*Ab Initio* Downfolding Based on the GW Approximation for Infinite-Layer Nickelates

^{1}QPEC, The University of Tokyo, Tokyo, Japan^{2}RIKEN Center for Emergent Matter Sciences (CEMS), Saitama, Japan^{3}Department of Applied Physics, The University of Tokyo, Tokyo, Japan

We derive an effective three-orbital model for the infinite-layer nickelates based on the band structure obtained by the GW approximation (GWA), where we consider the Ni 3*p* orbitals forming the *σ*-bond. In the GWA, the self-energy correction to the local density approximation (LDA) increases the energy difference between Ni *p*, which reduces the bandwidth of the antibonding 3

## Introduction

The discovery of nickel superconductors [1] has attracted renewed attention to superconductivity in strongly correlated electron systems [2–7]. So far, superconductivity has been found in film samples of doped infinite-layer nickelates *R*NiO_{2} (*R* = Nd, Pr, and La) [1, 8–16] and a quintuple-layer nickelate Nd_{6}Ni_{5}O_{12} [17]. Although the nature of the superconductivity is largely unknown, the pairing mechanism is likely to be unconventional: Theoretically, a phonon calculation for NdNiO_{2} has shown that the electron-phonon coupling is too weak to explain the superconductivity with a transition temperature on the order of 10 K [18]. Experimentally, both *U*- and *V*-shaped spectra have been observed using the scanning tunneling microscopy, depending on the location of the inhomogeneous surface of the doped NdNiO_{2} film [9]. Although the origin of the coexistence of the two different signals is controversial [19–23], the presence of the *V*-shape spectrum is consistent with an unconventional *d*-wave pairing. In fact, unconventional pairing mechanisms have been discussed since the early stages of the research [24–26].

In contrast with the conventional phonon-mediated superconductivity for which *ab initio* calculation based on density functional theory (DFT) plays a crucial role [27, 28], construction of low-energy models with few degrees of freedom is critically important for unconventional superconductivity since a detailed analysis of the correlation effects is mandatory. In the standard approach to derive a low-energy effective model from first principles, we first calculate the electronic structure with the local density approximation (LDA) or the generalized gradient approximation (GGA) in the framework of DFT. We then construct the maximally localized Wannier function (MLWF) [29, 30] for the low-energy states around the Fermi level and derive a tight-binding model. Next, we calculate the effective Coulomb interaction by the constrained random phase approximation (cRPA) [31, 32]. The matrix elements of the (partially) screened interaction are calculated for the Wannier basis, from which we estimate the Hubbard *U* and Hund coupling *J* in the multi-orbital Hubbard model [18, 24, 33, 34]. The cRPA is formulated in such a way that RPA calculation for the derived low-energy effective model reproduces a one-shot GW (*G*_{0}*W*_{0}) result [31, 32, 35].

To improve the accuracy of the parameters in the low-energy model, we can replace the Green’s function (*G*_{0}) constructed from the DFT/LDA eigenenergies with the dressed Green’s function in the GW approximation (GWA).^{1} Such a derivation based on the GWA has been recently performed for the celebrated cuprate superconductors [37, 38]. While two types of orbitals, i.e., the Cu 3*d* and O 2*p* orbitals, form low-energy bands near the Fermi level, the GW self-energy correction increases the energy difference between the *d* and *p* orbitals and reduce the bandwidth of the *d* band. With these modifications, it has been shown with an extensive variational Monte Carlo (VMC) calculation that the experimental values of the Mott gap and magnetic moment of La_{2}CuO_{4} are successfully reproduced [38, 39]. Given that the differences in the band structure between the DFT/LDA and that in the GWA are commonly seen in transition metal oxides where 3*d* and 2*p* orbitals with different correlation strengths coexist near the Fermi level, it would be of great interest to derive an effective low-energy model for infinite-layer nickelates based on the GWA.

In this study, we perform a first-principles derivation of the effective model for infinite-layer nickelates. In particular, we mainly focus on the *dpp* three-orbital models (single-orbital model is discussed in Appendix) because it is interesting to investigate how the GWA modifies the charge-transfer energy and correlation strength compared to the LDA-based downfolding.^{2} First, we calculate the band structure in the DFT/LDA and estimate the parameter of the effective model using the MLWF and cRPA technique. Next, we calculate the band structure in the GWA using the Green’s function of the LDA. We derive the effective model from the GW band structure and compare the results with those obtained from the LDA. We find that the GWA-based effective model is predicted to be more strongly-correlated with enhanced Mott-Hubbard type character. The model offers an interesting reference to be compared with that of the cuprates with the charge-transfer type character.

## Methods

In this study, we calculate the parameter of the Hubbard Hamiltonian for the low-energy degree of freedom,

Here, the hopping term is represented by

where *H* is the Hamiltonian in the LDA or GWA and *ϕ*_{ℓR} is the MLWF of the *ℓ*th orbital localized at the unit cell ** R**. The interaction term is given by

where *W*^{H} is the effective interaction for the low-energy degree of freedom,

We calculate the effective interaction from the one-shot GWA band. In the one-shot GWA, we calculate the self-energy from the Green’s function *G* and the fully-screened interaction *W*,

where *W* is calculated from all the polarizations in the RPA *P* as follows,

The quasiparticle approximation of the Hamiltonian in the GWA is expressed as

where *H*^{LDA} is the Hamiltonian in the LDA, *V*^{xc} is the exchange correlation potential in the LDA, and *Z*(*ϵ*^{LDA}) is the renormalization factor of Σ at the eigenenergy *ϵ*^{LDA}:

We calculate the electronic band structure of the YNiO_{2} using the experimental lattice parameters of LaNiO_{2}, where *f* orbital, here we use Y as the cation. The computational conditions for the DFT/LDA and GW are as follows. The calculation is based on the full-potential linear muffin-tin orbital implementation [70]. The exchange correlation functional is obtained by the local density approximation of the Ceperley-Alder type [71]. We neglect the spin-polarization. The self-consistent LDA calculation is done for the 12 × 12 × 12 *k*-mesh. The muffintin (MT) radii are as follows: *l* = 4 for all the atoms.

The cRPA and GW calculations use a mixed basis consisting of products of two atomic orbitals and interstitial plane waves [72]. In the cRPA and GW calculation, the 6 × 6 × 6 *k*-mesh is employed for YNiO_{2}. we interpolate the mesh using the tetrahedron method to treat the screening effect accurately [73, 74]. We disentangle the target band from other bands when the target band crosses another band and construct orthogonalized two separated Hilbert spaces [75]. We include bands about from −25 to 120 eV for calculation of the screened interaction and the self-energy.

## Result

Figure 1 shows the crystal structure of the infinite-layer nickelates. The block layer is a single lanthanide cation and has large interstitial regions surrounded by cations. This is one of the reasons for the formation of electron pockets originating from the block layer, as described below.

Figure 2 shows the band structure of YNiO_{2} in the LDA. The band structure of YNiO_{2} is very similar to that of NdNiO_{2} if we eliminate the Nd 4*f* bands. The *d* bands are almost fully occupied. However, differently from the cuprates, the infinite-layer nickelates have additional small electron pockets around the Γ and A points. These electron pockets originate from the *d*-orbital and the interstitial state in the block layer, respectively. The energy difference between the 3*d* bands of Ni^{1+} and the 2*p* bands of O^{2−} is larger than that between Cu^{2+} and O^{2−} in copper oxides, and they are energetically separated near −3 eV.

**FIGURE 2**. **(A)** DFT/LDA band structure for YNiO_{2} and **(B)** its magnified figure. The zero energy corresponds to the Fermi level.

The interstitial state is located at − 1.4 eV at the *A* point, and has a band inversion between *yz*/*zx* orbitals around the *A* point. Because of the inversion between bands with different numbers of degeneracies, the bands of the interstitial *s* and the *yz*/*zx* are continuously connected from the conduction band to the valence band. Since this band inversion is buried in the metallic band, it will be difficult to observe the surface state associated with the band inversion.

In this paper, we derive a three-orbital effective model consisting of the Ni *p* orbitals forming a *σ*-bonding. We first construct the maximally localized Wannier functions [29, 30] for these orbitals and evaluate the parameters in the tight-binding model (see Table 1 and Figure 3). The obtained model has a larger energy difference between the *p* orbitals than that of the cuprate, and is closer to the Mott-Hubbard type.

**TABLE 1**. Transfer integrals and effective interactions in the three-orbital Hamiltonian for YNiO_{2} (in eV). Both the one- and two-body part of the Hamiltonian are constructed based on the LDA band structure. *v*, *U* (0), *J*_{v}, and *J* (0) represent the bare Coulomb, the static values of the effective Coulomb, bare exchange interactions, and exchange interactions, respectively (at *ω* =0). The index “*n*” and “*nn*” represent the nearest unit cell (1,0,0) and the next-nearest unit cell (1,1,0), respectively.

**FIGURE 3**. Electronic band structure of the three-orbital model in the LDA (solid lines). The zero energy corresponds to the Fermi level. For comparison, the band structures in the LDA is also given (dotted lines).

We then calculate the effective interaction for the three-orbital model by the cRPA method. The obtained effective interactions are summarized in Table 1. The bare Coulomb interaction *v* is slightly smaller than that of the copper oxides (Ni *U*/*v* is smaller than that of the copper oxides partially due to the metallic screening from the block layer.

We next show the band structure in the GWA in Figure 4. In the GWA, the energy difference between the strongly correlated Ni 3*d* orbitals and the weakly correlated O 2*p* orbitals is enhanced [33, 76]. Thereby, the energy gap between the *d*- and *p*-bands around −3 eV is increased. On the other hand, the bandwidth of the antibonding orbitals of the *p* orbitals to the antibonding orbitals decreases due to the increase in the energy difference between the *d*- and *p*-orbitals. The bandwidth of the strongly correlated orbitals in the GWA is also reduced compared to that in the LDA due to the effect of the frequency dependence of the self-energy. The bandwidth of the O 2*p* orbitals remains approximately the same as that in the LDA.

**FIGURE 4**. **(A)** GW band structure for YNiO_{2} and **(B)** its magnified figure (solid lines). For comparison, the band structures in the LDA is also given (dotted lines). The zero energy corresponds to the Fermi level.

In the GWA, the position of the valence band is lifted up from that in the LDA. In particular, the electron pocket originating from the *d* orbital in the block layer near the Γ point disappears. On the other hand, the bottom of the band originating from the interstitial state still creates the electron pocket around the *A* point even in the GWA.

We derive the three-orbital model (see Figure 5) and summarize the hopping parameters in Table 2. The difference in the on-site potential between the *p* orbitals increase from 3.98 to 4.60 eV. The nearest-neighbor hopping between the *p* orbitals is almost the same (*p* orbitals results in an decrease of the oxygen contribution to the antibonding

**TABLE 2**. Transfer integrals and effective interactions in the three-band Hamiltonian for YNiO_{2} (in eV). The one-body part is obtained from the GW band structure, and the effective interaction is the result of the cRPA calculation for the GW bands. *v*, *U* (0), *J*_{v}, and *J* (0) represent the bare Coulomb, the static values of the effective Coulomb, bare exchange interactions, and exchange interactions, respectively (at *ω* =0). The index “*n*” and “*nn*” represent the nearest unit cell [1,0,0] and the next-nearest unit cell [1,1,0] respectively.

**FIGURE 5**. Electronic band structure of the three-orbital model in the GWA (solid lines). The zero energy corresponds to the Fermi level. For comparison, the band structures in the GWA is also given (dotted lines).

The screening effect of the system is reduced compared to that in the LDA mainly due to the increase of the charge-transfer energy, which increases the bare Coulomb interaction of the *p* bands. The bands originating from the block layer as well as the O 2*p* orbitals in the GWA move away from the Fermi level compared to the LDA, which makes the screening effect weaker. The disappearance of the metallic screening from the electron pocket at the Γ point also partially contribute to the reduction of the correlation. Therefore, the value of the effective interaction is increased from that in the LDA. For example, while the on-site interaction is 4.6 eV for the *p* orbital in the LDA-based cRPA calculation, the GWA-based cRPA gives 5.0 eV for the *p* orbital. The nearest-neighbor interactions also increase from 0.16 to 0.22 eV for the *A* point still remains even in the GWA.^{3}

## Conclusion

We derived a three-orbital low-energy model for the infinite-layer nickelates based on the GWA. In the GWA, the O 2*p* bands locate deeper below the Fermi level, and the bandwidth of the Ni *ab initio* downfolding gives a more correlated model than the LDA-based downfolding.

## Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

## Author Contributions

MH conducted calculations. All authors contributed to writing the article.

## Funding

We acknowledge funding through Grant-in-Aids for Scientific Research (JSPS KAKENHI) (Grant Nos 20K14423 (YN), 21H01041 (YN), and 19H05825 (RA)) and “Program for Promoting Researches on the Supercomputer Fugaku” (Basic Science for Emergence and Functionality in Quantum Matter —Innovative Strongly-Correlated Electron Science by Integration of “Fugaku” and Frontier Experiments) (Grant No. JPMXP1020200104) from MEXT.

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The reviewer AF declared a shared affiliation, with no collaboration, with the authors, RA, MH, to the handling editor at time of review.

## Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

## Acknowledgments

We thank Terumasa Tadano and Motoharu Kitatani for valuable discussions.

## Footnotes

^{1}It should be noted that although the cRPA method is free from the double counting problem for the interaction parameters, we have to apply the constrained GW (cGW) method to avoid the double counting in the self-energy [36].

^{2}We note that there are several other effective models for infinite-layer nickelates that have been discussed, including a multi-band model that includes 3*d* orbitals other than the 3*d*_{x2−y2} orbital [40–52], a model that includes the contribution of rare-earth 4*f* electrons [53–56], and a model that includes the self-doping bands [57–59]. Here, we focus on the debate [40, 57, 60–67] on the classification of the Mott-Hubbard or charge-transfer regimes in Zaanen-Sawatzky-Allen phase diagram [68].

^{3}We note that there is a proposal that the electron pocket at the *A* point can be eliminated by designing a different type of the block layer [33].

## Appendix

For reference, we summarize the parameters in the single-orbital model in Tables A1 and A2. Special attention should be paid to the strength of the interaction in the GWA-based effective single-orbital model (see Ref. [38]). In the copper oxides, the correlation effect beyond the RPA between the *d* and *p* orbitals in the three-orbital model is not small. Therefore, in order to calculate the single-orbital model accurately, it is necessary to treat the screening effect originating from the bonding and nonbonding bands beyond the RPA. To do so, we need to solve the three-orbital model once with a low-energy solver such as the VMC and estimate the energy corrections between the *d* and *p* orbitals beyond the GWA. By combining such a correction with the GW self-energy correction, we can calculate the band structure beyond the GWA, and can estimate a single-orbital model with high accuracy (See Ref. [38] for details of the method). Because the nickelates have a qualitatively similar band structure to the cuprates, the reliability of the GWA-based single-orbital model for the nickelates also needs to be carefully examined: in particular, the correlation strength |*U*/*t*| might be overestimated.

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Keywords: nickelate superconductivity, density functional theory, GW approximation, ab initio downfolding, multi-orbital Hubbard model

Citation: Hirayama M, Nomura Y and Arita R (2022) *Ab Initio* Downfolding Based on the GW Approximation for Infinite-Layer Nickelates. *Front. Phys.* 10:824144. doi: 10.3389/fphy.2022.824144

Received: 29 November 2021; Accepted: 17 January 2022;

Published: 23 February 2022.

Edited by:

Junjie Zhang, Shandong University, ChinaReviewed by:

Atsushi Fujimori, Waseda University, JapanLiviu Chioncel, University of Augsburg, Germany

Copyright © 2022 Hirayama, Nomura and Arita. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Motoaki Hirayama, hirayama@ap.t.u-tokyo.ac.jp