# The doping-dependent character and possible magnetic ordering of NdNiO2

Frank Lechermann  European XFEL, Holzkoppel 4, 22869 Schenefeld, Germany

###### Abstract

The novel nickelate superconductors of infinite-layer type feature challenging
electronic pecularities in the normal-state phase diagram with doping.
Distinct many-body behavior and different dispersion regimes of the entangled
$\\{$Ni-$d_{z^{2}}$, Ni-$d_{x^{2}-y^{2}}$$\\}$ orbital sector give rise to
highly rich physics, which is here studied for the case of the NdNiO2 system.
An analysis based on advanced realistic dynamical mean-field theory unveils
that the superconducting hole-doped region is the meeting place of a
(self-)doped Mott insulator from the underdoped side, and a bad Hund metal
from the overdoped side. Fermi-level crossing of the Ni-$d_{z^{2}}$ flat-band
ties both regimes together to form a singular arena for unconventional
superconductivity. We furthermore shed light on the intriguing problem of
elusive magnetism in infinite-layer nickelates. Antiferromagnetic (AFM) order
with small Ni moments is shown to be a vital competitor at low temperature. At
stoichiometry, C-AFM order with ferromagnetic spin-alignment along the
$c$-axis benefits from a conceivable coexistence with Kondo(-lattice)
screening.

##  I Introduction

Nickelate superconductivity in thin films of Sr-doped NdNiO2 below a critical
temperature $T_{\rm c}\sim 10$ K entered the scene in summer 2019 Li  _et al._
(2019). Recently, the same physics has also been identified in Sr-doped PrNiO2
thin films Osada  _et al._ (2020a, b). Because of the sophisticated sample
preparation and the so far lack of bulk single crystals, experimental progress
on these Ni($3d^{9}$)-based infinite-layer systems is challenging.

Several findings still enable further insight into a very rich phyenomenology.
Ordered magnetism is hard to measure in the thin-film geometry, but neutron
scattering on polycrystalline bulk samples provides no clear evidence for
long-range antiferromagnetic (AFM) order in NdNiO2 Hayward and Rosseinsky
(2003); Wang  _et al._ (2020a). However, recent nuclear-magnetic-resonance
(NMR) experiments Cui  _et al._ (2020) on similar samples suggest quasistatic
AFM ordering for $T<40$ K. This, together with a resistivity upturn Li  _et
al._ (2019) below $T\sim 70$ K, points to intriguing low-energy physics even
far from the superconducting region. The latter is sandwiched between weakly-
insulating doping regions, in the range $0.125\lesssim x\lesssim 0.25$ of the
the phase diagram for hole doping $x$ of Sr kind Li  _et al._ (2020); Zeng
_et al._ (2020). Doping-dependent Hall data Li  _et al._ (2020); Zeng  _et
al._ (2020) point to a two-band scenario with $x$. First spectroscopic results
reveal hybridization of Ni$(3d)$ states with other delocalized states and
reduced Ni-O intermixing compared to NiO and LaNiO3 Hepting  _et al._ (2020).
Furthermore for finite $x$, substantial Ni$(3d^{8})$ character is unveiled
Goodge  _et al._ (2020); Rossi  _et al._ (2020), in contrast to the dominant
Cu$(3d^{9}\underline{L})$, i.e. Cu$(3d^{9})$ with ligand hole on oxygen,
weight in high-$T_{\rm c}$ cuprates. This Ni$(3d^{8})$ signature is either
attributed to multi-orbital physics Goodge  _et al._ (2020) or to Ni($S=0$)
single-orbital physics of $d_{x^{2}-y^{2}}$ kind Rossi  _et al._ (2020).
Concerning the superconducting state, a small anisotropy in the upper critical
field and hints to a Pauli-limited coupling, as well as to a two-symmetry gap
nature are reported Xiang  _et al._ (2020); Wang  _et al._ (2020b); Gu  _et
al._ (2020a).

From the theoretical side, assessments based on density functional theory
(DFT) highlight the self-doping (SD) nature of an otherwise seemingly cuprate-
like band structure Wu  _et al._ (2020); Nomura  _et al._ (2019); Botana and
Norman (2020); Lechermann (2020a, b). The weakly-filled SD band originates
from Nd$(5d)$-Ni$(3d)$ hybridization and causes a multi-sheeted Fermi surface
at stoichiometry. There seems also agreement that the charge-transfer
character of NdNiO2 is reduced compared to estimates in akin cuprates Jiang
_et al._ (2020); Lechermann (2020a). Further beyond DFT, two main theory
concepts are pursued: either (decorated) single-Ni-orbital physics of
Ni-$d_{x^{2}-y^{2}}$ type Wu  _et al._ (2020); Nomura  _et al._ (2019); Botana
and Norman (2020); Zhang  _et al._ (2020a); Si  _et al._ (2020); Kitatani  _et
al._ (2020); Karp  _et al._ (2020); Adhikary  _et al._ (2020); Lang  _et al._
(2020), or multiorbital Ni($3d)$ processes Lee and Pickett (2004); Hu and Wu
(2019); Zhang and Vishwanath (2020); Werner and Hoshino (2020); Lechermann
(2020a, b); Chang  _et al._ (2020); Choi  _et al._ (2020a); Petocchi  _et al._
(2020); Wang  _et al._ (2020c); Kang  _et al._ (2020) are assumed key to the
correlation phenomenology. In two previous works Lechermann (2020a, b), we
showed that based on calculations utilizing the combination of DFT with
dynamical mean-field theory (DMFT) and furthermore including explicit Coulomb
interactions on oxygen, the relevance of Ni-multiorbital degrees of freedom is
indeed inevitable. Besides the indisputable Ni-$d_{x^{2}-y^{2}}$ orbital and
its related low-energy band, the Ni-$d_{z^{2}}$ orbital plays a further
crucial role. First, in mediating a Kondo(-lattice) coupling between Ni$(3d)$
and Nd$(5d)$ at stoichiometry. And second, for establishing a unique flat-band
scenario in the doping region of low-$T$ superconductivity.

In the present work, the goal is to extent our previous studies and shed
further light onto the Ni-multiorbital physics in infinite-layer nickelates.
In Refs. Lechermann, 2020a, b, paramagnetism has been assumed and doping has
been addressed by a supercell description Lechermann (2020a) as well as within
a minimal three-band model solved for a simplified Fermi-liquid setting
Lechermann (2020b). Here, we provide a detailed account of the doping-
dependent features of the correlated electronic strucuture based on the
virtual-crystal approximation, and additionally investigate the intriguing
problem of magnetic ordering in the NdNiO2 system.

##  II Methodology

Figure 1: (color online) Projected local orbitals of Ni-$d_{z^{2}}$ (left) and
Ni-$d_{x^{2}-y^{2}}$ (right) type in the NdNiO2 compound: Ni(grey),
Nd(large,green), O(small,red).

The charge self-consistent combination Savrasov  _et al._ (2001); Grieger  _et
al._ (2012) of DFT and DMFT with the additional incorporation of the self-
interaction correction (SIC) is used to investigate the normal-state
properties of pristine and doped neodynium nickelate in the infinite-layer
structure. The realistic part of this DFT+sicDMFT scheme Lechermann  _et al._
(2019) builds up on a mixed-basis pseudopotential representation Elsässer  _et
al._ (1990); Lechermann  _et al._ (2002); Meyer  _et al._ (1998) in the local
density approximation (LDA). Coulomb interactions on oxygen are addressed
beyond DFT within SIC applied to the O pseudopotential Körner and Elsässer
(2010). While the O$(2s)$ orbital is fully corrected with weight factor
$w_{p}=1.0$, the reasonable choice Körner and Elsässer (2010); Lechermann  _et
al._ (2019) $w_{p}=0.8$ is used for the O$(2p)$ orbitals.

The Nd$(4f)$ states are put in the pseudopotential frozen core, since they are
not decisive for the key physics of infinite-layer nickelates Zhang  _et al._
(2020a). This also means, that $f$-electron based magnetism is not considered
in this work. As in many other comparable compounds, due to the local nature
of the $4f$ moments and their weak exchange coupling, magnetic ordering of Nd
moments is believed to occur at very low temperatures. Note furthermore that
the $j$-resolved frozen occupation of the Nd$(4f)$ shell is here chosen with
small resulting moment. The role of spin-orbit effects in the overall crystal
calculations is neglected. Additional details for the computation of the
magnetically ordered phases in this work will be given in section IV.

To facilitate the description at finite doping, we here employ the virtual-
crystal approximation (VCA) in the fully charge self-consistent scheme. This
amounts to replace the Nd atom by a pseudo atom of nuclear (and electronic)
charge ${\cal Z}=Z_{\rm Nd}-\delta$ in the pristine unit cell. As a choice,
the charge content of the Nd$(4f)$ shell is here not modified by this
construction. It is then assumed that $\delta=x$ holds, i.e. the VCA doping
$\delta$ mimics the true Sr doping $x$ of effectively replacing Nd3+ by Sr2+.
As any theoretical doping description, also the VCA misses certain effects,
e.g. local-structural distortions or correct local charge states of the
impurity. But it usually provides a faithful description of the overall
effects of charge doping. Note that in the previous works Refs. Lechermann,
2020a, b, the other two standard options to describe doping, namely supercell
Lechermann (2020a) and chemical-potential shift Lechermann (2020a) have been
put into practise. Thus the present perspective completes the conventional
method spectrum in that regard.

The DMFT correlated subspace is governed by a full Slater Hamiltonian applied
to the Ni$(3d)$ projected-local orbitals Amadon  _et al._ (2008). The
projection is performed on the $6+5+1=12$ Kohn-Sham states above the dominant
O$(2s)$ bands, associated with O$(2p)$, Ni$(3d)$ and the self-doping band. A
Hubbard $U=10$ eV and a Hund exchange $J_{\rm H}=1$ eV prove reasonable for
this choice of the energy window in the given nickelate Lechermann (2020a, b);
Lechermann  _et al._ (2019). Continuous-time quantum Monte Carlo in
hybridzation expansion Werner  _et al._ (2006) as implemented in the TRIQS
code Parcollet  _et al._ (2015); Seth  _et al._ (2016) is utilized to handle
the DMFT problem. The fully-localized-limit double-counting scheme Anisimov
_et al._ (1993) is applied. Maximum-entropy and Padé methods are employed for
the analytical continuation from Matsubara space onto the real-frequency axis.
Stoichiometric lattice parameters are taken from experiment Li  _et al._
(2019), reading $a=3.92$ Å and $c=3.31$ Å. The system temperature is set to
$T=30$ K in order to properly approach the competition/cooperation between
quasiparticle (QP) formation, Mott criticality, Kondo physics and magnetic
order in these challenging nickelates. If not otherwise stated, all shown data
is obtained within charge-selfconsistent DFT+sicDMFT.

The inclusion of explicit oxygen-based correlations beyond conventional
DFT+DMFT for the NdNiO2 system is believed necessary for mainly two reasons.
First generally, in nickelates as late transition-metal (TM) oxides the
charge-transfer aspect is stronger as in early TM oxides, and therefore
Coulomb interactions from O$(2p)$ will have a larger impact Zaanen  _et al._
(1985); Imada  _et al._ (1998); Han  _et al._ (2011). Second, while
Ni-$d_{x^{2}-y^{2}}$ is strongly hybridized with O$(2p)$ in the infinite-layer
structure, Ni-$d_{z^{2}}$ is less so since there is no apical oxygen (see Fig.
1). Thus especially the Ni-$e_{g}$ $\\{z^{2},x^{2}-y^{2}\\}$ orbitals of the
DMFT-active Ni$(3d)$ shell are quite differently affected by O$(2p)$ and
therefore it should be insufficient to treat the Coulomb interactions from
oxygen only implicitly via a chemical-potential or Coulomb-on-Ni shift. In the
Appendix A, we provide a brief comparison between the DFT+sicDMFT and DFT+DMFT
picture of pristine and doped NdNiO2.

##  III Paramagnetic system

###  III.1 Stoichiometric compound

A basic characterization of the correlated electronic structure of
paramagnetic (PM) NdNiO2 has been given in Ref. Lechermann, 2020a for ambient
temperatures, and in Ref. Lechermann, 2020b for the lower-temperature regime.
To set the stage, we here recapitulate the main findings and discuss the
different site/orbital contributions within the correlated regime in some more
detail.

Figure 2: (color online) Spectral information for pristine NdNiO2. (a) ${\bf
k}$-resolved function $A({\bf k},\omega)$ along high-symmetry lines:
$\Gamma$-X-M describes a triangle in the $k_{z}=0$ plane and Z-R-A marks the
same triangle translated to the $k_{z}=1/2$ plane. Above the Fermi level, only
the single SD dispersion is shown and further dispersions discarded in the
plot. (b-f) ${\bf k}$-resolved orbital weights (i.e. fatbands) for the given
spectral function: (b) Ni-$d_{z^{2}}$, (c) Ni-$d_{x^{2}-y^{2}}$, (d)
Ni-$t_{2g}$, (e) Nd-$d_{z^{2}}$ and (f) Nd-$d_{xy}$. Note that with this
chosen intensity resolution, the nondispersing-level feature at
$\varepsilon_{\rm F}^{\hfill}$, giving rise to a Kondo(-lattice) mechanism
Lechermann (2020b) at lower $T$, is not visible in $A({\bf k},\omega)$. Figure
3: (color online) Effective Ni$(3d)$ occupation in doped NdNiO2 from
introducing $\delta$ holes/electrons. (a) Difference $n_{d,0}-n_{d}$ between
the total occupation at $n_{d,0}$ zero doping and at given doping $n_{d}$,
versus $\delta$. The dashed red line marks the regime where the complete
doping charge is transfered to Ni$(3d)$. (b) Orbital-resolved filling of the
Ni-$e_{g}$ states.

On the DFT(LDA) level, the $x^{2}$$-$$y^{2}$ orbital has the lowest crystal-
field level within the Ni$(3d)$ sector and the energy splittings to the other
levels read
$\\{\varepsilon_{z^{2}},\varepsilon_{x^{2}-y^{2}},\varepsilon_{xz},\varepsilon_{yz},\varepsilon_{xy}\\}=\\{356,0,286,286,536\\}$
meV. Then in essence, the compound is a self-doped Mott insulator in
DFT+sicDMFT, whereby the effectively Mott-insulating states are of half-filled
Ni-$d_{x^{2}-y^{2}}$ kind and the self doping originates from a weakly-
occupied band resulting from the hybridization between Nd$(5d)$ and Ni$(3d)$.
Thus there is no Ni-$d_{x^{2}-y^{2}}$ quasiparticle dispersion crossing the
Fermi level $\varepsilon_{\rm F}^{\hfill}$ in the ${\bf k}$-resolved spectral
function, as shown in Fig. 2a. At low energy, there are only SD-band based
electron pockets at $\Gamma$ and A in the Brillouin zone.

Figure 4: (color online) Spectral function $A({\bf k},\omega)$ with doping
$\delta$. Top row, from left to right: $\delta=-0.30,-0.15,-0.05,0$. Bottom
row, from left to right: $\delta=0.05,0.15,0.30,0.45$. Same setting as in Fig.
2a.

Correspondingly, orbital weights along the given high-symmetry lines are
provided in Fig. 2b-f for the most-relevant orbital sectors. The
Ni-$d_{z^{2}}$ contribution is twofold. First, it forms an incoherent part
around -3 eV for $k_{z}=0$, which becomes more coherent, flat and intense in
the $k_{z}=1/2$ plane. Second, it has its prominent share in the SD-band
formation for the $\Gamma$-pocket part. On the other hand due to Mottness, the
Ni-$d_{x^{2}-y^{2}}$ weight remains largely incoherent troughout the chosen
${\bf k}$-space regions, with the exception of a small part along Z-R in the
regime of the overall Ni-$d_{z^{2}}$-dominated spectrum. The Ni-$t_{2g}$
$\\{xz,yz,xy\\}$ orbitals cause dispersing features between [-3,-2] eV and
have weak contribution to the SD band around A. The Nd$(5d)$-based orbitals of
$z^{2}$ and $xy$ character are the main contributors to the SD band from the
rare-earth site; Nd-$d_{xy}$ for the A pocket and Nd-$d_{z^{2}}$ pretty much
everywhere else. The hybridization between Nd-$d_{xy}$ and Ni-$t_{2g}$ is also
evident from the joint weight at deeper energies. The coupling between
Ni-$d_{z^{2}}$ and Nd-$d_{z^{2}}$ is mostly confined to the $\Gamma$-X-M-Z
path. Mott-insulating Ni-$d_{x^{2}-y^{2}}$ appears disconnected from the
Nd$(5d)$-Ni$(3d)$ hybridization, and indeed nearest-neighbor hopping from both
relevant Nd$(5d)$ orbitals to the in-plane Ni-$e_{g}$ orbital is zero Nomura
_et al._ (2019); Lechermann (2020a, b). The O$(2p)$ contribution is surely
also vital in the given energy window, but a ${{\bf k}}$-resolved discussion
appears not highly instructive (see Ref. Lechermann, 2020a for the ${\bf
k}$-integrated weight).

With high spectral-resolution focus on the low-energy region, a single-level
feature at the Fermi level would become additionally visible for the here
chosen lower temperature regime Lechermann (2020b). It is stemming from
Ni-$d_{x^{2}-y^{2}}$ and is part of an intriguing Kondo(-lattice) mechanism,
involving furthermore Ni-$d_{z^{2}}$ and Nd-$d_{z^{2}}$. Basically, the
Ni-$d_{z^{2}}$ orbital mediates a Hund-assisted Kondo coupling between the $d$
sectors of Ni and Nd, by enabling fluctuations between Ni-$e_{g}$ and weakly-
correlated Nd-$d_{z^{2}}$.

###  III.2 Doped compound

Superconductivity in the nickelate thin films is reached by Sr doping Li  _et
al._ (2019), expected to introduce holes to the system. For completeness we
here discuss both, electron and hole doping within the VCA framework. From a
preparation perspective, electron doping, e.g. by substituting Nd3+ by Ce4+,
should be more challenging Hirsch and Marsiglio (2019): the already delicate
Ni+ ionic state is reduced even further and the charge imbalance within the
NiO2 plane becomes additionally stretched. Indeed, the DFT+sicDMFT
calculations show that the progress of the doping level $\delta$ toward the
Ni$(3d)$ shell is strongly hindered on the electron-doped side (see Fig. 3a).
On the other hand on the hole-doped side, the doping originating on the rare-
earth site is nearly exclusively transfered to the Ni site. This furthermore
underlines the weaker charge-transfer character compared to cuprates, where
most holes reside on oxygen.

Figure 5: (color online) Imaginary part ${\rm Im}\,\Sigma(i\omega_{n})$ of the
Ni-$e_{g}$ self energy on the Matsubara axis $\omega_{n}=(2n+1)\pi T$ for
different dopings $\delta$. Top: Ni-$d_{z^{2}}$ and bottom:
Ni-$d_{x^{2}-y^{2}}$. Left part: $\delta=-0.30,-0.15,-0.05,0$ and right part:
$\delta=0.05,0.15,0.30,0.45$.

The orbital-resolved filling of the Ni-$e_{g}$ states with doping $\delta$ in
Fig. 3b exhibits the strong hole attraction of Ni-$d_{z^{2}}$ and its weaker
additional electron filling on the electron-doped side. On the contrary, the
Ni-$d_{x^{2}-y^{2}}$ filling is only weakly affected by the charge doping of
the system and remains nearly inert. All these observations are in line with
previous findings in a minimal-Hamiltonian description Lechermann (2020b).
Note that the Ni-$t_{2g}$ states are always close to complete filling
throughout the doping landscape and seemingly do not play a key role for the
phase diagram with $\delta$.

Figure 6: (color online) Spectral information of hole-doped NdNiO2 with
$\delta=0.15$. (a-f) As in Figs. 2a-f. (g) Constant-energy surfaces in the
$k_{z}=1/2$ plane. Note that the maximum intensity for the surfaces in (g) is
reduced by a factor of three compared to the one for $A({\bf k},\omega)$ in
(a).

We now turn to the ${{\bf k}}$-resolved spectral evolution with doping,
displayed in Fig. 4. Electron doping mainly leads to an expected downshift of
the overall spectrum, resulting in an increased filling of the SD band. The
Mott-insulating character of the Ni-$d_{x^{2}-y^{2}}$ orbital remains
apparently robust, though the corresponding self-energy is considerably
weakened with electron doping (cf. lower-left part of Fig. 5). Note however
that this weakening proceeds even stronger with hole doping (cf. lower-right
part of Fig. 5), as observed already in the minimal-Hamiltonian picture
Lechermann (2020b). It leads to a partial melting of the
Ni-$d_{x^{2}-y^{2}}$-based Mott state for $\delta\geq 0.05$. Moreover, the SD
electron pockets are shifted into the unoccupied region by sizable amount
already for $\delta=0.05$. In line with this, the dominant Ni-$d_{z^{2}}$
spectral part in the occupied region gets shifted progressively toward the
Fermi level. The spectral evolution of the originally occupied Ni-$d_{z^{2}}$
part is in fact one striking feature with hole doping. Its flat-band part in
the $k_{z}=1/2$ plane crosses $\varepsilon_{\rm F}^{\hfill}$ here for
$\delta\geq 0.15$.

Let us therefore pause with the further evolution to higher hole dopings for a
moment and focus on the spectrum at $\delta=0.15$ (cf. Fig. 6). The
Ni-$d_{z^{2}}$ flat-band part is touching the Fermi level Lechermann (2020b);
Petocchi  _et al._ (2020) and the strength of the Ni-$d_{x^{2}-y^{2}}$ Mott
state is weakened, giving rise to minor dispersive spectral weight at low
energy. The latter is strongest close to the R point, where a van-Hove
singularity settles. The occupied Ni-$t_{2g}$, Nd-$d_{z^{2}}$ and Nd-$d_{xy}$
weights appear generally strenghthened compared to the stoichiometric case.
They are also shifted to lower energy, but do not participate on the singular
fermiology. The actual Fermi surface with obvious Ni-$d_{z^{2}}$ weight at the
A point is shown on the left of Fig. 6g. Because of the flat-band character,
the topology of the constant-energy surface changes quite significantly within
a $\sim 10$ meV range below $\varepsilon_{\rm F}^{\hfill}$. Going down in
energy from $\omega=0$, a hole surface around A transforms into an electron
surface around Z. The substantial hole character just below the Fermi level is
in line with experimental findings of hole-like transport in that doping
region Li  _et al._ (2020); Zeng  _et al._ (2020). But note that the “actual”
Fermi surface for a given $0.125\lesssim\delta\lesssim 0.25$ in the region
with low-$T$ superconductivity is still very delicate, and perfect agreement
between experiment and theory will be hard to reach. Nonetheless, we predict
that the experimental $\delta=0.15$ Fermi surface has the appearance of one of
the three $\omega=[0,-0.045]$ eV plots of Fig. 6g.

The Ni-$e_{g}$ self-energies have a rather intriguing development with hole
doping (cf. right part of Fig. 5). For Ni-$d_{z^{2}}$, the self-energy part
${\rm Im}\,\Sigma(i\omega_{n})$ grows with $\delta$, and the low-frequency
curvature increasingly departs from standard Fermi-liquid behavior. Fitting to
the function $F=A\,\omega_{n}^{\alpha}$, the values $\alpha_{\delta}$ read
$\\{\alpha_{0.05},\alpha_{0.15},\alpha_{0.40}\\}=\\{0.89,0.74,0.60\\}$, while
$\alpha=1$ holds in a Fermi liquid. Furthermore for $\delta=0.45$, the self-
energy attains a negative curvature, signalling strong bad-metal/insulating
tendencies.

Figure 7: (color online) Total spectral function after k-integration (top) and
local Ni-$e_{g}$ spectrum (bottom) of pristine and hole-doped NdNiO2 at low
energy.

A clearly larger Ni-$d_{x^{2}-y^{2}}$ self-energy shows nearly zero curvature
at $\delta=0.15$, i.e. just when the Ni-$d_{z^{2}}$ flat band starts to cross
$\varepsilon_{\rm F}^{\hfill}$. Increasing hole doping further, first
stabilizes a putative Fermi-liquid character with a low scattering rate
($\propto{\rm Im}\,\Sigma(0)$), but then jumps back to a high-scattering Mott-
critical regime at $\delta=0.45$. What the data for sure shows is that both
Ni-$e_{g}$ orbitals strongly affect each other in the hole-doped regime of
NdNiO2.

The development with hole doping can also be inspected from plotting the ${\bf
k}$-integrated total spectrum and the local Ni-$e_{g}$ spectrum at low energy
in Fig. 7. The total spectrum around $\varepsilon_{\rm F}^{\hfill}$ grows
significantly for $\delta\geq 0.15$, but weakens again for $\delta=0.45$. The
low-energy Ni-$e_{g}$ weight is small at stoichiometry; Ni-$d_{x^{2}-y^{2}}$
is essentially Mott insulating and Ni-$d_{z^{2}}$ only weakly contributing
based on its hybridization on the SD electron pockets. Those are shifted above
the Fermi level for small hole doping, and hence only Ni-$d_{x^{2}-y^{2}}$
weight from a pre-melted Mott state appears close to $\varepsilon_{\rm
F}^{\hfill}$. Once the flat band shifts into the low-energy region, the
corresponding Ni-$d_{z^{2}}$ weight starts to dominate the local spectrum. But
it develops a pseudogap structure due to specific dispersion topology of a
fully occupied(empty) flat branch below(above) the Fermi level. Note that the
increased QP character of Ni-$d_{x^{2}-y^{2}}$ between entry and departure of
the Ni-$d_{z^{2}}$ flat-band part from the Fermi level becomes here also
visible. As already derived from the self-energies, the high-doping region of
the hole-doped side does not resemble a good metal. But its semimetallic/bad-
metallic appearance is indeed in agreement with the weakly-insulating behavior
found in experiment Li  _et al._ (2020); Zeng  _et al._ (2020).

The filling scenario and ${\bf k}$-space topology behind the Ni-$d_{z^{2}}$
pseudogap structure for hole dopings beyond the superconducting region is one
reason for weak metallicity, especially for in-plane transport, as already
pointed out in Ref. Lechermann, 2020b. However in addition, the non-Fermi-
liquid and heavily bad-metallic character of the Ni-$d_{z^{2}}$-dominated
states, associated with the pseudogap structure and a specific Hund-metal
scenario of Ni-$e_{g}$, is a second vital reason.

magnetic order |  $m_{\rm Ni}\,(\mu_{\rm B})$ , $E_{\rm tot}^{\rm AFM}-E_{\rm tot}^{\rm PM}$ (meV/atom)
---|---
| $\delta=0$ | $\delta=0.15$
G-AFM | 0.076 , 105 | 0.059 , 126
C-AFM | 0.081 , 20 | 0.000 , $-$
Table 1: Ordered Ni magnetic moment in the G-AFM and C-AFM phase at $T=30$ K
and energy difference to the PM phase.

Let us finally note, that the present behavior with hole doping does not seem
perfectly commensurable with the experimental phase diagram. There, the
superconducting region stretches from $0.125\lesssim\delta\lesssim 0.25$,
whereas the prominent role of Ni-$d_{z^{2}}$ seemingly peaks here at
$\delta=0.30$. There may be several reasons for this difference. First, due to
our sparse doping sampling, a ’true’ peaking Ni-$d_{z^{2}}$ behavior within
the experimental superconducting doping range is still possible, as for
$\delta=0.30$ there is already the pseudogap onset. Second, we only assume
$\delta=x$ to hold, but since the present VCA approach has its limitations, a
small shift of behavior would not be surprising. Moreover throughout the
study, the stoichiometric lattice parameters are employed, and e.g. the
$c$-axis increase with hole doping Li  _et al._ (2019) is not included in the
calculations. And third, albeit the Ni-$d_{z^{2}}$ flat band neighorhood to
the Fermi level is believed to be the key driving force for superconductivity,
its $\varepsilon_{\rm F}^{\hfill}$ entry and departure points do not
necessarily have to agree exactly with the phase boundaries for
superconductivity.

Figure 8: (color online) Comparing PM and AFM-ordered NdNiO2. (a) G-AFM
ordering and (b) C-AFM ordering. (c) Total (top) and local Ni-$e_{g}$ (bottom)
spectral function for PM, G-AFM and C-AFM phase, respectively. (d-f) ${\bf
k}$-resolved spectral function along high symmetry lines for G-AFM (d), C-AFM
(e) and PM (f) phase. The $\sqrt{2}\times\sqrt{2}$ enlarged in-plane unit cell
translates into X$\leftrightarrow$M and R$\leftrightarrow$A for the (G,C)-AFM
structure. And the upper Ni-$d_{z^{2}}$ flat-band part is flipped with respect
to $k_{z}$ for the G-AFM structure, due to the doubled unit cell along $c$.

##  IV Antiferromagnetic system

A key feature of structurally akin high-$T_{\rm c}$ cuprates is the prominent
AFM phase at and near stoichiometry Keimer  _et al._ (1992); Dagotto (1994).
So far, static magnetic order in infinite-layer nickelates has been elusive in
experiment, but the investigations point to a low-temperature behavior that
deviates from straightforward paramagnetism Hayward and Rosseinsky (2003);
Wang  _et al._ (2020a); Cui  _et al._ (2020).

From theory, there are various studies Lee and Pickett (2004); Hepting  _et
al._ (2020); Botana and Norman (2020); Choi  _et al._ (2020b); Zhang  _et al._
(2020b); Gu  _et al._ (2020b); Liu  _et al._ (2020); Leonov  _et al._ (2020);
Wan  _et al._ (2020); Zhang  _et al._ (2020c); Kapeghian and Botana (2020);
Choi  _et al._ (2020a) that address ordered magnetism in these systems.
Besides more unlikely ferromagnetic (FM) order, the canonical orderings
examined are G-type and C-type antiferromagnetism. While the G-AFM order
displays antiparallel alignment between in-plane and $c$-axis nearest-neighbor
(NN) Ni spins, the C-AFM order deviates therefrom in showing ferro-alignment
along the $c$-axis (see Figs. 8a,b). Most results from DFT(+U) and model-
Hamiltonian DMFT calculations reveal a large ordered magnetic moment of
$m_{\rm Ni}\sim 1$ $\mu_{\rm B}$ on the Ni site for these patterns. At
stoichiometry, the C-AFM state appears energetically favorable against the
nonmagnetic state and against FM order within DFT+U Choi  _et al._ (2020a) and
additionally against G-AFM when utilizing the SCAN (i.e. meta generalized-
gradient approximation) functional in DFT Zhang  _et al._ (2020c). However
note that electronic correlations are only treated on a static level in these
approaches (e.g. there is no true PM state straightforwardly reachable), and
furthermore they formally describe the system at $T=0$. Nearest-neighbor in-
plane exchange constants between Ni spins are estimated 65 meV Zhang  _et al._
(2020c) and 82 meV Wan  _et al._ (2020), i.e. roughly half the value usually
revealed for high-$T_{\rm c}$ cuprates.

Here we report DFT+sicDMFT calculations for possible G-AFM and C-AFM ordering
at $T=30$ K. Two technical aspects have to be mentioned. First, the possible
additional ordering of the Nd spins is not considered in this investigation.
This is not unreasonable, as in most rare-earth compounds the corresponding
$4f$-based spins order at much lower temperature. Second, in a hybrid scheme
of DFT and DMFT, there are in principle two basic formalisms to handle spin
polarization: either it is treated both in the DFT and the DMFT part, or only
on the DMFT level. The former approach includes ligand-based exchange
mechanisms more directly, but the $T$-independent exchange splitting from the
DFT part usually ruins a reliable assessment of temperature effects. In other
words, especially for transition-metal oxides the robustness of the ordered
magnetic moments with temperature is most often severely overestimated.
Therefore, spin polarization is here treated only on the DMFT level, i.e. the
Ni$(3d)$ self-energies are solely active in creating Ni ordered moments in the
charge self-consistent DFT+sicDMFT computations.

With this in mind, let us discuss the results by first stating that the
calculations are quite demanding. For instance, we did so far not succeed to
stabilize a magnetically ordered phase in electron-doped NdNiO2. At
stoichiometry and with hole doping where this proves possible, the Ni magnetic
moments turn out generally very small, with values $m_{\rm Ni}\lesssim 0.1$
$\mu_{\rm B}$ (see Tab. 1). But importantly, the differentiation between an
actual ’zero moment’ and such a small one is always very manageable from the
present degree of convergence and accuracy. Though surprising in view of the
Mott-insulating Ni-$d_{x^{2}-y^{2}}$ orbital with formally sizable local
$S=1/2$ spin at stoichiometry, Hayward and coworkers Hayward and Rosseinsky
(2003); Hayward  _et al._ (1999) noted from neutron scattering on
polycrystalline bulk samples strong deviation from conventional $S=1/2$ local-
moment behavior of Curie-Weiss form. In Ref. Hayward and Rosseinsky, 2003, the
data analysis is performed by assuming G-AFM order with a sensitivity limit of
0.06 $\mu_{\rm B}$/Ni.

Figure 9: (color online) Comparison of PM and G-AFM low-energy spectral
functions at doping level $\delta=0.15$. (Top) Total spectrum and (bottom)
local Ni-$e_{g}$ spectrum.

In the present investigation, supercells of $\sqrt{2}\times\sqrt{2}\times 1$
kind for the C-AFM and of $\sqrt{2}\times\sqrt{2}\times 2$ for the G-AFM
structure are utilized. At stoichiometry, the PM phase has the lowest total
energy for the given temperature. But the energy difference $\Delta E=20$
meV/atom to the C-AFM phase is rather small, whereas the G-AFM phase with
$\Delta E=105$ meV/atom does not appear as a relevant competitor. For these
three different phases, Fig. 8c displays total and local Ni-$e_{g}$ spectral
functions, and Figs. 8d-f exhibit the corresponding ${\bf k}$-resolved
spectra. Inspection of the total spectrum renders it obvious, that though the
ordered moments are tiny, intriguingly, the low-energy spectra of the
different magnetic phases are actually quite different. Compared to the PM
phase, the small peak just below the Fermi level is substantially enhanced in
the C-AFM phase and more or less suppressed for the G-AFM phase. However, all
three phases still display the SD electron pockets at the Fermi level. Key
difference between the G-AFM and C-AFM low-energy physics is the enhancement
of the Ni-$d_{x^{2}-y^{2}}$ single-level feature at $\varepsilon_{\rm
F}^{\hfill}$ for the latter phase. This becomes clear from the brown curve in
the bottom part of Fig. 8c, as well as from the nondispersive feature in
$A({\bf k},\omega)$ at the Fermi level in Fig. 8e. That Kondo(-lattice)
resonance exists also in the PM phase Lechermann (2020b), yet with reduced
weight (and therefore not visible within the given resolution in Fig. 8e). The
C-AFM phase may thus be interpreted as an exotic phase with coexistence of
antiferromagnetic ordering and Kondo(-lattice) screening Zhang and Yu (2000);
Isaev and Vekhter (2013); Bernhard and Lacroix (2015); Li  _et al._ (2015).
Because of the weak filling of the the SD band, the latter Kondo physics takes
place in an underscreening scenario, leaving room for magnetic ordering.
Additionally, since the Kondo screening is mainly mediated by the $d_{z^{2}}$
orbitals of Ni and Nd Lechermann (2020b), the ferro-alignment of spins along
the $c$-axis in the C-AFM phase enables a screening action without additional
spin flip. This may explain as to why the G-AFM phase cannot easily coexist
with Kondo screening.

For finite hole doping $\delta=0.15$, the C-AFM phase has transformed into the
PM phase with zero ordered Ni moment. On the contrary, the G-AFM phase still
remains (meta)stable with somewhat reduced $m_{\rm Ni}$ compared to $\delta=0$
(see Tab. 1). Notably, as the SD pockets are shifted above the Fermi level for
that hole-doping level, partial Kondo screening of the pristine-scenario kind
has vanished. Figure 9 shows that close to the Fermi level, the Ni-$d_{z^{2}}$
character is enhanced and the Ni-$d_{x^{2}-y^{2}}$ one suppressed in the doped
G-AFM phase compared to the PM phase. Though energetically less favorable (see
Tab. 1), if NN-AFM ordering is indeed stable at low temperature and with
doping, there would be a $\delta$-induced transition from C-AFM to G-AFM
ordering. Note that a qualitative change of magnetic order with doping has
also been deduced by Leonov and Savrasov Leonov  _et al._ (2020) from DFT+DMFT
calculations for the system at ambient temperature, albeit differently with
G-AFM order at stoichiometry and C-AFM order at finite hole doping.

##  V Summary and discussion

The present DFT+sicDMFT study of pristine and doped NdNiO2 uncovers two key
results. First, the character of the paramagnetic electronic structure changes
distinctively with doping. On the electron-doped side, which may be
challenging to explore experimentally, the nickelate is a (Ni,Nd)-$d_{z^{2}}$
based one-band system in a Mott-critical background of Ni-$d_{x^{2}-y^{2}}$
kind. At stoichiometry, the latter background is of robust Mott-insulating
nature and the solid one-band part at the Fermi energy has evolved into a
weakly-filled self-doping entity. Since the SD dispersion carries relevant
Ni-$d_{z^{2}}$ itinerancy, one may also picture this phase as of orbital-
selective type, eventually decorated with partial Kondo screening at low
temperature Lechermann (2020b). Already small hole doping leads to a sizable
upward shift of the SD band into the unoccupied region, and the stabilization
of a Ni-$d_{x^{2}-y^{2}}$-based weakly-doped/pre-melted Mott-insulator in
which Ni-$d_{z^{2}}$ degrees of freedom are frozen. For hole doping
$\delta\gtrsim 0.15$, the Ni-$d_{z^{2}}$ flat-band part in the $k_{z}=1/2$
region crosses the Fermi level and the system displays intriguing
$\\{$Ni$-d_{z^{2}}$, Ni-$d_{x^{2}-y^{2}}\\}$ physics at low energy. As a
result thereof, unconventional superconductivity sets in at low temperature.
For even larger hole doping $\delta>0.30$, the nickelate does not attain a
conventional Fermi-liquid state, but enters a pseudogap and a bad Hund-metal
regime. In other words, the two-orbital Ni-$e_{g}$ subspace with filling
$n_{e_{g}}\sim 2.5$ becomes strongly correlated with vanishing coherence scale
due to the interplay of the Hund’s $J_{\rm H}$ with the remaining local
Coulomb interactions in the Ni$(3d)$ and O$(2p)$ shells. Take notice that
other recent theoretical works mark general Nd1-xSrxNiO2 exclusively as a
strong Hund metal, and suggest overall similarities with iron-pnictide physics
Wang  _et al._ (2020c); Kang  _et al._ (2020). However from our point of view,
the (near) Mott-insulating state of Ni-$d_{x^{2}-y^{2}}$ and large filling of
Ni-$d_{z^{2}}$ close to stoichiometry render this somewhat questionable. As
described above, strong Hund-metal physics occurs here only for larger hole
dopings, and therefore loosely speaking, superconductivity in doped NdNiO2
appears to take place when ’cuprates’ (from the underdoped side) and
’pnictides’ (from the overdoped side) meet and bind together through a flat-
band scenario.

Second, it is shown that magnetic order of AFM type may be a vital competitor
with and without doping at low-enough temperatures, albeit the associated
features are far from those of conventional antiferromagnetism in e.g.
stoichiometric high-$T_{\rm c}$ cuprates. At $T=30$ K, the ordered Ni moments
are already very small for all encountered cases, which either points to a
critical temperature in the same range or to strong quantum
fluctuations/screening. The size of the local moment surely also depends on
calculational settings, e.g. on the method to handle spin-symmetry breaking in
realistic DMFT and the choice of the correlated subspace. But the present
choices are believed physically sound, and therefore the general outcome of
significantly reduced Ni moments around $T=30$ K should be qualitatively
robust. The C-AFM phase is energetically more favorable than the G-AFM phase
at stoichiometry, obviously benefiting from a coexistence with partial Kondo
screening. This result sounds rather exotic for a TM($3d$) oxide, however the
experimental searches for magnetic order in infinite-layer nickelates also
point to singular low-$T$ behavior Hayward and Rosseinsky (2003); Wang  _et
al._ (2020a); Cui  _et al._ (2020). Furthermore, we here detect an apparent
transition from C-AFM to G-AFM order/correlations with hole doping. This would
be coherent with the fact that the Kondo-screening tendencies at stoichiometry
vanish with hole doping, because of the shift of the SD band above the Fermi
level.

This work complements our previous investigation for infinite-layer nickelates
Lechermann (2020a, b). It is worthwhile to note that the obtained essential
picture of relevant Ni multiorbital physics, and especially the importance of
the Ni-$d_{z^{2}}$ flat-band part for hole dopings in the superconducting
region, is very robust across these studies. For instance, three different
frameworks to describe the doped NdNiO2 compound, i.e. supercell with
impurities Lechermann (2020a), minimal Hamiltonian with chemical-potential
shift Lechermann (2020b) and eventually the present VCA approach, all lead to
the same qualitative result. Thereby, the role of explicit Coulomb
interactions on oxygen is not of ’decorative’ type, but truly decisive for the
revealed physics (see Appendix for further details). When abandoning the SIC
inclusion into the theoretical framework and employing conventional DFT+DMFT
calculations, the Ni-$d_{z^{2}}$ flat-band part remains well below the Fermi
level for the relevant dopings, in agreement with comparable studies of this
type Si  _et al._ (2020); Leonov  _et al._ (2020); Karp  _et al._ (2020); Wang
_et al._ (2020c). Moreover, a recent GW+EDMFT investigation of Petocchi et al.
Petocchi  _et al._ (2020), which also includes beyond-Ni$(3d)$-based
correlations, supports our picture.

Let us finally take the opportunity to selectively touch base with further
theoretical studies from other research groups. In DFT+U studies by Choi et
al. Choi  _et al._ (2020a), the C-AFM order also turns out to be a proper
candidate for magnetism in NdNiO2. Furthermore, notably at stoichiometry, the
Ni-$d_{z^{2}}$ flat-band part in the $k_{z}=1/2$ region is placed right at the
Fermi level for this order within the given static-correlation scheme. Though
at a first glance looking quite different to our result for the C-AFM phase
(compare with Fig. 8e), this outcome could carry similar physics. The DFT+U
scheme may mimic Kondo(-like) physics by placing a spin-resolved flat band
right at the Fermi level. In this way, the corresponding (local) spin becomes
highly susceptible, and thats more or less the best a static-correlation
scheme may achieve for Kondo-based spin fluctuations. The fact that this band
is of dominant Ni-$d_{z^{2}}$ character underlines our previous finding of the
crucial Ni-$d_{z^{2}}$ role in mediating Kondo-screening in NdNiO2 Lechermann
(2020b). However, this DFT+U result for Ni-$d_{z^{2}}$ at stoichiometry should
not be directly related to the DFT+sicDMFT result of the Ni-$d_{z^{2}}$ flat-
band part crossing the Fermi level with sizable hole doping.

In a different DFT+U assessment by Liu et al. Liu  _et al._ (2020), the G-AFM
order is found stable at stoichiometry, but the Ni magnetic moment and the
general AFM ordering is also easily weakened by the impact of the SD band. The
resistivity upturn below $T\sim 70$ K is associated with the transition into
this bad-AFM metal when lowering temperature from the PM phase. Albeit details
differ, that bad-AFM metal regime could resemble our AFM plus Kondo-screening
phase.

Finally, all our theoretical studies on the thin-film NdNiO2 system, i.e.
Refs. Lechermann, 2020a, b and this one, are conducted in a ’bulk-like’
fashion by using the thin-film lattice parameter for a bulk-crystal
calculation. Already several works Bernardini and Cano (2020); Geisler and
Pentcheva (2020); He  _et al._ (2020); Zhang  _et al._ (2020d) addressed the
explicit thin-film geometry on a SrTiO3 substrate via supercell DFT(+U)
calculations. However, while nickelate/SrTiO3 interface physics may have some
impact, from the comparison of the currently available experimental data and
the results of a theoretical bulk-like approach there is no pressing evidence
that key effects are missing in the latter. Furthermore, the experimental thin
films with $\sim 10$ nm thickness are rather bulky, accounting for more than
thirty vertical unit cells. The measured critical current densities Osada  _et
al._ (2020a) of more than 300 kA/cm2 would also be quite large for sole
twodimensional superconductivity in the interface. Nonetheless, investigating
NdNiO2 in an ultrathin limit of only a few unit cells would surely be
interesting.

In the end, different theoretical concepts for infinite-layer nickelates are
presently put to the test. Ultimately, only experiment can tell which of these
concepts is closest to the truth. The here presented evolution of the doping-
dependent phenomenology is already in line with various experimental findings
Li  _et al._ (2020); Zeng  _et al._ (2020); Goodge  _et al._ (2020), most
notably concerning the identifaction of weakly-insulating regions on either
side of the superconducting doping region as well as the dominance of hole-
like transport for $\delta>0.15$. In a next step, angle-resolved photoemission
(ARPES) measurements appear deciding, since therefrom the correlation strength
and the role of the Ni-$d_{z^{2}}$ flat-band part with doping is most clearly
revealed. But ARPES on these thin film architectures, together with the
necessary resolution for possible flat-band physics out of the $k_{z}=0$
plane, is surely demanding. By any means, the infinite-layer nickelate physics
is highly puzzling and appears to integrate various condensed-matter regimes
into one materials class, providing room for intense research in the future.

Figure 10: (color online) Comparison of DFT+DMFT and DFT+sicDMFT spectral
functions for pristine NdNiO2 (left) and at doping level $\delta=0.15$
(right). (Top) Total spectrum and (bottom) local Ni-$e_{g}$ spectrum.

###### Acknowledgements.

The author thanks I. Eremin, A. Hampel, J. Karp, and A. J. Millis for helpful
discussions. Computations were performed at the University of Hamburg and the
JUWELS Cluster of the Jülich Supercomputing Centre (JSC) under project number
hhh08.

*

##  Appendix A Role of explicit Coulomb interactions on oxygen

As discussed in more detail in Ref. Lechermann  _et al._ , 2019, the SIC
treatment for most relevant O$(2p)$ in the present form effectively introduces
a $U_{\rm pp}$ and a $U_{\rm pd}$ interaction term. As a result, the
$pd$-splitting between O$(2p)$ and Ni$(3d)$ is enhanced compared to LDA, and
the hoppings $t_{pd}$ between those site-orbitals are renormalized. In order
to assess the effect of including explicit Coulomb interactions on oxygen via
a SIC-modified O pseudpotential, we performed also calculations within the
conventional DFT+DMFT scheme for the paramagnetic system. There, the standard
O pseudpotential based on LDA is employed.

Figure 11: (color online) Comparison between DFT+DMFT and DFT+sicDMFT for
pristine (left) and $\delta=0.15$ hole-doped (right) NdNiO2. (a) ${\bf
k}$-resolved spectral function $A({\bf k},\omega)$ and (b) imaginary part of
the Ni-$e_{g}$ self-energy ${\rm Im}\,\Sigma(i\omega_{n})$.

In Fig. 10, the total spectral function $A_{\rm tot}(\omega)$ and the local
Ni-$e_{g}$ part are shown in comparison for both computational schemes. One
may infer that though the Hubbard $U$ on the Ni site is identical, the overall
correlation strength in the standard DFT+DMFT approach is much weaker than
with including additionally Coulomb interactions on oxygen. At stoichiometry,
the system is a strongly renormalized metal, i.e. with Ni-$d_{x^{2}-y^{2}}$
not reaching a Mott-insulating state, in line with other works Si  _et al._
(2020); Karp  _et al._ (2020); Wang  _et al._ (2020c). This can be explained
by the fact that a sizable renormalization of $t_{pd}$ is missing in DFT+DMFT.
Thus a sole focus on the Hubbard $U$ on Ni in order to assess the correlation
strength in infinite-layer nickelates may be misleading, since a vital
contribution is carried by the Coulomb interactions on oxygen because of the
late-TM-oxide context. Furthermore, the distance between oxygen and Ni$(3d)$
states, associated with both main peaks in the respective total spectrum, is
somewhat larger from the DFT+sicDMFT calculations, which reflects the increase
of $pd$ splitting.

For the hole-doped compound, it is additionally seen that the Ni-$d_{z^{2}}$
character does not show a peak close to the Fermi level for DFT+DMFT. This is
understood from Fig. 11, which displays the comparison between DFT+sicDMFT and
DFT+DMFT on the level of the ${\bf k}$-resolved spectral function $A({\bf
k},\omega)$ as well as the imaginary part of the Ni-$e_{g}$ self-energy ${\rm
Im}\,\Sigma(i\omega_{n})$. It becomes obvious that the previously discussed
key effect of hole doping, namely the shift of the Ni-$d_{z^{2}}$ flat-band
part in the $k_{z}=1/2$ plane is missing in DFT+DMFT. Moreover, the SD
electron pockets are much weaker shifted to higher (unoccupied) energies, also
in line with previous DFT+DMFT studies Si  _et al._ (2020); Leonov  _et al._
(2020); Karp  _et al._ (2020); Wang  _et al._ (2020c). This difference in
doping evolution of those specific dispersions has to be related to the fact,
that both individual Ni-$e_{g}$ orbitals hybridize very differently with
O$(2p)$ (see last paragraph in section II). Consequently, the
Ni-$d_{x^{2}-y^{2}}$ self-energy at $\delta=0.15$ also does not show a clean
low-frequency anomaly in DFT+DMFT, since that one should be linked to the
Ni-$d_{z^{2}}$ flat-band part residing at $\varepsilon_{\rm F}^{\hfill}$.
However, note that the Ni-$e_{g}$ occupations up to $\delta=0.15$ do not seem
to differ strongly between both correlation schemes.

In summary, there are strong qualitative differences between the DFT+DMFT and
the DFT+sicDMFT description of pristine and doped NdNiO2, which ask for a
rethinking of the sole TM-driven correlation picture of late transition-metal
oxides in many numerical approaches.

## References

  * Li  _et al._ (2019) D. Li, K. Lee, B. Y. Wang, M. Osada, S. Crossley, H. R. Lee, Y. Cui, Y. Hikita, and H. Hwang, Nature 572, 624 (2019).
  * Osada  _et al._ (2020a) M. Osada, B. Y. Wang, B. H. Goodge, K. Lee, H. Yoon, K. Sakuma, D. Li, M. Miura, L. F. Kourkoutis, and H. Y. Hwang, Nano Letters 20, 5735 (2020a).
  * Osada  _et al._ (2020b) M. Osada, B. Y. Wang, K. Lee, D. Li, and H. Y. Hwang, arXiv:2010.16101 (2020b).
  * Hayward and Rosseinsky (2003) M. A. Hayward and M. J. Rosseinsky, Solid State Sci. 5, 839 (2003).
  * Wang  _et al._ (2020a) B.-X. Wang, H. Zheng, E. Krivyakina, O. Chmaissem, P. P. Lopes, J. W. Lynn, L. C. Gallington, Y. Ren, S. Rosenkranz, J. F. Mitchell, and D. Phelan, Phys. Rev. Materials 4, 084409 (2020a).
  * Cui  _et al._ (2020) Y. Cui, C. Li, Q. Li, X. Zhu, Z. Hu, Y. feng Yang, J. S. Zhang, R. Yu, H.-H. Wen, and W. Yu, arXiv:2011.09610 (2020).
  * Li  _et al._ (2020) D. Li, B. Y. Wang, K. Lee, S. P. Harvey, M. Osada, B. H. Goodge, L. F. Kourkoutis, and H. Y. Hwang, Phys. Rev. Lett. 125, 027001 (2020).
  * Zeng  _et al._ (2020) S. Zeng, C. S. Tang, X. Yin, C. Li, Z. Huang, J. Hu, W. Liu, G. J. Omar, H. Jani, Z. S. Lim, K. Han, D. Wan, P. Yang, A. T. S. Wee, and A. Ariando, arXiv:2004.11281 (2020).
  * Hepting  _et al._ (2020) M. Hepting, D. Li, C. J. Jia, H. Lu, E. Paris, Y. Tseng, X. Feng, M. Osada, E. Been, Y. Hikita, Y. D. Chuang, Z. Hussain, K. J. Zhou, A. Nag, M. Garcia-Fernandez, M. Rossi, H. Y. Huang, D. J. Huang, Z. X. Shen, T. Schmitt, H. Y. Hwang, B. Moritz, J. Zaanen, T. P. Devereaux, and W. S. Lee, Nat. Mater. doi:10.1038/s41563-019-0585-z (2020).
  * Goodge  _et al._ (2020) B. H. Goodge, D. Li, M. Osada, B. Y. Wang, K. Lee, G. A. Sawatzky, H. Y. Hwang, and L. F. Kourkoutis, arXiv:2005.02847 (2020).
  * Rossi  _et al._ (2020) M. Rossi, H. Lu, A. Nag, D. Li, M. Osada, K. Lee, B. Y. Wang, S. Agrestini, M. Garcia-Fernandez, Y. D. Chuang, Z. X. Shen, H. Y. Hwang, B. Moritz, K.-J. Zhou, T. P. Devereaux, and W. S. Lee, arXiv:2011.00595 (2020).
  * Xiang  _et al._ (2020) Y. Xiang, Q. Li, Y. Li, H. Yang, Y. Nie, and H.-H. Wen, arXiv:2007.04884 (2020).
  * Wang  _et al._ (2020b) B. Y. Wang, D. Li, B. H. Goodge, K. Lee, M. Osada, S. P. Harvey, L. F. Kourkoutis, M. R. Beasley, and H. Y. Hwang, arXiv:2012.06560 (2020b).
  * Gu  _et al._ (2020a) Q. Gu, Y. Li, S. Wan, H. Li, W. Guo, H. Yang, Q. Li, X. Zhu, X. Pan, Y. Nie, and H.-H. Wen, Nat Commun 11, 6027 (2020a).
  * Wu  _et al._ (2020) X. Wu, D. D. Sante, T. Schwemmer, W. Hanke, H. Y. Hwang, S. Raghu, and R. Thomale, Phys. Rev. B 101, 060504(R) (2020).
  * Nomura  _et al._ (2019) Y. Nomura, M. Hirayama, T. Tadano, Y. Yoshimoto, K. Nakamura, and R. Arita, Phys. Rev. B 100, 205138 (2019).
  * Botana and Norman (2020) A. S. Botana and M. R. Norman, Phys. Rev. X 10, 011024 (2020).
  * Lechermann (2020a) F. Lechermann, Phys. Rev. B 101, 081110((R) (2020a).
  * Lechermann (2020b) F. Lechermann, Phys. Rev. X 10, 041002 (2020b).
  * Jiang  _et al._ (2020) M. Jiang, M. Berciu, and G. A. Sawatzky, Phys. Rev. Lett. 124, 207004 (2020).
  * Zhang  _et al._ (2020a) G.-M. Zhang, Y.-F. Yang, and F.-C. Zhang, Phys. Rev. B 101, 020501(R) (2020a).
  * Si  _et al._ (2020) L. Si, W. Xiao, J. Kaufmann, J. M. Tomczak, Y. Lu, Z. Zhong, and K. Held, Phys. Rev. Lett. 124, 166402 (2020).
  * Kitatani  _et al._ (2020) M. Kitatani, L. Si, O. Janson, R. Arita, Z. Zhong, and K. Held, arXiv:2002.12230 (2020).
  * Karp  _et al._ (2020) J. Karp, A. S. Botana, M. R. Norman, H. Park, M. Zingl, and A. Millis, Phys. Rev. X 10, 021061 (2020).
  * Adhikary  _et al._ (2020) P. Adhikary, S. Bandyopadhyay, T. Das, I. Dasgupta, and T. Saha-Dasgupta, Phys. Rev. B 102, 100501 (2020).
  * Lang  _et al._ (2020) Z.-J. Lang, R. Jiang, and W. Ku, arXiv:2005.00022 (2020).
  * Lee and Pickett (2004) K.-W. Lee and W. E. Pickett, Phys. Rev. B 70, 165109 (2004).
  * Hu and Wu (2019) L.-H. Hu and C. Wu, Phys. Rev. Research 1, 032046 (2019).
  * Zhang and Vishwanath (2020) Y.-H. Zhang and A. Vishwanath, Phys. Rev. Research 2, 023112 (2020).
  * Werner and Hoshino (2020) P. Werner and S. Hoshino, Phys. Rev. B 101, 041104(R) (2020).
  * Chang  _et al._ (2020) J. Chang, J. Zhao, and Y. Ding, Eur. Phys. J. B 93, 220 (2020).
  * Choi  _et al._ (2020a) M.-Y. Choi, W. E. Pickett, and K.-W. Lee, Phys. Rev. Research 2, 033445 (2020a).
  * Petocchi  _et al._ (2020) F. Petocchi, V. Christiansson, F. Nilsson, F. Aryasetiawan, and P. Werner, Phys. Rev. X 10, 041047 (2020).
  * Wang  _et al._ (2020c) Y. Wang, C.-J. Kang, H. Miao, and G. Kotliar, Phys. Rev. B 102, 161118 (2020c).
  * Kang  _et al._ (2020) B. Kang, C. Melnick, P. Semon, G. Kotliar, and S. Choi, arXiv:2007.14610 (2020).
  * Savrasov  _et al._ (2001) S. Y. Savrasov, G. Kotliar, and E. Abrahams, Nature 410, 793 (2001).
  * Grieger  _et al._ (2012) D. Grieger, C. Piefke, O. E. Peil, and F. Lechermann, Phys. Rev. B 86, 155121 (2012).
  * Lechermann  _et al._ (2019) F. Lechermann, W. Körner, D. F. Urban, and C. Elsässer, Phys. Rev. B 100, 115125 (2019).
  * Elsässer  _et al._ (1990) C. Elsässer, N. Takeuchi, K. M. Ho, C. T. Chan, P. Braun, and M. Fähnle, J. Phys.: Condens. Matter 2, 4371 (1990).
  * Lechermann  _et al._ (2002) F. Lechermann, F. Welsch, C. Elsässer, C. Ederer, M. Fähnle, J. M. Sanchez, and B. Meyer, Phys. Rev. B 65, 132104 (2002).
  * Meyer  _et al._ (1998) B. Meyer, C. Elsässer, F. Lechermann, and M. Fähnle,  _FORTRAN 90 Program for Mixed-Basis-Pseudopotential Calculations for Crystals_ , Max-Planck-Institut für Metallforschung, Stuttgart (1998).
  * Körner and Elsässer (2010) W. Körner and C. Elsässer, Phys. Rev. B 81, 085324 (2010).
  * Amadon  _et al._ (2008) B. Amadon, F. Lechermann, A. Georges, F. Jollet, T. O. Wehling, and A. I. Lichtenstein, Phys. Rev. B 77, 205112 (2008).
  * Werner  _et al._ (2006) P. Werner, A. Comanac, L. de’ Medici, M. Troyer, and A. J. Millis, Phys. Rev. Lett. 97, 076405 (2006).
  * Parcollet  _et al._ (2015) O. Parcollet, M. Ferrero, T. Ayral, H. Hafermann, I. Krivenko, L. Messio, and P. Seth, Comput. Phys. Commun. 196, 398 (2015).
  * Seth  _et al._ (2016) P. Seth, I. Krivenko, M. Ferrero, and O. Parcollet, Comput. Phys. Commun. 200, 274 (2016).
  * Anisimov  _et al._ (1993) V. I. Anisimov, I. V. Solovyev, M. A. Korotin, M. T. Czy$\dot{\text{z}}$yk, and G. A. Sawatzky, Phys. Rev. B 48, 16929 (1993).
  * Zaanen  _et al._ (1985) J. Zaanen, G. A. Sawatzky, and J. W. Allen, Phys. Rev. Lett. 55, 418 (1985).
  * Imada  _et al._ (1998) M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 70, 1039 (1998).
  * Han  _et al._ (2011) M. J. Han, X. Wang, C. A. Marianetti, and A. J. Millis, Phys. Rev. Lett. 107, 206804 (2011).
  * Hirsch and Marsiglio (2019) J. Hirsch and F. Marsiglio, Physica C: Superconductivity and its Applications 566, 1353534 (2019).
  * Keimer  _et al._ (1992) B. Keimer, N. Belk, R. J. Birgeneau, A. Cassanho, C. Y. Chen, M. Greven, M. A. Kastner, A. Aharony, Y. Endoh, R. W. Erwin, and G. Shirane, Phys. Rev. B 46, 14034 (1992).
  * Dagotto (1994) E. Dagotto, Rev. Mod. Phys. 66, 763 (1994).
  * Choi  _et al._ (2020b) M.-Y. Choi, K.-W. Lee, and W. E. Pickett, Phys. Rev. B 101, 020503 (2020b).
  * Zhang  _et al._ (2020b) H. Zhang, L. Jin, S. Wang, B. Xi, X. Shi, F. Ye, and J.-W. Mei, Phys. Rev. Research 2, 013214 (2020b).
  * Gu  _et al._ (2020b) Y. Gu, S. Zhu, X. Wang, J. Hu, and H. Chen, Commun Phys 3, 84 (2020b).
  * Liu  _et al._ (2020) Z. Liu, Z. Ren, W. Zhu, Z. F. Wang, and J. Yang, npj Quantum Mater. 5, 31 (2020).
  * Leonov  _et al._ (2020) I. Leonov, S. L. Skornyakov, and S. Y. Savrasov, Phys. Rev. B 101, ”241108(R)” (2020).
  * Wan  _et al._ (2020) X. Wan, V. Ivanov, G. Resta, I. Leonov, and S. Y. Savrasov, arXiv:2008.07465 (2020).
  * Zhang  _et al._ (2020c) R. Zhang, C. Lane, B. Singh, J. Nokelainen, B. Barbiellini, R. S. Markiewicz, A. Bansil, and J. Sun, arXiv:2009.05816 (2020c).
  * Kapeghian and Botana (2020) J. Kapeghian and A. S. Botana, Phys. Rev. B 102, 205130 (2020).
  * Hayward  _et al._ (1999) M. A. Hayward, M. A. Green, M. J. Rosseinsky, and J. Sloan, J. Am. Chem. Soc. 121, 8843 (1999).
  * Zhang and Yu (2000) G.-M. Zhang and L. Yu, Phys. Rev. B 62, 76 (2000).
  * Isaev and Vekhter (2013) L. Isaev and I. Vekhter, Phys. Rev. Lett. 110, 026403 (2013).
  * Bernhard and Lacroix (2015) B. H. Bernhard and C. Lacroix, Phys. Rev. B 92, 094401 (2015).
  * Li  _et al._ (2015) H. Li, Y. Liu, G.-M. Zhang, and L. Yu, J. Phys.: Condens. Matter 27, 425601 (2015).
  * Bernardini and Cano (2020) F. Bernardini and A. Cano, J. Phys. Mater. 3, 03LT01 (2020).
  * Geisler and Pentcheva (2020) B. Geisler and R. Pentcheva, Phys. Rev. B 102, 020502 (2020).
  * He  _et al._ (2020) R. He, P. Jiang, Y. Lu, Y. Song, M. Chen, M. Jin, L. Shui, and Z. Zhong, Phys. Rev. B 102, 035118 (2020).
  * Zhang  _et al._ (2020d) Y. Zhang, L.-F. Lin, W. Hu, A. Moreo, S. Dong, and E. Dagotto, Phys. Rev. B 102, 195117 (2020d).