Effective Hamiltonian for superconducting Ni oxides Nd$_{1-x}$Sr$_x$NiO$_2$

We derive the effective single-band Hamiltonian in the flat NiO$_2$ planes for nickelate compounds Nd$_{1-x}$Sr$_x$NiO$_2$. We first implement the first-principles calculation to study electronic structures of nickelates using the Heyd-Scuseria-Ernzerhof hybrid density functional and derive a three-band Hubbard model for Ni-O $pd\sigma$ bands of Ni$^+$ $3d_{x^2-y^2}$ and O$^{2-}$ $2p_{x/y}$ orbitals in the NiO$_2$ planes. To obtain the effective one-band $t$-$t'$-$J$ model Hamiltonian, we perform the exact diagonalization of the three-band Hubbard model for the Ni$_5$O$_{16}$ cluster and map the low-energy spectra onto the effective one-band models. We find that the undoped NiO$_2$ plane is a Hubbard Mott insulator, and the doped holes primarily locate on Ni sites. The physics of the NiO$_2$ plane is a doped Mott insulator, described by the one-band $t$-$t'$-$J$ model with $t=265$~meV, $t'=-21$~meV and $J=28.6$~meV. We also discuss the electronic structure for the"self-doping"effect and heavy fermion behavior of electron pockets of Nd$^{3+}$ $5d$ character in Nd$_{1-x}$Sr$_x$NiO$_2$.


I. INTRODUCTION
The layered high-temperature superconductors in copper oxides and iron pnictides have motivated the search for new superconductivity compounds with layered structures [1][2][3][4] . Due to the similar crystal and electronic structure, LaNiO 2 has been theoretical studies as the possible analogs to the cuprates 5,6 . Recently, the superconductivity with the critical temperature up to T c ∼ 15 K is indeed discovered in Nd 0.8 Sr 0.2 NiO 2 thin film 7 , albeit the presence of the superconductivity remains debated 8,9 . Similar to copper oxides, perovskite nickelates (RNiO 3 , where R is rare earth or heavy metal such as Tl or Bi) display lots of strongly correlated physics properties, such as the sharp metal-insulator transitions, particular magnetic order, and charge order [10][11][12][13] . The reduced form of RNiO 3 leads to the infinite layered phase RNiO 2 14-20 , which has a very flat NiO 2 plane of the square lattice for Ni + with one hole in the d x 2 −y 2 orbital. The superconductivity likely occurs in the NiO 2 plane with the charged carrier doping. In the cuprate superconductor compounds, the strong electronic interactions play a significant role in the electronic structure 21 , and the effective one-band Hamiltonian has been proposed to describe the low energy physics of the correlation effects for 3d 9 electrons [21][22][23] . To understand the strongly correlated electronic structures of the nickelate oxides Nd 1−x Sr x NiO 2 , we need to find out the proper effective (one-band) Hamiltonian to explore the similarity and difference from the cuprate compounds. NdNiO 2 crystallizes in the P 4/mmm (No. 123) space group, as depicted in Fig. 1 (a). Four oxygens surround the nickel in a planar square environment ( Fig. 1 (d)), and the crystal field splits the d orbitals as shown in Fig. 1 (e). Ni + has the d 9 electronic state configuration, and the highest partially occupied d orbital is 3d x2−y2 . The rare-earth ion Nd 3+ sits in the center of cuboid formed by eight oxygen ions as shown in Fig. 1 (c). As Nd 3+ in Nd 2 CuO 4 24 and Ho 3+ in HoNiO 3 25 , Nd 3+ has the local 4f moment far below the Fermi energy level. Nd 5d orbitals have the split energy levels, as shown in Fig. 1 (e), and near the Fermi energy, the lowest 5d orbital in Nd 3+ is d z 2 . Therefore, in the simple reckoning for the relevant electronic structure, there are 12 bands near Fermi energy level corresponding to mainly Ni d (5 states) and O p (2×3 states) and Nd d z 2 (1 state). Previous density functional theory (DFT) studies within the local density approximation (LDA) on LaNiO2 6 supports the rough impression of the nonmagnetic electronic structures.
In this paper, we first implement the first-principles simulations to derive a three-band Hubbard model for Ni-O pdσ bands of Ni + 3d x 2 −y 2 and O 2− 2p x/y orbitals in the NiO2 planes. Based on the three-band Hubbard model, we perform the exact diagonalization for the Ni 5 O 16 cluster and obtain the low-energy one-band effective Hamiltonian for the NiO 2 planes. We also discuss the electronic structure for the "selfdoping" effect and the heavy-fermion behavior of electron pockets of Nd 3+ 5d character in Nd 1−x Sr x NiO 2 . We present our main results in the two successive stages in Sec. II. We discuss the physics of the one-band t − J model in the NiO2 planes in Sec. III. In the Appendix, we present the results for La 1−x Sr x NiO 2 , implying the generic electronic structures of RNiO 2 series. We also include other supplementary results for Nd 1−x Sr x NiO 2 in the Appendix.
The main results are summarized as follows. In Sec. II A, we first perform DFT calculations of Nd 1−x Sr x NiO 2 within the Heyd-Scuseria-Ernzerhof (HSE) hybrid density functional. We notice that in the previous study of the nickelates RNiO 3 , the HSE hybrid functional method is essential to reproduce the experimentally observed magnetic ground state 26 . On the generalized gradient approximation (GGA) level, DFT simulations suggest the G-type antiferromagnetic (AFM within the NiO 2 plane and AFM between NiO 2 planes along the c direction) ground state of the moment on Ni sites in the parent compound NdNiO2. The HSE results for Nd 1−x Sr x NiO 2 adopt the G-type spin configuration to mimic strong spin correlations. In Sec. II B, we describe the electronic structures based on the DFT simulations. Threedimensional electron Fermi pockets of Nd 5d character are found near Fermi energy and behave as a heavy-fermion system due to its coupling with the local moments of Nd 4f , similar to the electron-doped cuprate Nd 1.8 Ce 0.2 CuO 4 27 . We also obtain the three-band Hubbard model for Ni-O pdσ bands of Ni + 3d x 2 −y 2 and O 2− 2p x /p y orbitals in the NiO 2 planes. In Sec. II C, we derive the effective one-band Hamiltonian of the NiO2 plane, following the standard procedure in the cuprates 22,23 . We derive the parameters for the threeband Hubbard model of the Ni-O pdσ bands for Ni + 3d x 2 −y 2 and O 2− 2p x/y orbitals from the LDA results from the firstprinciples simulations for the non-magnetic ground state for NdNiO 2 . Exact diagonalization (ED) studies of the Ni 5 O 16 cluster within the three-band Hubbard model are used to select and map the low-energy spectra onto the effective oneband t-t -J model. According to ED results on finite clusters, in hole-doped nickelates, the doped holes primarily locate on Ni sites in a good agreement with the HSE results in Sec.II A and the experiment 28 , while for cuprate the holes mainly locate on oxygen site 22,23 . The physics of the NiO 2 is a doped Mott insulator described by the effective on-band t-t -J model with t = 265 meV, t = −21 meV, and J = 28.6 meV. The one-band effective Hubbard model is also given.

A. DFT results of Nd1−xSrxNiO2
We performed first-principles calculations based on DFT 29 with the HSE06 hybrid functional 30 as implemented in the Vienna Ab Initio Simulation Package (VASP) [31][32][33] . We also implement PerdewBurkeErnzerhoff (PBE) functional in gen-eralized gradient approximation (GGA) 34 and strongly constrained and appropriately normed semilocal density functional (SCAN) in meta-GGA 35 for comparisons. We use an energy cutoff of 500 eV and 12×12×12, 4×4×4, and 4×4×4 Monkhorst-Pack grids 36 in the PBE, SCAN and HSE06 calculations, respectively. The 4f orbitals in Nd 3+ are expected to display the local magnetic moment as Nd 3+ in Nd 2 CuO 4 24 and Ho 3+ in HoNiO 3 25 , and we treat them as the core-level electrons in the Nd pseudopotential in the HSE06 calculations. We check the results with 4f valence electrons pseudopotential within GGA+U (U 4f = 10 eV) scheme 37 in Appendix A. We also check that the spin-orbit coupling does not significantly change the DFT results in the Appendix. Therefore, we only present the simulations by taking 4f the corelevel electrons in Nd pseudopotential and do not include the spin-orbit coupling in our HSE06 hybrid functional simulations.
We use the bulk lattice constants a = b = 3.9208Å, c = 3.281Å 16 for Nd 1−x Sr x NiO 2 and didn't optimize the crystal structure during the DFT simulations. In this setup, the NiO 2 plane remains flat in the doped case Nd 0.75 Sr 0.25 NiO 2 . We didn't consider the lattice distortion effect upon doping in the doped nickelate oxides. Figure. 1 (b) is the HSE06 band structure for the nonmagnetic state in NdNiO2. The band ordering is different from the crystal field theory in Fig. 1 (e) due to the covalent effect in the hybridizations between d orbitals (Nd and Ni) and 2p orbitals (O). There are 12 electronic bands near the Fermi energy E F , corresponding to orbitals mainly of Ni d (5 states) and O p (2×3 states) and Nd d z 2 (1 state) for the relevant electronic structure. The O 2p bands extend from about -10 to -5 eV. The Ni 3d bands are distributed from -5 to 2 eV, while the broad Nd 5d states range from -1 to 10 eV. The Ni 3d x 2 −y 2 and Nd 5d z 2 cross the Fermi energy E F as expected from the crystal field splitting, as shown in Fig. 1 (e). Ni 3d x 2 −y 2 is very broad along the Γ-X-M direction due to the strong dpσ antibonding interaction with oxygen p x/y states and encloses holes centered at the M point. Ni 3d xy/xz/yz bands localize near -4 eV due to the weak dpπ hybridization with O 2p states. Around A point, The Nd 5d z 2 state forms an electron pocket around the Γ point, goes up along Γ-Z direction, and lies above the Fermi energy level with k z = π/c. The Nd 5d xy lowers down and crosses the Fermi level, forming an electron pocket around A point.
Along Γ-Z direction, four O 2p x/y bands and four Ni+ (3dxz/yz, 3d xy , and 3d x 2 −y 2 ) bands have weak dispersions, indicating the two-dimensional features of these bands. Ni 3d z 2 and Nd 5d states are dispersive along with Γ-Z direc- tions and three-dimensional extended. The two electron pockets around Γ and A points of the Nd 5d orbital character have the Ni orbitals mixing. However, such mixing is not quickly resolved in Fig. 1 (b) since the Nd 5d characters dominate the electron pockets and cover up the contributions from Ni orbitals. To further clarify the mixing character, we also present the band structure for the simulated SrNiO 2 in with the same structure as NdNiO 2 , in order to eliminate Nd 5d orbitals in Fig. 2. From the comparing between Fig. 1 (b) and Fig. 2, we can see that Nd 5d z 2 band crosses E F around Γ point with Ni 4s mixing, while Nd 5d xy band crosses E F around A point with Ni 4p z mixing.
The GGA band structure of NdNiO 2 in Appendix A is quite similar to the LDA band structure of LaNiO 2 6 , indicating a generic electronic structure in the RNiO 2 family. Compared with the GGA band structure (Appendix A), for the nonmagnetic state, the mixing of the exact exchange in the HSE06 hybrid functional separates the two bands crossing the Fermi energy E F away from other bands, without significant change of dispersions and relative positions of the bands far from E F . The separation of two bands can also be reproduced in the LDA+U results 6 . However, in the LDA+U scheme, Ni 3d z 2 is raised by U and crosses E F when U is large 6 , different from the HSE06 hybrid functional simulation for the non-magnetic state.
The magnetization measurement and neutron powder diffraction didn't reveal the long-magnetic order in LaNiO 2 and NdNiO 2 ; however, the paramagnetic susceptibilities imply the (at least short) spin correlations 15,16 . The absence of long-range magnetic order may be due to poor sample qualities, or due to the "self-doping" effects of the Nd 5d electron pockets. The strong correlation for electrons on Ni + induce the magnetism in the system.
To demonstrate the correlation and magnetism in NdNiO 2 , we calculate the magnetic moment and compare the ground state with the non-magnetic one within different functionals (GGA, SCAN, and HSE06). To save computation time, we consider the ferromagnetic spin configuration in the primary unit cell. The results are list in TABLE I. Than the nonmagnetic state, the magnetic states have increased moments on Ni with increasingly lower ground state energies from GGA, SCAN to HSE06. For GGA, the magnetic state has even higher energy than the non-magnetic state, indicating the non-magnetic ground state within the GGA functional, consistent with the previous study in LaNiO2 6 . The SCAN functional includes more correlation effects, and the magnetism is significantly enhanced. The magnetic ground state has further lower energy than the non-magnetic state in the HSE06 functional. The fact that the correlation achieves the magnetism is a strong indication that NdNiO 2 is magnetic if we include the correlations.
Even though there is no long-range magnetic order, we still impose static magnetic configurations on Ni ions in Nd 1−x Sr x NiO 2 to mimic the spin correlations in the DFT simulations for electronic structures. We find the G-type AFM magnetic state has the lowest ground state energy within the GGA DFT simulations. Therefore, in the HSE06 simulations, we adopt the G-type AFM spin configurations on Ni using the √ 2 × √ 2 × 2 supercell to study the electronic structures of Nd 1−x Sr x NiO 2 (x = 0, 0.25). Therefore, the Brillouin zone is folded by the G-type AFM spin configuration. However, without any confusion, we still use notations, Γ, X, M, Z, R, A, for high symmetry k-points in the folded Brillouin zone. Figure Fig. 1 (b). sults, the AFM magnetic configuration not only folds the band structure, but also dramatically change the bands near the Fermi energy E F , very different from the GGA result for the G-type AFM magnetic configuration in the Appendix A. The significant change of band structures in the AFM states between GGA and HSE06 implies strong correlations in Nd 1−x Sr x NiO 2 .
In the G-type AFM state of NdNiO 2 (Fig. 3 (a)), the Nd 5d z 2 band is raised above the Fermi energy E F by the correlation, eliminating the electron pocket of the Nd 5d z 2 character in the non-magnetic state ( Fig. 1 (b)). The Nd 5d xy band is also raised, but the electron pocket of this band still exists. We remind here again that the electron pocket of the Nd 5d xy character has the Ni 4p z orbital mixing. The Nd 5d xy electron pocket locates around Γ point in the folded magnetic Brillouin zone. Therefore, the AFM spin correlation has an essential influence on the "self-doping" effect of Nd 5d electron pockets. For the Ni 3d band, the AFM correlation significantly renormalizes the bandwidth. The Ni 3d x 2 −y 2 bands split into upper and lower Hubbard bands separated by around 5 eV due to the AFM spin configuration, indicating the strong correlation in NdNiO 2 . The lower Hubbard 3d x 2 −y 2 band locates lower than the Ni 3d z 2 band, bring the doping problem into a complicated situation.
When one Sr 2+ ion substitutes Nd 3+ in NdNiO 2 , we dope an extra hole into the NiO 2 planes. To simulate the doped nickelate oxide, we calculate the band structures of Nd 0.75 Sr 0.25 NiO 2 (Nd 3 SrNi 4 O 8 ) in the √ 2 × √ 2 × 2 supercell with the G-type AFM spin configuration on Ni ions. Fig. 3 (b) is the HSE06 band structure of Nd 0.75 Sr 0.25 NiO 2 . The Nd 5d xy band goes up above the Fermi energy level, and the band minimum locates 0.2 eV. We can expect that Nd 5d xy band contributes Hall coefficient at high temperatures at low dopings 7 . The doped hole does not go to the Ni 3d z 2 band. Instead, it locates on the 3d x 2 −y 2 band, which goes up above 3d z 2 in Nd 0.75 Sr 0.25 NiO 2 . Therefore, the doped hole does not polarize Ni 2+ into the S = 1 local moment, but create an S = 0 hole-like doped charge carrier. The Ni 3d x 2 −y 2 encloses the hole pockets around X point in the folded Brillouin zone (M in the original non-magnetic Brillouin zone). The HSE06 band structures in Fig. 3 suggest that the undoped NiO 2 plane is Hubbard-like Mott insulator, not the chargetransfer-like one as in the cuprate. The doped hole goes into Ni 3d x 2 −y 2 orbital, creating an S = 0 hole-like charge carrier, rather than into the O 2− 2p x/y orbitals.

B. Electronic structures of Nd1−xSrxNiO2
The presence of the electron pockets of Nd 5d bands suggests the "self-doping" effect, implying a small charge transfer from these pockets to the Ni-O sheets even without chemical doping. The "self-doping" effect also exists in the cuprate family, e.g., YBa 2 Cu 3 O 7 and Bi 2 Sr 2 CaCu 2 O 8 38 . The threedimensional electron pockets of Nd 5d xy states have the mixing with Ni 4p z orbitals. The "self-doping" effect allows some charge transfer and changes the hole count in the NiO 2 planes, resulting in the metallic behavior even without chemical dop-ing. In the Appendix A, we present the GGA band structure with the 4f valence electron pseudopotential within the GGA+U scheme. We can see that Nd 4f states have the mixing with Nd 5d states. Due to the presence of Nd 4f orbitals, the electron pockets of the Nd 5d xy character hybridizes with the 4f local moments, behaving as a heavy-fermion system where a kσ creates fermion with momentum k on the electron pockets and s i and S 4f i are the spin operators for the 5d electrons and 4f local moments, respectively.
We now turn to the physics of the NiO 2 plane with the hole doping, following the process for the CuO 2 planes in cuprates 22,23 . The HSE06 band structures in Fig. 3 suggest that the undoped NiO 2 plane is Hubbard-like Mott insulator, and the doped hole goes into Ni 3d x 2 −y 2 orbital, creating an S = 0 hole-like charge carrier. Therefore, the physics of the NiO 2 plane with the charge doping is described by the threeband Hubbard model for the dpσ bands of Ni 3d x 2 −y 2 and O 2p x/y orbitals. Fig. 4 schematically presents the orbitals of Ni 3d x 2 −y 2 and O 2p x/y with the green and brown colors corresponding to the positive and negative signs of wave functions, respectively.
We assume a vacuum state d 10 p 6 and introduce the operators d † iσ and p † lσ creating the Ni 3d x 2 −y 2 hole and the O 2p x/y hole at the i-the Ni site and the l-th O site, respectively, with the spin σ =↑ / ↓. The holes hop between Ni 3d x 2 −y 2 and O 2p x/y orbitals with the amplitude t dp , and among O 2p x/y orbitals with the amplitude t pp . We set the on-site potential of Ni 3d x 2 −y 2 as d = 0, and the chemical potential difference dp pp FIG between Ni 3d x 2 −y 2 and O 2p x/y orbitals as = p − d . The strong correlations involve the on-site interactions U p and U d , and inter-site interactions U dp and U pp for holes on O 2p x/y and Ni 3d x 2 −y 2 orbitals. The three-band Hubbard model of the dpsigma bands reads out Here · · · denotes the nearest neighbor bonds.

C. Effective t-t -J model Hamiltonian of NiO2 planes
The tight-binding parameters t dp , t pp , and can be obtained from the Wannier fitting 39,40 of the 12 bands in the LDA simulations for the non-magnetic band structure of NdNiO 2 . It is noteworthy that the HSE06 band structure in Fig. 1 (b) already contains the renormalization effect due to the strong correlations. There is double-counting for the correlations if we obtain the tight-binding parameters from the HSE06 band structure. The parameters are given as = 4.2 eV, t pd = 1.3 eV, t pp = 0.6 eV. We choose a proper basis of the 3d x 2 −y 2 and 2p x/y for all positive t pd and t pp . To match the similar sizable band splitting of the lower and upper Hubbard bands of the Ni 3d x 2 −y 2 orbitals in the HSE06 band structure in Fig 3, we set the interaction terms as U d = 7.5 eV, U p = 5.0 eV, U pd = 4.0 eV, and U pp = 2.0 eV in the ED calculation for the derivation of the effective one-band Hamiltonian.
We follow the process in the cuprates 22,23 to obtain the effective one-band Hamiltonian of the NiO 2 planes. We perform direct ED studies of the three-band Hubbard model H dp in Eq. 2 and find the effective one-band Hamiltonian by the low-energy spectrum mapping. We carry out the ED calculation for the Ni 5 O 16 cluster as shown in Fig. 4 with five and six holes for the undoped and hole-doped NiO 2 planes, respectively. The Ni 5 O 16 cluster is embedded in an array of Ni 3d 9 sites which shift the effective on-site energy of the outer O orbitals due to the inter-site Coulomb energy 23 . Figure 5 is the low-energy spectrum mapping for the Ni5O16 cluster with five holes in the insulating ground state of the undoped NiO 2 planes. The spin-1/2 Heisenberg model with the nearest neighbor and next nearest neighbor exchange interactions J = 28.6 meV and J = 0.4 meV well reproduces the low-energy spectrum for the three-band Hubbard model as shown in Fig. 5. Therefore, we obtain the effective Heisenberg model for the undoped NiO 2 planes. For the hole-doped phase, the calculated low-energy spectra for six holes in the Ni5O16 cluster is shown in Fig 6. The chemical character of the hole in the ground state of the doped NiO2 planes is Ni (76%) and O (24%) character, indicating that NdNiO2 is in the regime of a Hubbard-Mott insulator, different from the cuprate where the doped holes primarily locate on O 22,23 . The significant difference comes from a smaller U d , but a larger = p − d in NdNiO 2 than those in La 2 CuO 4 .
We map the three-band Hubbard model in Eq. (2) onto the single-band t-t -J model where c iσ is the electron operator on the Ni sites connected by the solid (t and J) and dashed (t and J ) lines in Fig. 4. We first fix J = 28.6 meV and J = 0.4 meV as in the undoped case and then tune the hopping parameters t and t . The suitable values for t = 265 meV, t = −21 meV with the mapping spectra shown in Fig. 6. The signs for t, t refer to hole notation. We notice that the low-energy spectra has the spectrum width around 58 meV, approximately twice of the exchange strength J = 28.6 meV, in a good agreement with t-J model calculations for the cuprate, which show that, independent of the value of t, the dressing of the hole moving in the antiferromagnetic back-ground reduces the quasi-particle bandwidth to the twice of J 41 . We also map the low-energy spectra onto one-band Hubbard model with t = 254 meV, t = −30 meV and U = 6 eV in Fig. 6. The one-band Hubbard model gives the similar chargetransfer gap as the three-band Hubbard model in the Ni 5 O 16 cluster.

III. DISCUSSIONS AND CONCLUSIONS
From the above derivation of the effective one-band t − t − J model Hamiltonian, we find that the physics of NiO 2 planes in Nd 1−x Sr x NiO 2 is a doped Mott insulator, even further away from a conventional Fermi liquid than the superconducting cuprates. The undoped CuO 2 plane in the cuprate is a charge-transfer type Mott insulator, but the undoped NiO 2 plane in the nickelate is a Hubbard Mott insulator. Both the NiO 2 and CuO 2 planes have the same effective one-band t-t -J model Hamiltonians, and they have the similar physics of the superconductivity.
The discovering of the superconductivity in the nickelate oxide 7 certainly motivates the studies of the effective Hamiltonian in this paper. However, two recent experimental papers reported non-superconductivity results in Nd 1−x Sr x NiO 2 8,9 , getting the presence of superconductivity into controversy. The existence of the superconductivity in the doped nickelate oxides is a crucial issue and needs further exploring in the experiments. In this paper, our study is the consequence of a quantum-chemical description of Nd 1−x Sr x NiO 2 ; however, we cannot provide direct theoretical implication of the existence of the superconductivity. The effective Hamiltonian in this work provides essential support for, and constraint on, models to describe the low-energy physics of the nickelate oxides, regardless of the presence of the superconductivity. In our recent experimental work 42 , we perform Raman scattering on NdNiO 2 single crystal and measure the Heisenberg superexchange strength J = 25 meV from the two-magnon peak, in a good agreement with our present work. Although the current situation is not clear, we hope that our work will help us understand the electronic structure of nickelate oxides.
In conclusion, we have explicitly derived a single-band effective t-t -J model Hamiltonian for Ni-O based compounds starting from a three-band model based on the density functional theory.
Note added -At the stage of finishing the manuscript, we noticed related theoretical works for nickelates [43][44][45][46][47][48][49][50] . The paring of hole carriers in the oxygen pπ orbitals is discussed in Ref. 45 . First-principles simulations within GGA have been worked out in Refs. 43,44,[47][48][49][50] . DMFT is carried out in Ref. 50 , and RPA analysis of the pairing symmetry is done in Ref. 44,47 . The hoping parameters for the three-band Hubbard model are (implicitly or explicitly) given from the Wannier fitting in Refs. 43,44,[47][48][49][50] , with similar values to our work. The hoping parameters (t and t ) of the effective Ni 3d x 2 −y 2 band are also given in Refs. 43,[47][48][49] with t ∼ 370 meV and t ∼ −100 meV, larger than the values t = 265 meV and t = −21 meV renormalized by correlations in our work. The exchange interaction is estimated as J = 100 meV in Ref. 47 , larger than our values J = 28.6 meV. The Hubbard-Mott scenario is also proposed in Ref. 46 and the charge-transfer gap is estimated = 7 ∼ 9 eV, larger than our value = 4.2 eV. In the Ref. 46 , the S = 1 Ni 2+ state is proposed when the hope is doped into the NiO 2 planes, different from the S = 0 Ni 2+ state in our work. We notice that if the charge-transfer gap is taken as the value we use in the work, = 4.2 eV, the S = 0 Ni 2+ state is favored according to the calculation in Ref. 46 . In this part, we provide the supplementary DFT results for Nd 1−x Sr x NiO 2 . The main purpose of the supplementary results is two-folded: (a) check the validity of the Nd pseudopotential with the core-level 4f electrons; (b) compare the GGA and SCAN band structures to the HSE06 results in the main text. The spin-orbital coupling is also checked in this section.

GGA+U calculations for the Nd 4f valence electron pseudopotential
As Nd 3+ in Nd 2 CuO 4 24 and Ho 3+ in HoNiO 3 25 , Nd 3+ has the local 4f moment far below the Fermi energy level E F . In the main text, we treat the 4f electrons in Nd 3+ as the corelevel electrons in the Nd pseudopotential. In this subsection, we verify the validity of this treatment. Figure 7 are the band structure of G-type AFM states for NdNiO 2 and Nd 0.75 Sr 0.25 NiO 2 calculated with 4f valence electron pseudopotential within GGA+U scheme. Without the U term, the 4f electrons form very localized bands near the Fermi level. In the G-type AFM states, the local moments of 4f electrons are also AFM within the same Nd plane and between Nd planes along the c direction. We take the on-site interaction for 4f electrons U 4f = 10 eV in the GGA+U calculation which splits the 4f bands above and below the Fermi energy level E F . During the calculations, we also include the spin-orbital couplings.
According to Fig. 7, we can see that the 4f electron states couple to Nd 5d bands, however, doesn't significantly change the band structures near the Fermi level. Thus we can treat 4f orbitals as the core-level electrons in the Nd pseudopotential.

GGA and SCAN band structures
In this subsection, we present the band structures for Nd 1−x Sr x NiO 2 within GGA and SCAN functionals without/with spin-orbital couplings in Fig. 8 and Fig. 9, respectively. Again, once soc is taken into account, there is no significant change in the band structures. The main features of SCAN band structures are very close to those in HSE06 band structures.