The emergence of antiferromagnetic correlations and Kondo-like features in a two-band model for infinite-layer nickelates

We report a determinant quantum Monte Carlo study of a two-band model, inspired by infinite-layer nickelates, focusing on the influence of interlayer hybridization between $3d_{x^2-y^2}$ orbitals derived from Ni (or Ni and O) in one layer and rare-earth ($R$) 5d orbitals in the other layer, hereafter the NI and $R$ layers, respectively. For a filling with one electron shared between the two layers on average, interlayer hybridization leads to"self-doped"holes in the Ni layer and the absence of antiferromagnetic ordering, but rather the appearance of spin-density and charge-density stripe-like states. As the interlayer hybridization increases, both the Ni and $R$ layers develop antiferromagnetic correlations, even though either layer individually remains away from half-filling. For hybridization within an intermediate range, roughly comparable to the intralayer nearest-neighbor hopping $t_{\text{Ni}}$, the model develops signatures of Kondo-like physics.


I. INTRODUCTION
The recent discovery of superconducting infinite-layer nickelates [1] has sparked an inquiry into the relationship between the fundamental physics in infinite-layer nickelates and in cuprates concerning the superconducting mechanism.Various microscopic models have been proposed to capture essential features of infinite-layer nickelates, and a consensus has focused on the role of multiorbital physics [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].However, controversy still exists over which orbitals matter in a minimum model.Given the similar crystal and electronic structures of cuprates and infinite-layer nickelates, it is natural to adapt the effective low-energy model of curprates to describe nickelates, incorporating modifications to account for observed experimental discrepancies between nickelates and cuprates.Several studies have explored the superconducting mechanism in doped nickelates [3,13].Unlike in cuprates, where doped holes enters into oxygen sites facilitating Zhang-Rice singlet formation [20], in nickelates, it is proposed that doped holes enter into the Ni 3d orbitals, which leads to the exploration of a coupling between the localized moment in 3d x 2 −y 2 orbital and the doped hole in 3d z 2 orbital, with the emergence of Kondo resonance under mean-field calculations [13].The observed upturn behavior of resistivity in the undoped parent compounds and lightly doped nickelates, along with the logarithmic temperature dependence of both resistivity and Hall coefficient at the intermediate temperatures [1,[21][22][23], provides clear evidence of magnetic Kondo scattering, which supports the inclusion of Kondo coupling to cuprate-like models [12].Based on the absence of antiferromagnetic long-range order in the parent compounds, attention has also been given to the self-doping effect brought by rare-earth orbitals, which emerges proposals of many cupratelike models featuring the coupling between R 5d orbital and Ni 3d x 2 −y 2 orbital [5,8,12,14,24], or Ni 3d x 2 orbital [25][26][27].
When focusing on the undoped parent compounds or lightly-doped nickelates, the minimal two-band model, with one 3d x 2 −y 2 band representing physics within the NiO 2 plane and the other featuring itinerant electrons from the rare-earth elements, has shown to be a strong candidate.Previous density functional theory (DFT) [8,14,24] and density-matrix renormalization group (DMRG) [28] studies on this minimal two-band model encourage the view that nickelates are distinct from the cuprates in terms of the specific electronic structure and potential charge density wave (CDW) behavior.The distinction is exemplified by the presence of carriers in both the Ni 3d x 2 −y 2 and rare-earth 5d layers at a filling of one electron shared between the Ni and R layers per site, and when moderately doped, as well as the emergence of charge density waves with a periodicity locked between the NiO 2 and rare-earth layers [28].These numerical results have a close correspondence to observations from various spectroscopic experiments [29][30][31][32][33].
In this series of studies, the key to explaining the differences between nickelates and cuprates lies in the interlayer hybridization between the Ni layer and the R layer: it leads to "self-doping", with holes entering into the Ni layer and electrons into the R layer, which may partially explain why no antiferromagnetic order has been observed experimentally in the parent compounds.However, the small hybridization between Ni 3d x 2 −y 2 and R 5d orbitals, as estimated by DFT calculations [24], does not account for the Kondo effect observed in exper-iments [1] within this two-band model.Also, some DFT calculations indicate that the self-doping effect and hybridization between Ni 3d and R 5d bands are reduced by increasing the doping level [14,15,17].Nevertheless, studies [34,35] suggest that the hybridization between Ni 3d and interstitial s orbitals can significantly enhance the hybridization between the NiO 2 plane and itinerant electrons.Furthermore, this hybridization could be further enhanced by rare-earth substitution and/or pressure effects [24,36,37].In light of this complex scenario, conducting model studies on this two-band model under various levels of hybridization could be insightful.
In this paper, we focus on the influence of interlayer hybridization on the two-band Hubbard model, with selfdoping of 1/8 holes in the Ni layer and 1/8 electrons in the R layer.For small interlayer hybridization, this self-doping leads to the absence of long-range antiferromagnetic correlations, but rather the emergence of spin and charge stripe-like features, phenomena supported by our numerical measurements.For larger interlayer hybridization, the role of the R layer may go beyond that of just an electron reservoir, potentially leading to the involvement of Kondo-like physics [12,38,39].A shift in the predominantly R band, eventually crossing the Fermi level, leaves the system in an antiferromagnetic insulating state.Antiferromagnetic correlations start to develop in both layers when the interlayer hybridization becomes comparable to the intralayer nearest-neighbor hopping t Ni , resulting in a Kondo-like coupling close to the strength of intralayer Heisenberg coupling in the Ni layer, as predicted by perturbation theory.A notable suppression in the antiferromagnetic spin susceptibility, as well as in the uniform d-wave pairing susceptibility mediated by spin fluctuations, has been observed near the cross-over from spin stripes to the antiferromagnetic state.Such non-monotonic behaviors are themselves suggestive of underlying Kondo-like physics.

II. MODEL AND METHODOLOGY
We consider a simplified bilayer two-orbital Hubbard model, which resembles a low-energy effective model obtained from Wannier downfolding the DFT-derived band structure for RNiO 2 [8,24].The Hamiltonian can be written as Here i,σ ) denotes the creation (annihilation) operator of an electron with spin σ (σ =↑, ↓) at site i within the ℓ (ℓ = Ni, R) layer, and n i,σ is the electron density.The overall electron concentration is defined as n = ℓ n ℓ , where i,σ ⟩ and N is the number of sites in a single layer.t ℓ and t ′ ℓ denote the hopping integrals between nearest-neighbor sites and next-nearest-neighbor sites in the ℓ layer, respectively, and t Ni−R is the hopping integral between the two layers.The parameters U Ni and µ ℓ are described in the following paragraph.
We perform numerically exact determinant quantum Monte Carlo (DQMC) simulations [40] on N = 16 × 4 rectangular clusters (and 2 layers) with periodic boundary conditions along the x and y directions.In this paper, we measure all energies in the unit of t Ni and most other parameters, such as t R = 0.5 and t ′ Ni = −0.25, are extracted from Wannier downfolding of the RNiO 2 bandstructure as in Refs.[8,24].We consider the local (on-site) Coulomb interaction in the Ni layer to be U Ni = 6.µ ℓ is the chemical potential that controls the electron densities and the on-site energy difference between the R and Ni layers, While in the undoped parent compound NdNiO 2 , the electron concentration in the R layer is approximately 8% as determined by Hall coefficient measurements [23], we choose a commonly selected hole concentration of 12.5% in the Ni layer and 12.5% electrons concentration in the R layer to facilitate comparison with measurements in single-band Hubbard model [41,42].
In this study, we aim to explore the dependence of various physical properties on the interlayer hopping t Ni−R , considering a range of values from 0.1t Ni (typical Ni 3d x 2 −y 2 and R 5d hybridization from DFT calculations [8,24]), to 2t Ni (considering Ni 3d and interstitial s hybridization [34]) and even higher values.Up to 80 independently seeded Markov chains and 250000 measurement sweeps are used for each parameter set.The smallest average sign encountered in our simulations is 0.117 for inverse temperature β = t Ni /k B T = 6 and t Ni−R ∼ 0.8.Error bars displayed in DQMC results represent a range that extends one standard error above and below the mean values, as determined by jackknife resampling.

A. Spin Susceptibility
Our desire to understand how the strong interactions of electrons are affected by the interlayer hybridization with itinerant electrons has motivated us to investigate the spin and charge density waves, which were proposed to exist in the doped single-band Hubbard model [41,42].
The dynamical spin susceptibility relates to the imag- inary time spin correlation functions by where χ ℓ (q, τ ) = 1 N i,j e iq•(ri−rj ) ⟨ŝ z iℓ (τ )ŝ z jℓ (0)⟩.Here we only consider the z-component of spin ŝz i↓ ) due to SU (2) symmetry.In the following, we will discuss the in-plane static spin susceptibilities χ ℓ (q, ω = 0) = β 0 dτ χ ℓ (q, τ ) in the ℓ layer (ℓ = Ni, R) to study the low-energy magnetic behavior [43].Here, we note that within the range of t Ni−R that we explored, the features observed in the equal-time spin correlation function are consistent with those of the static spin susceptibility.
Figure 1 displays the real-space staggered static spin susceptibilities within each layer at finite temperatures, i.e., χℓ (r) = (−1) rx+ry χ ℓ (r), with r = (r x , r y ), as well as their momentum-space counterparts, for different values of the interlayer hybridization t Ni−R .In Fig. 1a, for small values of t Ni−R , spin stripes -antiferromagnetic regions separated by anti-phase domain walls -are found to exist in the Ni layer.It closely resembles the behavior of a doped single-band Hubbard layer, with the periodicity of the antiphase domain walls in the Ni layer matching that of the 1/8 hole-doped single-band Hubbard model, as found in Ref. [41,42].In contrast, the R layer closely resembles the spin susceptibility for a non-interacting single band with the same t Ni−R .As t Ni−R increases, the AF domains of χNi (r) widen, and the peak of χ Ni (q) becomes sharper, as shown in Figure 1c, indicating an increase in quasi-long-range spin correlations within the Ni layer.
As shown in Fig. 1b and 1d, the static spin susceptibility in the R layer transitions from a non-interacting pattern to a quasi-long-range antiferromagnetic pattern for t Ni−R > 1.4 (see more detailed data in Supplementary Note 1).This transition shifts to smaller t Ni−R as the temperature decreases.As illustrated in Fig. 2, while the R layer with t Ni−R = 1.2 at temperature T = 1/2 displays notable non-interacting features, it is dominated by uniform antiferromagnetic correlations as the tempera- ture decreases to T = 1/6.There is a crossover between spin stripe and uniform antiferromagnetic correlations, which curves downward towards smaller t Ni−R as temperature decreases.Charge susceptibility measurements in Supplementary Note 1 suggest that charge-stripe behavior, which dominates the Ni layer as t Ni−R is turned off, gradually disappears with increasing t Ni−R .

B. Transport Properties
To further investigate the mechanism by which these two layers become antiferromagnetically correlated at large t Ni−R , we next study the change in transport properties.To distinguish between metallic and insulating behavior, we can use the density of states through the single-particle spectral function A(ω).The imaginary time Green's function and analytic continuation can be used to determine the single-particle density of states N (ω) = k A(k, ω)/N from the imaginary-time QMC data.To avoid numerical complications associated with analytic continuation, we can estimate a low temperature approximation The imaginary time current-current correlator Λ(τ ) at zero momentum can be obtained by measuring where Ĵ = i ijσ t ij (r i − r j )ĉ † iσ ĉjσ .To obtain the dc conductivity σ dc from the imaginary time response, we can either perform analytic continuation to compute ImΛ(ω) for all frequencies, which is related to Λ(τ ) through the following relation: or estimate the low frequency behavior of ImΛ xx ≈ ωσ dc at sufficiently low temperatures from [44] Λ As Fig. 3a shows, the dc conductivity and the proxy for the density of states at the Fermi level for both layers manifest a decrease as t Ni−R increases at T = 1/3.As expected, both layers are metallic as long as the interlayer hybridization remains small.In the range where the R layer goes from non-interacting to antiferromagnetic (t Ni−R = 1.2 ∼ 1.6), the R layer becomes an insulator, while the metallicity of the Ni layer exhibits a non-monotonic dependence on temperature, initially increasing with decreasing T before eventually decreasing.When t Ni−R is large enough, e.g., t Ni−R = 4, both layers are unambiguously insulators.
The momentum dependence of βG ℓ (k, β/2)/π with ℓ = Ni, R along high-symmetry cuts is shown in Fig. 3b, changing with t Ni−R .As the proxy of the density of state at the Fermi energy, it encodes the shape of the Fermi surface.At small t Ni−R , the proxy suggests that the R layer is likely to be a conduction band, while the Ni layer closely resembles the Hubbard band.At large t Ni−R , as shown in the inset, the proxies for both layers are similar in shape and differ by a constant factor.

C. Kondo Screening Regime
To explain the previous observations, we propose a heuristic understanding for the small and large interlayer hybridization regimes.When t Ni−R is small, all electrons in the R layer (accounting for 1/8 of the electrons) are itinerant, while 7/8 of the electrons in the Ni layer participate in forming spin stripes.For sufficiently strong interlayer hybridization, both a bondingand anti-bonding-like band form, and 'layer' no longer serves as a conserved index because both interacting and non-interacting bands contribute to the Ni layer and R layer.The anti-bonding-like band is well above the Fermi energy, requiring the bonding-like band to be half-filled in order to maintain the overall electron density.This bonding-like band tends to be gapped, and consequently both layers become insulating.However, we will show that the system displays distinct behaviors in the intermediate t Ni−R regime, necessitating an understanding in this specific scenario.
In this section the nature of d−wave superconducting correlations in conjunction with antiferromagnetic correlations is explored in this model.The operator for uniform d-wave pairing in layer ℓ is commonly expressed as with the pair-field susceptibility as Its disconnected part is denoted as Pd,ℓ .To reveal the dominant pairing interaction in the system, we compute the pairing vertex Γ using the Dyson-like expression Γ = P −1 − P −1 .We measure the pairing estimator λ * d,ℓ = Γ d,ℓ Pd,ℓ to represent d-wave pairing in layer ℓ. observations in Fig. 3.

IV. DISCUSSION
A clear cross-over from spin and charge stripe-like behavior to uniform antiferromagnetic correlations in the interacting Ni layer occurs as one increases the interlayer hybridization t Ni−R in the model.Even the noninteracting R layer develops uniform antiferromagnetic correlations as the interlayer hybridization becomes large enough for a given set of parameters.Focusing on the region near the cross-over, the antiferromagnetic spin susceptibility, as well as the proxy of uniform d-wave pair-field susceptibility (−λ * d,Ni ), exhibit a suppression in the Ni layer, where an effective (perturbative) interlayer Kondo coupling J K ∼ 2t 2  Ni−R /U Ni becomes competitive with an effective (perturbative) intralayer Heisenberg coupling J H ∼ 4t 2 Ni /U Ni .Such a scenario may lead to screening of the electrons in the Ni layer by all of the available itinerant electrons in the R layer, resulting in the formation of Kondo singlets and quenching the itinerant charge degrees of freedom.Our observations of the dramatic decrease in conductivity in the R layer, the suppression in intralayer antiferromagnetic susceptibility and d-wave pair fields induced by magnetic fluctuations, and more importantly, an enhancement of the on-site interlayer spin-spin correlation, support this hypothesis.For sufficiently strong t Ni−R , the Hubbard interaction projected into the half-filled bonding-orbital band induces strong antiferromagnetic correlations and insulating behavior.
We observe a two-fold degenerate ground state at the n = 1 sector.We find that the magnetic field lifts the degeneracy of the ground state, so it leads to the ground state with a positive ⟨ Ŝz R ⟩ having the largest Boltzmann weight, which results in a positive ensemble average of S z R .Therefore, we conclude that magnetic field favors a spin-up electron in the Ni layer.Focusing on the n = 1 sector, the eigenstate with |ψ 0 ⟩ = 0.936 ↑ 0 + 0.352 0 ↑ emerges as having the lowest eigenvalue, which leads to a positive value of on-site interlayer spin susceptibility χ Ni−R (r = (0, 0)).The other notable observation is that χ Ni−R (r = (0, 0)) calculated through Lehmann representation in the system without perturbative applied field is also positive at all finite temperatures.
In conclusion, the single-site ED analysis offers a complementary explanation for the distinct behaviors of the on-site interlayer spin correlation S Ni−R (r = (0, 0)) and spin susceptibility χ Ni−R (r = (0, 0)).Specifically, the eigenstates with two electrons per site collectively yield a negative value of S Ni−R (r = (0, 0)), while the eigenstates with one electron per site result in a positive value of χ Ni−R (r = (0, 0)).Essentially, this difference arises from the fact that the equal-time correlation is more affected by high-energy excitations than the static susceptibility.
Focusing solely on the ground state, both ⟨ Ŝz Ŝz R ⟩ are zero.The fourth row of the table demonstrates that the eigenstates in the n = 2 sector contribute to a negative value to ⟨ Ŝz Consequently, the ensemble average of S z Ni S z R is negative, while that of S z R remains zero.Herein, we introduce a perturbative magnetic field h Ni to the Ni layer.We then observe the resulting spin magnetization in the R layer, ⟨ Ŝz R ⟩, and then the spin susceptibility between the Ni layer and the R layer can be calculated byχ Ni−R = ∂⟨ Ŝz R ⟩ ∂hNi (it is equivalent to χ Ni−R (ω = 0) = β 0 dτ ⟨ Ŝz Ni (τ ) Ŝz R(0)⟩ as defined in the main text based on fluctuation-dissipation theorem).The following table displays the eigenstates and their corresponding physical observables with h Ni = 0.1: