## Abstract

The recently discovered infinite-layer nickelates show great promise in helping to disentangle the various cooperative mechanisms responsible for high-temperature superconductivity. However, lack of antiferromagnetic order in the pristine nickelates presents a challenge for connecting the physics of the cuprates and nickelates. Here, by using a quantum many-body Green’s function-based approach to treat the electronic and magnetic structures, we unveil the presence of many two- and three-dimensional magnetic stripe instabilities that are shown to persist across the phase diagram of LaNiO_{2}. Our analysis indicates that the magnetic properties of the infinite-layer nickelates are closer to those of the doped cuprates, which host a stripe ground state, rather than the undoped cuprates. The computed longitudinal-spin, transverse-spin, and charge spectra of LaNiO_{2} are found to contain an admixture of contributions from localized and itinerant carriers. Theoretically obtained dispersion of magnetic excitations (spin-flip) is found to be in good accord with the results of recent resonant inelastic X-ray scattering experiments. Our study gives insight into the origin of strong magnetic competition in the infinite-layer nickelates and their relationship with the cuprates.

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## Introduction

The common thread linking the family of high-temperature superconductors is that competing interactions involving charge, spin, lattice, and orbital degrees of freedom conspire with electronic correlation effects to produce many complex properties in this materials family^{1}. The phase diagrams of transition-metal oxides, for example, are astonishingly complex and exhibit unconventional superconducting pairing, pseudogap, and glassy phases, and colossal magnetoresistance in stark contrast to the standard metals^{2,3,4,5}. The high-T_{c} cuprates have been of intense interest, where a wide variety of experiments have reported the presence of competing and intertwined inhomogeneous orders that could contribute to the pairing mechanism^{6,7,8,9}. However, deconstructing the mechanism of high-Tc superconductivity and possible contributions involved in the pairing process has remained a challenge. Understanding the electronic and magnetic properties of related materials can provide insight into the mechanism of superconductivity in the cuprates as well as other high-Tc superconductors.

Recently, superconductivity was discovered in doped infinite-layer nickelates^{10}. The *R*NiO_{2} (*R* = Nd,Pr,La) family of compounds is isostructural to the infinite-layer parent cuprate CaCuO_{2}^{11}, where the two-dimensional (2D) NiO_{2} planes are separated by rare earth spacer layers^{12,13,14}. Due to the missing apical oxygens in *R*NiO_{2}, the nickel atoms take a 3*d*^{9} configuration that is equivalent to Cu^{2+} in the cuprates, thereby strengthening the hypothesis that these two materials’ families are electronically analogous^{15,16}.

A number of experimental techniques have been employed to elucidate possible connections between nickel and copper-based superconductivity and the role of competing orders in their phase diagrams. Transport measurements on the nickelates report key departures from the cuprate phenomenology. Specifically, no Mott or antiferromagnetic (AFM) parent phase is observed, with both the underdoped and overdoped nickelates only displaying a weakly insulating state^{13,14,17,18,19,20,21,22,23,24,25}. The Hall coefficient *R*_{H} is found to change sign at optimal doping (*x* ~ 0.17) in all nickelates that have been investigated, signaling the existence and importance of both hole and electron pockets at the Fermi level^{13,14,18,22,23,24,25,26,27}. X-ray absorption spectroscopy (XAS) and resonant inelastic X-ray scattering (RIXS) experiments find the doped holes to reside on the Ni-\({d}_{{x}^{2}-{y}^{2}}\) orbitals, with minor 5*d* electron doping due to Ni-3*d*/La-5*d* hybridization^{18,28,29,30}, suggesting the coexistence of multiple active orbitals at the Fermi level. Despite these differences, the nickelates exhibit signatures of stripe formation and singlets in the ground state, similar to the underdoped curpates^{14,30,31,32}.

The magnetic properties of the nickelates differ substantially from those of the cuprates in that the pristine undoped cuprates exhibit commensurate AFM order. In sharp contrast, no long-range AFM order is found in *R*NiO_{2}^{33,34,35,36}, despite the existence of robust two-dimensional (2D) AFM spin wave dispersions observed in RIXS^{37,38}. Instead, strong non-local magnetic correlations and weak to intermediate glassy short-range behavior appear to dominate the ground state^{35,39,40}. Moreover, a recent muon spin rotation/relaxation experiment finds the infinite-layer nickelates to be intrinsically magnetic and further provides direct evidence for the coexistence of superconductivity and magnetism, with phase separation only possible on the nano scale^{41}. This suggests that the magnetic properties of the infinite-layer nickelates are closer to those of the doped cuprates, where inhomogeneities comprise the ground state rather than the undoped cuprates, which present a pristine ordered phase.

Many density functional theory (DFT)^{11,42,43,44,45,46,47,48} and DFT + dynamical mean-field theory (DMFT)^{28,39,45,49,50,51} studies yield a magnetic ground state in the infinite-layer nickelates that are at odds with the experimental evidence. Dynamical vertex approximation (DΓA) and cellular dynamical-mean field theory (CDMFT) calculations^{39} capture the short-range correlations but do not offer a transparent picture of the instabilities at play. In this connection, a variety of tight-binding models have been invoked^{52,53,54,55,56} to understand the low-energy physics and magnetic instabilities in nickelates. These models, however, are fundamentally limited because the relevant orbitals involved remain unclear.

In this article, we propose that the absence of long-range order in the infinite-layer nickelates originates from a large number of competing symmetry-breaking magnetic instabilities in the pristine phase, which strongly couple at low temperatures as is common in glassy phases of matter. Upon hole doping, these instabilities, originating from the three-dimensionality (3D) of the Fermi surface, pass through a 3D–2D crossover in the magnetic fluctuation spectrum near the onset of the superconducting dome. By using the C-type AFM state as a model system, we simulate the charge and magnetic excitation spectra. By comparing our theoretical spectra to the results from RIXS experiments, we find that the observed dispersion can be reproduced with a renormalization factor of 0.63. Our analysis of the spin-wave spectrum reveals a dominant Ni-\(3{d}_{{x}^{2}-{y}^{2}}\) contribution along with an admixture of contributions from Ni-\(3{d}_{{z}^{2}}\) and La-4*f* orbitals. Our strongly-constrained-and-appropriately-normed (SCAN) meta-GGA-based results are similar to those obtained using DMFT. A two-band model provides an adequate representation of both charge and magnetic instabilities and excitations.

## Results and discussion

### Electronic structure and competing fluctuations

Figure 1a shows the interpolated electronic band structure for LaNiO_{2} in the non-magnetic (NM) phase [Fig. 1b]. In order to conduct an objective analysis of the relevant orbital degrees of freedom at the two-particle level, our atomic orbital model of the NM phase explicitly considers the full set of Ni-3*d*, La-5*d*, and La-4*f* atomic orbitals. We find the quality of the fit to be quite sensitive to the set of included orbitals. Specifically, due to mixing between the Ni and La states within a 2 eV window of the Fermi level, the removal of even a single 4*f* or 5*d* state results in a poor fit of the Ni-\(3{d}_{{x}^{2}-{y}^{2}}\) band at the M and A points in the Brillouin zone.

In the lanthanum-based infinite-layer nickelates, three distinct bands cross the Fermi level: one is of nearly pure Ni-\(3{d}_{{x}^{2}-{y}^{2}}\) character, while the other two are derived from Ni-3*d*_{xy/yz} and Ni-\(3{d}_{{z}^{2}}\)/La-5*d* orbitals. The latter bands give rise to spherical electron Fermi pockets at the Γ and A symmetry points in the Brillouin zone [Fig. 1c], and may be responsible for the experimentally observed metallic behavior of the resistivity at high temperature^{10,12,13,14,24,25,35,39}, along with a negative Hall coefficient^{10,13,14,24,25,35}. In contrast, the Ni-\(3{d}_{{x}^{2}-{y}^{2}}\) band generates a large, slightly warped quasi-2D cylindrical Fermi surface similar to the cuprates. The relatively isolated Ni-\(3{d}_{{x}^{2}-{y}^{2}}\) state is the result of the oxygen de-intercalation process used to convert *R*NiO_{3} to *R*NiO_{2}, thereby reorganizing the electronic states from that of an octahedral crystal field to a square-planar geometry. The half-filled Ni-\({d}_{{x}^{2}-{y}^{2}}\) band closely resembles the corresponding band in the cuprates^{57}, except for an enhanced *k*_{z} dispersion due to the three-dimensional (3D) nature of the crystal structure. This results in a shift in the position of the van Hove singularity (VHS) from below to infinitesimally above the Fermi level along the *k*_{z} direction in the Brillouin zone. Concomitantly, the Fermi surface transitions from being hole-like (open) in the *k*_{z} = 0 plane to becoming electron-like (closed) in the *k*_{z} = *π*/*c* plane.

A key difference between the parent compounds of the cuprates and nickelates is the lack of long-range magnetic order in the nickelates. Instead, strong AFM correlations and glassy dynamics are observed to dominate throughout the phase diagram of *R*NiO_{2}. To gain insight into the landscape of charge and magnetic instabilities in the ground state, we examine the response (*δ**ρ*) of the system to an infinitesimal perturbing source field (*δ**π*). The associated response function is given by,

where the orbital indices have been suppressed, the spin indices (*I*, *J*, *L*, *M*) are given in the Pauli basis, *v*^{ML} is the generalized electron–electron interaction, and the polarizability is defined as:

Here, \(\bar{{{\Gamma }}}\) describes a multiple scattering process involving two quasiparticles with the vertex^{58}. Assuming the material exhibits sufficient screening, the vertex correction \(\bar{{{\Gamma }}}\) can be considered negligible and ignored. Then Eq. (1) can be solved outright, producing a generalized RPA-type matrix equation

Note that in order to solve for *χ*^{IJ}, we have introduced the matrix inverse of \(\left[1-\bar{F}\right]\). Therefore, extra care must be taken when interpreting the response function. For a system exhibiting an ordered phase, e.g. AFM order, the poles \(1-\bar{F}\) generated in the various spin and orbital channels predict bosonic quasiparticles, such as magnons. In a non-ordered system, if *χ*^{IJ}(**q**, *ω* = 0) ≫ 1, then the ground state is unstable to a broken-symmetry phase. The specific charge and spin instabilities of the system can be made transparent by diagonalizing the kernel \(\bar{F}\),

where Λ_{F} is a diagonal matrix, and *V* is the eigenvector. Then

where *α* enumerates the instability ‘bands.’ Now, as the instability strength \({{{\Lambda }}}_{F}^{\alpha }({{{{{{{\bf{q}}}}}}}},\omega =0)\) approaches 1, *χ*^{IJ}(**q**, *ω* = 0) becomes singular, or physically, the ground state becomes unstable to an ordered phase. Additionally, the momentum satisfying \({{{\Lambda }}}_{F}^{{\alpha }_{{{{{\rm{max}}}}}}}({{{{{{{\bf{Q}}}}}}}},\omega =0)=1\), where *α*_{max} is the index of the maximum instability band, is the propagating vector **Q** of the emerging Stoner instability in the multiorbital spin-dependent system. The character of this instability may then be obtained by analyzing the associated eigenvectors, *V*.

Figure 2a presents the momentum dependence of the maximum instability \({{{\Lambda }}}_{F}^{0}({{{{{{{\bf{q}}}}}}}},\omega =0)\) for pristine LaNiO_{2} in the NM phase for various slices along *q*_{z} in the Brillouin zone. The overall peak structure in \({{{\Lambda }}}_{F}^{0}\) follows the folded Fermi surface of the Ni-\(3{d}_{{x}^{2}-{y}^{2}}\) band, with the main ridges displaying minimal *q*_{z} dispersion. The dominant peaks are concentrated around the M − A edge of the Brillouin zone, with weaker satellites along the path connecting the edge and the zone center. The momenta **Q**^{*} of the largest instability (blue and red ‘x’ marks) in each slice is found to evolve along the *q*_{z}-axis, taking positions at (*π* − *δ*, *π*, 0), (*π* − *δ*, *π*, *q*_{z}), (*π* − *δ*, *π* − *η*, *q*_{z}), (*π* − *ξ*, *π* − *ξ*, *q*_{z}), and (*π*, *π*, *π*). These momentum points yield nearly degenerate instability strengths, where only 0.0178 separates the critical momenta in the *q*_{z} = 0 and *q*_{z} = *π*/*c* planes. A similar near degeneracy is found for various in-plane momenta surrounding the M − A edge of the zone [Fig. 2b], with an instability strength difference of 0.0889 and 0.0023 between *Q*_{1} and *Q*_{2} in the Γ and Z planes, respectively. By analyzing the eigenvectors at the various marked **Q**^{*} points, we find magnetic fluctuations to dominate by order of magnitude over the charge sector. Moreover, the transverse and longitudinal magnetic fluctuations are predominantly composed of intra-orbital Ni-\(3{d}_{{x}^{2}-{y}^{2}}\) weight, with additional weak inter-orbital contributions from Ni-\({d}_{{x}^{2}-{y}^{2}}\)/Ni-*d*_{xy/xz/yz} and Ni-\({d}_{{x}^{2}-{y}^{2}}\)/La-4*f* (\(4{f}_{x({x}^{-}3{y}^{2})}\) and \(4{f}_{y(3{x}^{2}-{y}^{2})}\)) hybridization. In contrast, the charge channel is comprised of inter-orbital hybridization between Ni-\(3{d}_{{x}^{2}-{y}^{2}}\) and Ni-3*d*_{xy} primarily, with smaller overlaps from Ni-3*d*_{xz} and Ni-3*d*_{yz} orbitals. An additional weak matrix element is found in the Ni-\(3{d}_{{x}^{2}-{y}^{2}}\)/La-\(5{d}_{{x}^{2}-{y}^{2}}\) channel. These results clearly show that the leading G-type antiferromagnetic instability [**Q**^{*} = (*π*, *π*, *π*)] is virtually degenerate with a dense manifold of 2D and 3D incommensurate magnetic stripe orders. This is consistent with total-energy-DFT and DFT+DMFT results yielding a myriad of nearly degenerate magnetic configurations that lower the total energy with respect to the NM phase^{45,46,47,48,50,53}.

In Fig. 2a, b, the momentum dependence of the maximum instability is found to be quite flat throughout the Brillouin zone, producing a clear pile-up of various magnetic configurations within an infinitesimally small instability strength. To make this statement more precise and accurately count the total number of competing magnetic configurations, we introduce the density of instabilities *λ*(Ω), where *λ*(Ω)*δ*Ω is the number of instabilities in the system whose strengths lie in the range from Ω to Ω + *δ*Ω. That is, *λ*(Ω) is defined as

where Ω is the instability strength and *α* enumerates the instability eigenvalues defined in Eq. (4). We further emphasize that *λ*(Ω) contains the instability information for all eigenvalues, not just for the maximum.

Figure 2c shows the density of instabilities for pristine LaNiO_{2} in the NM phase, along with an inset illustrating the region of origin of the various key features. The spectrum reveals two clear Van Hove-like singularities, one close to the maximum instability (\({\mathbb{I}}\)) and the other in the body just above 0.8 instability strength (\({\mathbb{II}}\)), along with a large step edge at the bottom of the spectrum (\({\mathbb{III}}\)). Furthermore, by decomposing the density of instabilities into the various band contributions, these peaks are found to originate from dense points in **q**-space (Van Hove-like) rather than a clustering of instability “bands”; see Supplementary Note 2 for details. As pointed out by Léon Van Hove^{59} in 1953, the appearance of singular features in the density of states of either electrons or phonons is intimately connected with the topology of the underlying band structure. Here, the presence of these singularities implies the existence of saddle points in the momentum-dependent instability “bands” \({{{\Lambda }}}_{F}^{\alpha }\). For example, peak \({\mathbb{I}}\) originates from the very subtle change of \({{{\Lambda }}}_{F}^{0}\) from being a local minimum at (*π*, *π*, 0) to a global maximum at (*π*, *π*, *π*) [Fig. 2a], implying the existence of a critical \({q}_{z}^{* }\) where the concavity changes sign (saddle point). Peak \({\mathbb{II}}\) emerges from the flat plateau between the instability band edges, marked in the inset as the tips of the four-pointed star. The relative placement of the singularities, along with the step-edge \({\mathbb{III}}\), suggests that the fluctuations in LaNiO_{2} are 3D in nature. However, since peak \({\mathbb{I}}\) is in very close proximity to the maximum instability, the system is very close to a 3D–2D transition.

Much like the presence of a Van Hove singularity near the Fermi level can modify and enhance correlated physics of an interacting electron liquid^{60,61,62,63,64,65,66}, a similar “Van Hove scenario” arises when a saddle point in the density of instabilities nearly fulfills the Stoner criteria, Λ_{F} ~ 1. In the latter case, a large population of charge and magnetic instabilities with different propagating **q**-vectors are able to interact and compete with one another, thus inducing strong correlation corrections to the low-temperature behavior of the system^{67,68}. In line with this scenario, a complimentary study applying DMFT, DΓA, and CDMFT to a single-band Hubbard model for NdNiO_{2} finds the inclusion of vertex corrections to suppress long-range order, leaving strong short-range correlations to dominate down to very low temperatures^{39}. This study gives credence to the important role that the density of instabilities can play in giving way to strong AFM fluctuations, local magnetism, and a pseudogap-like weakly insulating state.

Figure 3 shows the evolution of the momentum dependence of \({{{\Lambda }}}_{F}^{0}\) and the corresponding density of instabilities for LaNiO_{2} under various hole dopings. As hole carriers are added, the Ni-\(3{d}_{{x}^{2}-{y}^{2}}\) Fermi surface sheet expands in volume, gradually reducing and eliminating the electron-like Fermi surface in the *k*_{z} = *π*/*c* plane. Consequently, the areas of regions \({\mathbb{I}}\) and \({\mathbb{II}}\) in \({{{\Lambda }}}_{F}^{0}\) grow with increased hole concentration, as shown in Fig. 2c. Moreover, the concavity of \({{{\Lambda }}}_{F}^{0}\) at (*π*, *π*, *π*) goes from negative to positive around 10% hole doping, as illustrated by the critical momenta (red ‘x’ marks), which change location from (*π*, *π*, *π*) to (*π* − *δ*, *π* − *η*, *π*). This process is reflected in the density of instabilities where peak \({\mathbb{I}}\) transitions from a saddle point (*x* = 0.0) to a step edge (*x* = 0.30). The Van Hove-like singularity thus precipitates an effective dimensionality reduction of the fluctuations from 3D to 2D. Curiously, this transition appears just before the sign change in the Hall coefficient at the start of the superconducting dome^{24}. Finally, at *x* = 0.4, the leading edge of the density-of-instabilities softens, exhibiting the presence of a small number of instabilities and suggesting a severe reduction in the competition between the various magnetic configurations.

The persistence of the peak \({\mathbb{I}}\) at or near the maximum instability edge for both the underdoped and overdoped regimes suggests the preservation of strong competition between the magnetic states, despite the systematic reduction in the fluctuation strength with doping. The 3D to quasi-2D transition in the nature of the fluctuations just before optimal doping makes LaNiO_{2} distinct from the cuprates, which display predominantly 2D fluctuations for all hole dopings. Furthermore, this suggests that 2D magnetic fluctuations, in particular, are important for Cooper pairing, with optimal *T*_{c} arising from the delicate balance between the dimensionality and strength of the fluctuations. In this connection, recent magnetotransport measurements find superconductivity to be quasi-two-dimensional in the infinite-layer nickelates^{69}.

### Spin excitation spectra

So far, we have examined the manifold of possible magnetic states that may emerge from the non-magnetic state of pristine and doped LaNiO_{2}. We now turn to examine the charge and spin excitations that occur within the magnetic state to gain insight into the measured magnetic excitation spectrum^{38,70}.

Figure 4a presents the band structure of LaNiO_{2} in the C-AFM phase. Our model of the C-AFM phase explicitly considers the full set of Ni-3*d*, La-\(5{d}_{{z}^{2}}\), La-5*d*_{xy}, La-\(4{f}_{{z}^{3}}\), and La-\(4{f}_{x({x}^{2}-3{y}^{2})}\) orbitals. Reducing the orbital projections further resulted in a poor fit. The AFM state stabilizes in the Ni-\(3{d}_{{x}^{2}-{y}^{2}}\) band with a gap of approximately 2 eV that opens up around the Fermi energy of the NM system. The partially filled Ni-3*d*_{xy/yz} and Ni-\(3{d}_{{z}^{2}}\) bands remain pinned to the Fermi level, with the Ni-\(3{d}_{{z}^{2}}\) state becoming flat in the Z plane. Unlike the parent cuprate compounds, itinerant excitations are expected in addition to those of the local magnetic moments.

The charge \(-{{{{{{{\rm{Im}}}}}}}}{\chi }^{00}\), longitudinal-spin \(-{{{{{{{\rm{Im}}}}}}}}{\chi }^{zz}\) and transverse-spin (spin-flip) \(-{{{{{{{\rm{Im}}}}}}}}{\chi }^{+-}\) excitation spectra are obtained from the corresponding components of the dynamical response *χ*^{IJ} [Eq. (3)] applied to the magnetic ground state. It is also useful to define the *total spectrum *\({\sum }_{\nu {\nu }^{{\prime} };\mu {\mu }^{{\prime} }}{\chi }_{\nu {\nu }^{{\prime} };\mu \mu {\prime} }^{IJ}\) where all four indices are integrated out and the *maximum spectra tensor *\({\max }_{{{{{{{{\bf{q}}}}}}}},\omega }\left\{\left\vert {\chi }_{\nu {\nu }^{{\prime} };\mu \mu {\prime} }^{IJ}({{{{{{{\bf{q}}}}}}}},\omega )\right\vert \right\}\) for the various charge (spin) components *I*, *J*. Finally, we introduce a scaled unit of energy \(\bar{\omega }={\varepsilon }_{r}\cdot \omega\) to evaluate the excitation energy under renormalization effects, which are common when itinerant carriers are present in an antiferromagnet^{71}. Moreover, *ε*_{r} is inversely proportional to the quasiparticle renormalization factor *Z*^{72}. Comparing our spectrum to the reported RIXS spectra^{38,70}, we find a renormalization *ε*_{r} value of 0.63 (*Z* = 1.58), which suggests LaNiO_{2} to be an intermediately coupled material. Interestingly, this renormalization also corresponds to ~26% hole-doped La_{2}CuO_{4}^{73} and optimally doped BaFe_{2}As_{2}^{74}. We note that the dipole matrix elements of the RIXS measurement process may also be included to capture polarization effects^{75}.

Figure 4b shows the total transverse-spin excitation spectrum along the various high-symmetry lines in the NM Brillouin zone, with the experimental data overlaid^{38,70}. Highly dispersive magnetic excitations are found following the spin-1/2 AFM magnons on a square lattice. Specifically, they disperse strongly with maxima at X and M/2, and linearly soften toward the AFM ordering wave vector at M in very good accord with the experimental spectra. Our theoretical excitations display a gap in the Γ plane compared to those in the plane through Z. This behavior is typical of infinite-layer perovskites, such as CaCuO_{2}^{76}, where a pronounced in- and out-of-plane exchange coupling anisotropy can gap out magnetic excitations. By inspection of our electronic band’s structure, this gap can be attributed to the very small electron pockets at the Fermi level in the Γ-plane compared to those in the Z-plane. Additional small anisotropies within the *x**y*-plane are also noticed.

Figure 4c, d presents the total longitudinal-spin spectrum and the charge spectrum, respectively, along the various high-symmetry lines in the NM Brillouin zone. Both spectra are very similar, exhibiting a broad continuum of excitations. The excitation spectrum is amplified in the *q*_{z} = *π* plane as a consequence of the flat Ni-\(3{d}_{{z}^{2}}\) band at the Fermi level. The incorporation of spin-orbit coupling effects in the electronic structure gives rise to significant mixing between the transverse-spin and charge sectors, as indicated by the non-zero *χ*^{0−} and *χ*^{0+} susceptibility components, see Sec. S3 of Supplementary Note 3 for details, and yield a “shadow” of the spin-flip excitations in the Γ plane and along Γ–Z in *χ*^{00}.

Figure 5a displays the maximum spectra tensor of the transverse-spin susceptibility as a heat map with the \(\nu ,\mu \,({\nu }^{{\prime} },{\mu }^{{\prime} })\) indices unrolled along the horizontal (vertical) axis. To help identify key atomic-site orbital components and asymmetries in the tensor, the left (right) distribution curve—given by the sum of each row (column)—is presented in the right (top) sub-panel. The right and left distribution curves can be thought of as being similar to the spin-density \(({S}_{\mu \nu })\) amplitude, and their overlap \(\langle {S}_{\mu \nu }^{+}{S}_{{\mu }^{{\prime} }{\nu }^{{\prime} }}^{-}\rangle\) is akin to the heat map.

Since the maximum-spectra-tensor heat map is symmetric about the anti-diagonal, the left and right distribution curves are equivalent. The distribution curves are quite sparse, exhibiting only a few prominent peaks in the (Ni-\(3{d}_{{x}^{2}-{y}^{2}}\),Ni-\(3{d}_{{x}^{2}-{y}^{2}}\)) channel, with smaller peaks present in the (Ni-\(3{d}_{{z}^{2}}\),Ni-\(3{d}_{{z}^{2}}\)) and (La-\(4{f}_{x({x}^{2}-3{y}^{2})}\),Ni-\(3{d}_{{x}^{2}-{y}^{2}}\)) components. In the heat map, these peaks appear as ‘hot’ horizontal and vertical lines, with maxima occurring at their intersection points (white arrows). These maxima identify the key atomic-site orbital components of the transverse-spin susceptibility. Specifically, our orbital analysis reveals that the magnetic excitations are dominated by Ni-\(3{d}_{{x}^{2}-{y}^{2}}\,\)\({\chi }_{\mu \mu ;\mu \mu }^{+-}\) character, suggesting that the Ni-\(3{d}_{{x}^{2}-{y}^{2}}\) orbital plays an important role in the local magnetic moment. Ab initio calculations show a similar trend where the Ni-\(3{d}_{{x}^{2}-{y}^{2}}\) contributes 75% to the total Ni magnetic moment^{48}. The corresponding excitation spectrum of the Ni-\(3{d}_{{x}^{2}-{y}^{2}}\) component is given in Fig. 5b.

The sub-dominant spin-excitation components arise as off-diagonal components of the heat map that mix Ni-\(3{d}_{{x}^{2}-{y}^{2}}\) with other Ni-3*d* and Ni-4*f* states. The intra-site \({\chi }_{\mu \nu ;\mu \nu }^{+-}\) term capturing \({}^{1}{{{{{{{\rm{Ni}}}}}}}}-3{d}_{{x}^{2}-{y}^{2}}{\to }^{1}{{{{{{{\rm{Ni}}}}}}}}-3{d}_{{z}^{2}}\) transitions [Fig. 5c] contribute to the low-energy part of the total spectrum near the center Γ(Z) and corner M(A) of the Brillouin zone. In contrast, long-range inter-site \({\chi }_{\mu \nu ;\nu \mu }^{+-}\) \({}^{1}{{{{{{{\rm{Ni}}}}}}}}-3{d}_{{x}^{2}-{y}^{2}}{\to }^{2}{{{{{{{\rm{Ni}}}}}}}}-3{d}_{{x}^{2}-{y}^{2}}\) excitations [Fig. 5e] have the greatest amplitude around 0.20 meV/*ε*_{r} centered around X(R) and M/2(A/2). The mixed term \({\chi }_{\mu \mu ;\nu \mu }^{+-}\) captures the coupling of a local \({}^{1}{{{{{{{\rm{Ni}}}}}}}}-3{d}_{{x}^{2}-{y}^{2}}\) excitation to a non-local \({}^{1}{{{{{{{\rm{Ni}}}}}}}}-3{d}_{{x}^{2}-{y}^{2}}{\to }^{1}{{{{{{{\rm{Ni}}}}}}}}-4{f}_{x({x}^{2}-3{y}^{2})}\) transition [Fig. 5d]. This term indicates that non-trivial Ni–La hybridization can play a role (order of 2%) in influencing the magnon dispersion throughout the Brillouin zone and could contribute to the long-range behavior of the Heisenberg exchange parameters^{37}. This is in contrast to the cuprates, where spin excitations are highly localized on the copper atoms^{77}.

### Comparison with other approaches

The discovery of a new class of superconductors not only brings forth new insights into the microscopic pairing mechanisms involved but it also provides a new data point for benchmarking and comparing various theoretical approaches toward superconductivity. Much of the existing theoretical analysis of the infinite-layer nickelates has been based on the standard DFT and DFT + DMFT frameworks. In this connection, it is useful to compare and contrast our results based on the more recently constructed SCAN meta-GGA density functional, which has been shown to yield a systematic improvement in capturing the properties of wide classes of materials^{78,79}, with electronic structures of the infinite-layer nickelates available in the literature.

In the NM state, SCAN and the local density approximation (LDA)^{16}—used to initialize the DMFT—yield very similar results in a ± 2eV energy window around the Fermi energy, with small differences appearing in band dispersions outside this energy window due to La-Ni (Ni-*t*_{2g} and Ni-\(3{d}_{{z}^{2}}\)) hybridization. LDA and SCAN are thus expected to yield the same instabilities. When the spin degrees of freedom are allowed to relax in SCAN, however, a C-AFM ground state is obtained, whereas when spin-polarized DMFT corrections are applied, a G-AFM ground state is found with a C-AFM order only ~6 meV higher in energy^{80}. This small difference in total energies can be explained by the additional lattice relaxations involved in the SCAN calculations. The DMFT instabilities are similar to those found via our RPA instability analysis. Both SCAN and DFT + DMFT^{80} (spin-polarized) find the Ni-\(3{d}_{{z}^{2}}\) states pushed to the Fermi level forming a flat band, as was also reported using DFT + U^{81}. The spin-polarized (C-AFM) SCAN electronic dispersions are also similar to the PM DFT + DMFT and DFT + self-interaction corrected DMFT (sicDMFT) results^{82}. Specifically, the atomically-resolved band dispersions from SCAN are comparable to the Ni-\(3{d}_{{t}_{2g}}\) states of DFT + DMFT and the Ni-\(3{d}_{{x}^{2}-{y}^{2}}\) and flat Ni-\(3{d}_{{z}^{2}}\) states of DFT + sicDMFT, where Ni-\(3{d}_{{x}^{2}-{y}^{2}}\) bands are completely absent at the Fermi level. This suggests that SCAN is adequately capturing the spin correlations. Further studies involving larger crystallographic unit cells to reveal the emergence of short-range spin-symmetry breaking would be interesting^{83}.

We turn next to compare the transverse-spin response within SCAN and DMFT. The PM DFT + sicDMFT spin response spectra^{84} appear to capture the linear softening of the magnon dispersion near Γ but fail to display the high-energy features at X and M/2, which are seen clearly in the experimental dispersion. The SCAN-based results match the RIXS curves over the full Brillouin zone when a renormalization factor is applied, as is typical for doped antiferromagnets. This is consistent with the reports of short-range magnetic dynamics over pure local moment fluctuations in LaNiO_{2}^{39}. Since both approaches use the same RPA framework to obtain the spin response, it will be interesting to evaluate corrections to the ground state using CDMFT or SCAN based on supercells involving large special quasirandom structures (SQS)^{85} to better capture the short-range correlations.

Low-energy models of the electronic structure are useful for disentangling the relationship between physical phenomena and the local chemistry, and indeed many such models have been invoked for the infinite-layer nickelates. Since we employ a multiorbital model spanning Ni-3*d*, La-5*d*, and La-4*f* orbitals, we are in a position to help identify a minimal set of orbitals necessary to reproduce the fluctuation instabilities and excitations. Our analysis of the NM phase shows that a simple model composed of a single Ni-\(3{d}_{{x}^{2}-{y}^{2}}\) orbital can capture the magnetic fluctuations, consistent with the observations of Kitatani et al.^{54,86}. To reproduce both the charge and magnetic fluctuations, however, a minimum of two orbitals is required. The minimal Hamiltonian proposed by Hu et al.^{87} using Ni-\(3{d}_{{x}^{2}-{y}^{2}}\) and Ni-3*d*_{xy} states is in good accord with our work, whereas the proposed Hamiltonians using Ni-\(3{d}_{{z}^{2}}\)^{45,88,89,90}, Ni-*s*^{91}, or La-5*d*^{29,92} states instead of Ni-3*d*_{xy} are not well-suited for modeling the NM state. However, if one wishes to self-consistently include magnetic correlation effects into the model—a prerequisite for capturing the magnetic excitation spectrum—, two-band Hamiltonians using Ni-\(3{d}_{{z}^{2}}\) states should be considered instead of those employing Ni-3*d*_{xy} orbitals in order to reproduce the partially occupied flat band at the Fermi level. Finally, if capturing the subtle effects arising from non-trivial Ni-rare earth hybridization are important, three-^{56,93,94,95} and four-band^{53,96} models provide a more faithful description.

## Conclusions

Our study demonstrates that LaNiO_{2} supports myriad competing, incommensurate spin fluctuations and magnetic excitations that are spread over multiple atomic sites. The reduction in the dimensionality of the magnetic fluctuations (3D to 2D) is found to coincide with the emergence of superconductivity. With further hole doping, fluctuations weaken, and superconductivity disappears, suggesting a tradeoff between the dimensionality and the strength of magnetic fluctuations in controlling the value of *T*_{c}. This behavior is similar to that of the doped cuprates where 2D inhomogeneous magnetic stripes are manifest already in the ground state^{97}. Our study gives insight into the nature of strong correlations in nickelates and provides a pathway for investigating complex correlated materials more generally.

## Methods

### First-principles calculations

Ab initio calculations were carried out by using the pseudopotential projector-augmented wave method^{98} implemented in the Vienna ab initio simulation package (VASP) ^{99,100} with an energy cutoff of 520 eV for the plane-wave basis set. Exchange-correlation effects were treated by using the strongly-constrained-and-appropriately-normed (SCAN) meta-GGA density functional^{101}. A 15 × 15 × 17 Γ-centered *k*-point mesh was used to sample the Brillouin zone. Spin–orbit coupling effects were included self-consistently. We used the experimental low-temperature P4/mmm crystal structure to initialize our computations^{34}. All atomic sites in the NM and C-AFM unit cells, along with the cell dimensions, were relaxed using a conjugate gradient algorithm to minimize the energy with an atomic force tolerance of 0.01 eV/Å. A total energy tolerance of 10^{−5} eV was used to determine the self-consistent charge density.

### Tight-binding Hamiltonian details

The many-body theory calculations of the spin–orbital fluctuations and magnetic excitations were performed by employing a real-space tight-binding model Hamiltonian, which was obtained by using the VASP2WANNIER90^{102} interface. For LaNiO_{2}, the full manifold of Ni-3*d*, La-5*d*, and La-4*f* states (a total of 17 orbitals) was included in generating the orbital projections, whereas in the C-AFM phase the \({d}_{{z}^{2}}\), *d*_{xy}, \({f}_{{z}^{3}}\), and \({f}_{x({x}^{2}-3{y}^{2})}\) were retained on the La sites (a total of 18 orbitals). We considered models employing a smaller number of basis functions for both the NM and C-AFM phase, but the resulting fits were quite poor. The Fourier transform convention *H*_{ij}(**k**) = ∑_{R}*e*^{ik⋅R}*H*_{ij}(**R**), is used throughout this work.

### Response function calculations

In the NM state, a 51 × 51 × 51Γ-centered *k*-mesh was used to evaluate the response functions in the Brillouin zone. Only the 4 bands at the Fermi level were used in the summation over the bands in the Lindhard susceptibility. In the C-AFM phase, a 23 × 23 × 23Γ-centered *k*-mesh was employed to sample the Brillouin zone, along with 101(2001)*ω*-mesh(*X*-mesh) energy points spanning 0–1 eV (−10 eV to 10 eV). See Supplementary Note 1 for a detailed definition of the *ω* and *X* meshes used in the “binning” method^{103,104,105}. All the bands were used in the eigenvalue integration for the C-AFM phase. A small broadening of 0.01 eV was used to simulate experimental conditions. Since our susceptibility code allows for finite temperatures, a very low value of the temperature of *T* = 0.001 K was used to mimic the DFT conditions and approximate a Heaviside step function. Coulomb interactions were included on the Ni-d orbitals and assumed to be rotationally symmetric, i.e. fulfilling the constraint \({U}^{{\prime} }=U-2J\) and \(J={J}^{{\prime} }\)^{58}. The maximum instability \({{{\Lambda }}}_{F}^{0}({{{{{{{{\bf{Q}}}}}}}}}^{* })\) was found to equal 1 for *U*, \({U}^{{\prime} }\), and *J* values of 1.22 eV, 0.732 eV, and 0.244 eV, respectively. Our results are insensitive to changes in *J* and show the dominance of Ni-\({d}_{{x}^{2}-{y}^{2}}\) orbital at the Fermi level. We also find negligible effect of including *U*, \({U}^{{\prime} }\), and *J* Coulomb parameters on La-5*d* states. The values of *U*, \({U}^{{\prime} }\), and *J* used were 3.675, 2.109, 0.782 eV, respectively, to simulate the magnon dispersion, where the ratio *U*/*J* = 5 was taken from SCAN ground state calculations^{48}.

## Data availability

All data supporting the findings of this study are available from the corresponding author upon a reasonable request.

## Code availability

The code used to calculate the spin-orbital fluctuations and magnetic excitations is currently in the process of becoming open source. Requests about the code should be made to the corresponding author.

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## Acknowledgements

The authors would like to thank Dr. Peter Mistark for many fruitful discussions. The work at Los Alamos National Laboratory was carried out under the auspices of the US Department of Energy (DOE) National Nuclear Security Administration under Contract No. 89233218CNA000001. It was supported by the LANL LDRD Program, the Quantum Science Center, a U.S. DOE Office of Science National Quantum Information Science Research Center, and in part by the Center for Integrated Nanotechnologies, a DOE BES user facility, in partnership with the LANL Institutional Computing Program for computational resources. Additional computations were performed at the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility located at Lawrence Berkeley National Laboratory, operated under Contract No. DE-AC02-05CH11231 using NERSC award ERCAP0020494. The work at Tulane University was supported by the U.S. Office of Naval Research (ONR) Grant No. N00014-22-1-2673. The work at Tulane University was also supported by start-up funding from Tulane University, the Cypress Computational Cluster at Tulane, the Extreme Science and Engineering Discovery Environment (XSEDE), and the National Energy Research Scientific Computing Center. The work at Northeastern University was supported by the US Department of Energy, Office of Science, Basic Energy Sciences Grant No. DE-SC0022216 and benefited from Northeastern University’s Advanced Scientific Computation Center and the Discovery Cluster, and the National Energy Research Scientific Computing Center through DOE Grant No. DE-AC02-05CH11231. B.B. acknowledges support from the COST Action CA16218.

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C.L. and R.Z. performed computations, and C.L., R.Z., B.B., R.S.M., and J.Z. analyzed the data. C.L., A.B., J.S., and J.Z. led the investigations, designed the computational approaches, provided computational infrastructure, and analyzed the results. All authors contributed to the writing of the paper.

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Lane, C., Zhang, R., Barbiellini, B. *et al.* Competing incommensurate spin fluctuations and magnetic excitations in infinite-layer nickelate superconductors.
*Commun Phys* **6**, 90 (2023). https://doi.org/10.1038/s42005-023-01213-0

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DOI: https://doi.org/10.1038/s42005-023-01213-0

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