, we find that Nd yields a high energy peak which is essentially fixed at ω ∼ 3.5 eV, regardless of the doping concentration, so that the spectral weight transfer originates from Ni (see SM for the site-resolved σ yy at various doping levels). These observations demonstrate a more metallic behaviour of the Ni sub-system with increasing hole-doping, despite the almost constant occupation and the increase in the interaction parameters (Eq. (1)). The weaker correlations result from the stronger increase in the interorbital interactions, compared to the intraorbital interactions, and the corresponding weakening of the Hund couplings, and hence are a nontrivial manifestation of multi-orbital effects. To quantify the degree of metallicity of the correlated multiorbital system, we extract the Drude weight by fitting the low energy part of σ yy (ω) to Eq. (2). This weight is inversely proportional to an effective mass defined for the entire system and is shown in Fig. 5b. The result indicates that doping indeed brings the system into a more metallic, less correlated state. This is consistent with several measurements 2,3,16 which report a decrease in the resistivity with hole doping on the underdoped side of the T c dome, which appears to be least affected by disorder. In Fig. 5b we also plot the quasiparticle weight obtained from the local self-energy of the Ni-3d x 2 −y 2 orbital. Both estimates are in qualitative agreement with the DFT+DMFT results reported by Kitatani et al. 13 .
While for the standard conductivity one needs to take the derivative of the free energy with respect to the vector potential twice, the Hall conductivity σ yxz involves a third order process, which requires an additional vertex insertion. In deriving the Hall current-current correlator we extended (specifically for the GW +EDMFT implementation) the approach described in Ref. 3 to the multiorbital case, as explained in the SM. The Hall coefficient, defined as

Discussion
To address the physics of doped NdNiO 2 in the normal state, and in particular the self-doping effect, we extended the recently developed GW +EDMFT approach to multi-site systems. This method has the significant advantage of being free from ill-defined double countings or arbitrary choices of interaction parameters. It captures the dynamical screening due to long ranged and local charge fluctuations and self-consistently computes the local and nonlocal interactions appropriate for the low-energy model. Our results allow us to assert with confidence that undoped and hole doped NdNiO 2 represent genuine multi-orbital systems. Orbitals that, within a DFT description, are expected to be fully occupied and es-sentially inert, are lifted closer to the Fermi level by the interactions. As a consequence, also other orbitals than the naively expected Ni-3d x 2 −y 2 become involved in low-energy charge fluctuations, and in the formation of high-spin states. We find prominent contributions from

Methods
Our simulations are based on the parameter-free multi-tier GW +DMFT scheme, [17][18][19] which treats different energy scales with appropriate levels of accuracy, and which self- Within the low-energy space with two Nd and five Ni orbitals, we perform a self-consistent GW +EDMFT calculation. In this approach local self-energiesΣ imp and polarizationsΠ imp for Nd and Ni are computed from separate EDMFT impurity problems with self-consistently optimized fermionic and bosonic Weiss fields G and U, respectively. To these local impurity self-energies and polarizations, we add the nonlocal GW components, where W acbd denotes the elements of the screened interaction, and G ab those of the interacting Green's function. The GW +EDMFT cycle starts from an initial guess forΣ imp andΠ imp .
Then, given the noninteracting lattice HamiltonianĤ(k) for the localized Wannier orbitals of the low-energy space, non-local GW self-energies and polarizations are computed. The sum of these two contributions yields the momentum-dependent self-energy Σ k and polarization Π q entering the lattice sums,Ĝ whereĜ (0) k is the noninteracting propagator of the low energy subspace, which incorporates also the G 0 W 0 contribution (see Eq. (12) below), andÛ q the "bare" Coulomb interaction resulting from the initial G 0 W 0 downfolding. These local Green's functions and screened interactions are obtained by inversion in the full (7×7) orbital space. The EDMFT selfconsistency condition then demands that the projections ofĜ loc andŴ loc onto the Ni and Nd sites are equal to the impurity Green's functions and screened interactions for these sites.
Using these G imp and W imp and the impurity self-energies and polarizations, the fermionic and bosonic Weiss fields are computed aŝ and used as inputs for the two EDMFT impurity problems.
Since all the self-energy and polarization contributions in this multi-tier scheme are diagrammatically defined, we can connect the different subspaces in a consistent way, without any double countings. The explicit expressions for the interacting lattice Green's functions and screened interactions are: where C refers to the strongly correlated local subspaces (treated with EDMFT) and I to the full 7-orbital subspace in which the GW calculation is performed. ∆V H is the Hartree contribution to the self-energy, v q is the bare interaction and V XC the exchange-correlation potential (which is replaced by the G 0 W 0 self-energy). A detailed derivation of the multi-tier GW+EDMFT approach can be found in Refs. 18 and 19. 27 This is not in general valid for realistic dispersions but, considering that the GW vertex function is a delta function and the EDMFT contribution is local, we consider this approximation accurate enough.
The optical conductivity σ yy has been computed for several real materials using DFT+DMFT implementations. [6][7][8][9] This is possible thanks to simplifications that occur in the limit , and then perform the analytical continuation to the real frequency axis using the maximum entropy method. 1 In Fig. 8 we report the site-resolved conductivities for the optimally doped and overdoped systems. The inter-site Ni-Nd contributions turn out to be negligible compared to the intra-site contributions. It is also interesting to notice how the feature located at ∼1.8eV in the undoped system vanishes in favour of a more prominent Drude peak. This is a clear indication that the Nickel becomes less correlated and more metallic if the doping is increased and, as noted in the main text, represents a fingerprint of the multi-orbital nature of the system.  While the correlator which defines the optical conductivity describes a second-order process, the Hall conductivity σ yxz (ω) stems from a third order process. Strictly following the notation of Ref. 3, we report in Fig. 9 the non-vanishing diagrams linear both in the electric A Ex n and magnetic A Bz q field components of the total vector potential, where k ± ≡ k ± q 2 . The sum of the reported diagrams vanishes at q = 0 and an expansion up to the linear order in q of both the vertex current v α,k and the Green's function G k ± q 2 , iω n is needed. In standard DMFT calculations the locality of the self-energy implies that G k ± q 2 , iω n = G (k, iω n ) ∓ q 2 v k G 2 (k, iω n ). This allows for the compact notation used in Refs. 4 and 5, but is not a viable option in our case, where the k-dependence of the Green's function does not come solely from the dispersion. Following the procedure outlined for the optical conductivity, we start with the evaluation of the three correlators in imaginary time at small wavevector: where we sum over all the possible orbitals affected by the insertion at τ 1 . Then we nu- merically differentiate this correlator with respect to δq. The results for the undoped and optimally doped compound are reported in Fig. 10. We then Fourier transform to Matsubara frequency and perform the analytic continuation to real frequency via the Padé algorithm. 2 We checked that the asymptotic behaviour of both the real and imaginary parts of the resulting Hall conductivity are compatible with the results derived in Ref. 4. Since analytic continuation algorithms might produce artefacts, to obtain σ yxz (0), we assume that Π yxz ααββ (iν n ) is purely imaginary and vanishes at ν n = 0 so that 2 e 3 σ yxz (0) = d Π (ν n ) dν n 0 , which allows for a more precise estimate of the Hall coefficient, as plotted in Fig. 5d of the main text.
character for each band. In the LDA case this is directly evaluated, while at the G 0 W 0 and GW +EDMFT level they are compute by analytical continuation at each k point. Figure 11. Nickel spectral functions for the undoped setup.     Interacting Fermi surfaces. In Fig. 18 we report the Fermi surfaces for the Ni-3d z 2 and Ni-3d x 2 −y 2 with the same arrangement as in the main text. The color intensities are normalized to the maximum value for each orbital, and the different k z panels (rows of the plot ) use the same normalization. We notice that at k z = 0.5 the Ni-3d