Proposed ordering of textured spin singlets in a bulk infinite layer nickelate

The gap values of ODS, FM, and AFM states versus $U$ are shown in Fig. 2(a). Not surprisingly, the gap opens for much smaller $U=1$eV for AFM order due to its smaller bandwidth; for the ODS the gap opens only at $U$=5 eV. In all cases the gap remains small for a transition metal oxide, even for $U$=9 eV. Next, we compare energies of the FM and AFM states relative to the ODS state in Fig. 2(b). NM (trivial $S=0$ ${}^{1}A$ singlet) lacks spin polarization energy and can be neglected. Both FM and AFM ordered states have $S=1$ Ni ions with their Hund’s rule energy gains, with AFM being slightly favored over FM with a difference that decreases with increasing $U$. The ODS state, which is spin-orbital polarized but $S$=0, above $U_{c}$=4 eV becomes favored over both states having $S$=1 ions, with the energy difference growing rapidly as $U$ increases. The Hund’s exchange energy is overcome by other energy changes, as discussed in Sec. VI.D and the Appendix.

The values of the FM and AFM moments at GGA level are $\sim$1.1$\mu_{B}$, as shown in the large lattice constant $a$ region of the upper panel of Fig 3. Including $U$, these grow (not shown) to around 1.9$\mu_{B}$, a value that is characteristic of $S$=1 reduced somewhat by hybridization. To emphasize: for the ODS ions, the orbital moment is around 0.16$\mu_{B}$ while the net spin moment vanishes.

Matsumoto and coauthorsmatsumoto have drawn a connection between the Ni-O bond length and the low-spin and high-spin states in layered $d^{8}$ compounds, identifying a critical separation in the 2.00-2.05Å range and ascribing the transition to crystal field splitting. This critical distance led us to study the lattice constant dependence, taking the FM state as an example. We performed the corresponding calculations, with results shown in Fig. 3 using fplo within GGA (wien2k results are similar). As $a$ is decreased from 4.2 Å to 3.9 Å with the cell volume fixed and internal parameters of Ba and Se being optimized (the observed value is $a$=4.21 Å), an electronic transition at Ni-O is observed. The Ni moment initially decreases ($\Delta M/\Delta a\approx 1.6\mu_{B}$/Å). When the Ni-O separation is lowered below the critical value of $a_{c}/2\equiv d^{Ni-O}_{c}$=2.03Å, the rate of decrease of the Ni moment doubles.

At this same separation, the (magnetic) energy undergoes a dramatic change. The energy increases slowly and quadratically down to 4.06 Å. Below $a=4.06$ Å, where Matsumoto and collaborators suggest a crossing of magnetic and nonmagnetic Ni ions, the energy difference rises steeply with a slope of –300 meV/Å. This electronic transition involves a reconfiguration of how orbital occupations change with volume.

This behavior is consistent with the critical Ni-O separation proposed by Matsumoto et al. They observed that most NiO₂ compounds with only square planar Ni (no apical oxygens) are nonmagnetic, but if the lattice constant is pushed above 2.0Å, the Ni ion becomes magnetic. For example Sr₂NiO₂(AgSe)₂, with $a$=4.094Å, is magnetic; BaNiO₂ and LaNiO_2.5 are not. The significant new aspect is that while our Ni ion is spin-polarized, i.e. magnetism must be considered in the electronic structure, it forms a singlet and is not magnetic to most experimental probes.

The band structure at the GGA level, which forms the basis for most beyond-GGA studies, is presented in Fig. 4 with the atom-projected density of states (PDOS). The $d_{z^{2}}$ band is degenerate with the $t_{2g}$ bands, all with 0.5-1 eV bandwidths. The Ni $d_{x^{2}-y^{2}}$ dispersion is 3 eV (see the $\Gamma-M$ line) with center of weight around 0.5 eV, giving an $e_{g}$ crystal field splitting of 1-1.5 eV. The lowest conduction band at $\Gamma$ has Se $p_{z}$ and Ag $s$ characters; this will be referred to as the Se-Ag band and attains importance for electron doping, discussed below.

Dimensionality of the band structure is one central point of interest in nickelates as well as cuprates. $k_{z}$ dispersion along $\Gamma-Z$ of valence bands is small except for (1) the Ni $d_{z^{2}}$ band just below E_F, with dispersion of roughly 0.5 eV, accommodated by hybridization with the Se $p$ orbitals, i.e. through the electronic polarization of the blocking layer, and (2) the Se-Ag conduction band, which shows in-plane dispersion of 1 eV and $k_{z}$ dispersion from $\Gamma$ of nearly 0.5 eV. The two $3d$ holes are accommodated mostly in the $d_{x^{2}-y^{2}}$ orbitals with a minor amount in the small Fermi surfaces with mostly $d_{z^{2}}$ character. These differences are central factors in how the Coulomb repulsion $U$ will produce changes in the spectral positions.

The PDOS displayed in the top panel of Fig. 5 reveals the paradigm-shifting effect of $U$. The ODS state is obtained already for $U$=1 eV, though not yet favored energetically. The gap opens for $U$ in the 4-5 eV range, beyond which the singlet is increasingly strongly favored (see Figs. 2(a,b)). The spin density shown in Fig. 1(b) is strongly anisotropic (maximally so), similar to that obtained previouslylanio2 for $d^{9}$ LaNiO₂, where large $U$ drove the system toward an unexpected $d^{8}$ configuration. $U$=6-8 eV was judged to be too large for conducting LaNiO₂, which is observed to be conducting and would have a more strongly screened (smaller) value of $U$.kaneko09

Comparison with the FM progression ($S$=1) in the lower panel of the PDOS in Fig. 5 reveals few differences aside from the flipped $d_{z^{2}}$ spin. At $U$=1eV, the occupied (unoccupied) $d_{z^{2}}$ states are at -2 eV (+0.2 eV, slightly overlapping E_F). There is little difference in positions and widths of most of the occupied $d$ bands. The noticeable difference is that the occupied $d_{z^{2}}$ band is 1.5 eV lower for the FM case at $U$=2 eV. A close look indicates that the $t_{2g}$ PDOS is lower for FM than for ODS. At $U$=1 eV and above, the $d_{x^{2}-y^{2}}$ and $d_{z^{2}}$ ‘Mott gaps’ increase proportional to $U$ with most of the separation being absorbed by increased binding of the occupied orbital.

In Fig. 5 the orbital-projected density of states (PDOS) of the ODS was compared to that of the FM state, for $U$=0-6 eV. In Fig. 6 we make the analogous comparison between the AFM and FM alignments of Ni ($S$=1) spins, which we find to be instructive. Bandwidths change, because majority states can only hop only to minority states on nearest Ni which have different energies; thus AFM bandwidths are narrower than FM as usual. Differences in energy position of orbital occupations may or may not change and can be judged by this information.

Figure 6 shows the progression of the Ni orbital projected DOS as $U$ is increased from zero to 6 eV, for AFM alignment (above) and FM alignment (below). Each of these corresponds to a high spin Ni ion. The $d_{z^{2}}$ PDOS, which is of most interest, is emphasized in yellow. For AFM alignment, at $U$=0 holes reside in primarily in $d_{x^{2}-y^{2}}$. The $d_{z^{2}}$ PDOS (occupied) is spread across 3 eV. At $U$=2 eV, the gap opens giving the full high spin state of the Ni ion. By $U$=4 eV, the (inactive) $t_{2g}$ states stay in the -1 eV to -2 eV region. The occupied $e_{g}$ majority orbitals move to lower energy and become localized, so something nonlinear has occurred. $U$=6 eV is like 4 eV, except Mott splitting has increased. The nonlinearity is due to the $d$ states moving below the Se and O $p$ bands, and is not of much consequence.

For FM alignment, at $U$=0 the state is metallic but is approximating the $S$=1 high spin state. At $U$=2 eV, tails of bands are still crossing $E_{F}$ but the $S$=1 configuration is emerging. Changes can be seen in the occupied $d$ region, as for AFM. Increasing to $U$=4 eV, the spectrum is essentially gapped. $t_{2g}$ states have moved to -4 eV and become localized. At $U$=6 eV the spin splitting of $t_{2g}$ has increased, and the full $S$=1 state is attained. There is a 1 eV difference in position between the two minority holes. From Fig. 2(b) the AFM and FM energies are not very different.

This progression with $U$ seems about as expected: $U$ increases the separations between occupied and unoccupied orbitals, and pushes orbital occupations toward zero or unity. There is “nonlinearity” in the movement of occupied $d$ states between 2 eV and 4 eV, but this is probably due to moving out, then being pushed out, by the O $2p$ or Se $4p$ bands in the -3 eV to -1 eV region. The linear progression of the unoccupied orbitals, widening the gap as $U$ increases, is usual.

The band structure of the ODS state is shown in Fig. 7, with the corresponding PDOS of relevant orbitals pictured in Fig. 8. Both $e_{g}$ orbitals are split by roughly $U$=7 eV (the Mott-Hubbard gap between opposite spin orbitals), with unoccupied $d_{x^{2}-y^{2}}$ bands 1.5 eV higher in energy due to the crystal field. A substantial effect of $U$ is to remove the $t_{2g}$ orbitals (both spins occupied) from just below the GGA Fermi level to –4 to –5 eV below the gap, where they can as usual be ignored. This displacement serves to leave the Se $4p$ states at the bottom of the gap at $\Gamma$, with the O $2p$ bands somewhat lower. Note: with zero net ODS spin, we have chosen that ‘up’ is the direction of the $d_{x^{2}-y^{2}}$ spin, ‘down’ is of course the opposite.

The gap opening in both spin directions at $U_{c}\approx$ 5 eV signals a metal-insulator transition and ensures that the net moment vanishes, giving the pure ODS state. This state violates Hund’s first rule, signaling that other (intra-atomic and environmental) contributions to the energy are compensating. We propose that one factor arises from the differences in crystal field splittings from neighboring divalent cations (Ba or Sr) versus the tripositive ions in $d^{9}$ LaNiO₂ and NdNiO₂. There are only minor differences between FPLO and wien2k, what is important is that the transition to the ODS singlet and gap opening occurs in both codes, so it is a robust feature.

The unoccupied Ni $d_{z^{2}}$ hole band is, in first approximation, flat at +1.3 eV. The dispersion that is seen in the Fig. 7 fatbands is due to rather strong mixing with the Se-Ag conduction band that is dipping down at $\Gamma$. This hybridized band disperses about 1 eV from $\Gamma$ to the zone edge $X$, and 0.5 eV width along $k_{z}$. The gap increases to 0.5 eV for $U$=8 eV (see Fig. 2), as occupancies attain essentially integer values. The spin density shown in Fig. 1(b) vividly illustrates the spin polarization that is a combination of $d_{z^{2}}$ and $d_{x^{2}-y^{2}}$ contributions of opposite spins, with $d_{x^{2}-y^{2}}$ orbitals mixed with oxygen $p_{\sigma}$ orbitals. We return in Sec. V to further implications of this ODS band structure.

The electron-hole asymmetry for $U>U_{c}$ is anomalous for a transition metal oxide, rendering the usual Mott insulator versus charge transfer insulator inapplicable. Oxygen $2p$ states do hybridize with $d_{x^{2}-y^{2}}$ but not as strongly as conventionally, since they lie 2-4 eV below the gap. Hole states, see Fig. 7, are instead Se $4p$ states in the down spin bands that are highly itinerant. O $2p$ bands are somewhat more tightly bound.

Electron states also present new behavior: electrons initially go into the 3D dispersive band at $\Gamma$ of the itinerant Se-Ag $4s$ band mixed with up spin $d_{z^{2}}$ character. There is a 0.2 eV exchange splitting, and down spin Se-Ag band does not mix with other bands. At a still small concentration electrons begin to occupy this spin down strongly itinerant band. Significant doping levels will therefore produce up-carriers in the NiO₂ layer and down-carriers in the AgSe layer, with associated spin polarization. The carriers can be expected to disturb the singlet, likely enabling further moments to emerge. A serious study of doping would require a separate study.

Variation of spin-resolved Ni occupancies with $U$ are displayed in Fig. 9, comparing ODS with FM alignment. The occupation numbers $n_{m}$ of the active $e_{g}$ orbitals $m$ (top panels) show similar behavior especially in the physically relevant region $U>3$, except that $d_{z^{2}}$ is flipped in spin in the ODS state. At the more detailed level, the hole occupation numbers for $S$=1 are constant for $U>$3 eV, while for the singlet $S$=0 there is a small but continuing increase by 0.02-0.04 up to $U$=9 eV. Somewhat unexpected is that it is $d_{x^{2}-y^{2}}$ that changes most, but is understandable because that dispersive band crosses E_F at small $U$ and $U$=5 eV is required to completely empty it.

The atomic orbital basis of fplo provides atomic Mulliken charges (MCs), in which contributions to the density from overlapping orbitals of neighboring atoms is divided equally between the atoms; the sum of MCs equals the total charge. The change of MCs with $U$, shown in Fig. 9 (middle and bottom panels), provide a complementary view of correlation effects, since they reflect changes in bonding (viz. relative orbital energies) as well as actual charge rearrangement. The atomic MCs (neutral atom values) for Ba, Ni, O, Ag, and Se are

respectively. These are quite reasonable values for MCs, and are not expected to correspond closely to the formal charges. All do differ from the neutral atom value in the ‘right direction’, and by roughly half of the formal valence.

We focus on the Ni MC variation (any change on Ni must go somewhere in the Mulliken decomposition, and that somewhere is the atoms O and Se that hybridize with Ni). The changes in total Ni Mulliken ionicity with increasing $U$ (lower two panels) are similar for ODS and FM states, saturating for Ni $S$=0 around $U$=6 at a value of –0.2 while for $S$=1 there is less tendency toward saturation. The spin-resolved MC ionicities (middle panels) differ qualitatively between FM and ODS states. For ODS $S$=0 the change in down MC is a small fraction of that of the up, which is in the -0.18 range. For FM $S$=1 the down MC charge becomes twice as large, while that of up is roughly half as large and of opposite sign. At $U=1$ eV, the Ni moment is –0.18 $\mu_{B}$, but the net moment is zero. With increasing $U$, the down spin charge moves to the other spin and other ions. Above $U\approx 4$ eV, the magnitude of the Ni moment vanishes, entering a true ODS phase. We are not aware of nickelate or cuprate values of MCs to compare with.

The insulating ground state ODS that is discussed above provides a system in which the elementary excitations are spin waves. We suggest here a minimal model of the spin system. Let the $d_{x^{2}-y^{2}}$ and $d_{z^{2}}$ spin operators be denoted by the Pauli matrices $\vec{\sigma}$ and $\vec{\tau}$ respectively. Factors of $\frac{1}{2}$ will be folded into the constants. The minimal spin Hamiltonian for the insulating state contains three parameters:

$J\equiv J_{\sigma}$ is the positive in-plane near-neighbor ($<i,j>$) superexchange constant between $d_{x^{2}-y^{2}}~{}\sigma$ spins, and $K$ is the on-site spin coupling between $\vec{\sigma}$ and “Kondo-like” $\vec{\tau}$ spins, with contributions from Hund’s coupling, Hubbard $U$ correlation, and environmental influences. The on-site singlet-triplet splitting is $2K$, and the sign of interest here (making it Kondo-like) being positive $K$. $J_{z}$ is the (spacer layer assisted) coupling between $\tau$-spin neighbors $[i,j]$ along $\hat{z}$, and the total on-site spin is $\vec{S_{i}}=\vec{\sigma}_{i}+\vec{\tau}_{i}$ couples to the magnetic field $\vec{B}$. For discussion in this section we set units such that $J$=1.

$H_{KS}$, which we refer to as the Kondo sieve model, is an extension to 2D of the Kondo necklace model introduced by Doniach.doniach It was derived from the 1D Kondo Hamiltonian by applying the Luther-Pelcher transformationluther from 1D spinless itinerant particles to on-site spins. Quantum Monte Carlo calculations indicate that the 1D Kondo necklace does not display magnetic order at any finite value of the exchange coupling constant.scalettar1985 The extension to 2D by Brenigbrenig2006 has been known (somewhat contradictorily) as the 2D Kondo necklace. This two spins per site model can also be viewed as the large Hubbard $U$ regime of the Kondo-Hubbard model at half filling.brenig2007 We picture BNOAS, in first approximation, as a realization of the Kondo sieve model, then with small and mostly frustrated interlayer coupling that may become important in emergence of ordered phases.

Quite a bit is known from Brenig about the Kondo sieve model. For $J$=0, $H_{KS}$ consists of isolated singlets; for $K$=0 it is the exhaustively studied spin-half 2D AFM Heisenberg model, which does not order at finite T (Mermin-Wagner theorem). Using stochastic series expansion methods, Brenig found that weak AFM order exists for $H_{KS}$ at low T=0.05 up to a quantum critical point (QCP)brenig2006 ; brenig2007 $K_{c}$=1.4. Below $K_{c}$ the $\tau$-projected order parameter is around 35% larger than that of the neighbor-coupled $\sigma$ ($d_{x^{2}-y^{2}}$) spins.

The ordering reveals that the Kondo spin promotes long range order even in 2D, and becomes a central influence even for a small Kondo coupling, reflecting apparent smaller quantum fluctuations than the $\sigma$ spin. The uniform susceptibility decreases with increasing $K$, vanishing at $K_{c}$ above which the singlets control the physics. Site dilution of the “Kondo” moments, as by non-stoichiometry or doping, results in structure in both $\chi(Q_{AFM},T)$ and $S(Q_{AFM},T)$ around T=0.1 or below, depending on concentration. The QCP vanishes into a crossover, i.e. weak AFM order extends to larger $K$, for which an order-by-disorder mechanism was proposed.brenig2007 These results will also impact the doping experiments proposed below.

Due to the body-centered stacking of NiO₂ layers both the $\tau$ and also much smaller $\sigma$ couplings between AFM layers will be frustrated. The 0.5 eV $k_{z}$ dispersion of the $d_{z^{2}}$ band imbues importance to this coupling, since it is the largest coupling available to enable 3D magnetic correlations and potential long-range 3D order. An in-plane term $\sum_{<i,j>}\vec{\tau}_{i}\cdot\vec{\tau}_{j}$ and interlayer coupling $\sum_{[i,j]}\vec{\sigma}_{i}\cdot\vec{\sigma}_{j}$ are symmetry-allowed, but both should be small.

Of special interest, insightful for the large $K$ that our calculations indicate, is that the on-site ($i$) term can be diagonalized in total spin $\vec{S}_{i}=\vec{\sigma}_{i}+\vec{\tau}_{i}$ space, with singlet $S_{i}$=0 and triplet $S_{i}$=1 sectors. For the isolated ion $K$ is negative, giving the familiar high-spin Hund’s coupling. In BNOAS we find instead from total energy results that the low-spin ODS configuration is favored (rather strongly) for physical values of $U$ [see Fig. 2(b)]. The non-magnetic ${}^{1}A$ state with two holes in the $d_{x^{2}-y^{2}}$ orbital is highly disfavored and is not included in the Kondo sieve model.

Given that the low-spin ODS state is favored in BNOAS, in zero-th approximation configurations are confined to the ODS sector $S_{i}=0$ for all sites $i$. Then the second term in $H$ becomes diagonal and the fourth term in the Hamiltonian vanishes (and also hence the linear susceptibility as in experiment), and $\vec{\tau}_{i}=\vec{S}_{i}-\vec{\sigma}_{i}$ can be eliminated to give (up to a constant)

The in-plane superexchange $J$ is ubiquitously calculated to be positive and large in nickelates, so the first term is the one that is commonly applied to the AFM insulating phase of cuprates. Since the $\sigma$ spins are antialigned (or at least strongly correlated), the body-centering of the BNOAS structure determines that interlayer $\vec{\tau}_{i}\cdot\vec{\tau}_{j}$ – here transformed to $\vec{\sigma}_{i}\cdot\vec{\sigma}_{j}$ – coupling is frustrated.

The result from the Kondo sieve model without considering further symmetry breaking (such as single ion anisotropy or structural distortion) is that the $\sigma$ spins are strongly correlated in-plane and become ordered below $K_{c}/J=1.4$, with experimental properties becoming nonstandard for a magnetically ordered system. The substantial Kondo coupling indicates that weak AFM should be considered in the interpretation of magnetization data. Ordering across the layers due to additional interactions may be a lower temperature possibility that is left for further work. For comparison with experimental data, the sensitivity of the QCP to defects should be kept in mind, since the synthesis methods of this class of compounds have been found to allow the incorporation of impurities and produce some minority phases.matsumoto2020

Above the characteristic temperature T${}_{m}=130$ K of BNOAS the $S$=0 singlets are magnetically inert, which naturally accounts for the lack of a Curie-Weiss term. The internal structure presumably is correlated due to antiferromagnetic $d_{x^{2}-y^{2}}$ coupling, but contributing a bit to $\chi(T)$ due to mixing of the $S$=1 triplet, consistent with experiment. In the language of the Kondo sieve model, the ordering below T_m means that $K<K_{c}=1.4J$, i.e. below the QCP where ordering occurs. As long range ordering of the $d_{x^{2}-y^{2}}$ internal structure emerges at and below $T_{m}$, magnetic behavior emerges apparently requiring involvement of the $S$=1 sector, only to return (in zero field cooled data) to invisibility in $\chi$ in the ordered state. This long-range correlation and ordering of weak moments (including a small orbital moment, which we calculate to be 0.16$\mu_{B}$ parallel to the $d_{x^{2}-y^{2}}$ moment) should be detectable, though possibly challengingly so, by neutron diffraction, by utilizing the magnetic (spin + orbital) structure factor that contributes to the magnetic density correlation function. Muon spin resonance is also a particularly sensitive method to detect magnetic order, though less so for magnetic correlations.

Probably more telling is that the susceptibility will be distinct from that of truly non-magnetic ions, with some T-dependence arising from a van Vleck contribution from the proximity in energy of the $S$=1 states. The field-cooled susceptibility below T_m may arise from domain structure in which $S$=1 moments at the domain boundaries come into play. To repeat from above: extrinsic phases may interfere in the data with intrinsic behavior, complicating interpretation.

The preceding discussion has assumed the integrity of the quantum ODS singlet. This is textbook behavior as long as the two spins remain in a coherent singlet, which conventionally assumes degenerate orbitals and negligible coupling to the environment. Neither of these conditions applies strictly to BNOAS. At elevated temperature these “perturbations” may average out so the singlet survives, but may become more influential as temperature is lowered. Similarly to the quenching of atomic orbital angular momentum by environmental effects (non-spherical potential), the ODS may revert to the non-quantum pair of individual but coupled $\vec{\sigma}$ and $\vec{\tau}$ spins (as described by DFT). Coupling then will strictly antialign the individual orbital spins, which can then separately respond also to an applied magnetic field, and have separate (and unequal) orbital contributions and provide a distinct susceptibility though probably still small. The magnetic transition at T_m could, in this scenario, correspond to the quenching of the quantum singlet, with consequences that may not have been studied previously.

Electron doping, which would drive the Ni ion toward the $d^{9}$ configuration where high temperature superconductivity (HTS) occurs in cuprates, is of primary interest. Figure 7 indicates an unusual band structure for such a strongly layered nickelate discussed in Sec. III.D, with the characters of the bands bordering the gap described in Sec. IV.A. At very low doping carriers go into a band that disperses out of plane as much as in-plane: spheroidal Fermi surfaces. With increased doping the spheres will join along the short $k_{z}$ dimension of the zone, leaving roughly cylindrical Fermi surfaces. At higher dopings continued conduction can be expected, arising from very different spin-up and -down contributions as discussed in Sec. IV.A: spin-up electrons in a largely $d_{z^{2}}$ band, spin-down electrons in the AgSe layer.

The most obvious candidate is La³⁺ substitution for Ba²⁺, which gave the initial HTS (La,Ba)₂CuO₄. Such La$\leftrightarrow$Ba substitutions are common, and with their electropositivites they do little other than change the carrier concentration and to affect the volume slightly. Similarly, Ag⁺ replaced by Cd²⁺ might have similar effects.

Substituting on the anion sites is the other possibility. Replacement of O^2- by Cl^- can achieve similar effects, with the difference that bonding with Ni is affected to some degree. Se^2- replacement by Br^- is less commonly tried (or at least achieved). Finally, one can consider Ni²⁺ replaced by Co or Cu. These might assume the 2+ charge state, which would give no doping but disturb the singlet character. Chemical behavior at high pressure may determine which type of replacement is most likely to retain the crystal structure and provide electron doping.

Hole doping seems less compelling. The states are primarily Se $4p$ in character and essentially spin degenerate. This itinerant charge in the AgSe layer does not change the Ni valence, and does not even screen the Ni ion, i.e. does not change $U$ on Ni nor the gap in the NiO₂ layer. In clean crystalline samples it might produce a high mobility 2D electron gas, with its well studied properties. Having this 2D hole gas interlayered with (ordered) ODSs would however be a novel system to study.

Related behavior was observed in DFT+U studies of MnO under pressure, which both experiment and theory identify as a high spin to low spin (but not zero spin) transitionyoo2005 in the vicinity of an insulator-metal transition. In DFT+U calculationsdeepa2007 , the low pressure $S=\frac{5}{2}$ Mn ion first undergoes, under pressure, an insulator-to-insulator transition in which two Mn $3d$ orbitals flip spin direction whilst each orbital remains fully spin polarized, i.e. an $S=\frac{5}{2}\rightarrow S=\frac{1}{2}$ spin collapse without change in orbital occupation. This spin collapse amounts to an even more drastic violation of Hund’s first rule than in BNOAS that is discussed here. The origin of the energy gain was traced to the persisting strong (maximal) anisotropy in spin and charge, and to a change in $pd\sigma$ hybridization. In that system MnO was also in proximity to a critical metal-oxygen separation. Both of these characteristics (anisotropy and a critical bond length) are also apparent in BNOAS.

A prime interest in BNOAS will be how it may inform the electronic structure and magnetic behavior in superconducting (Nd,Sr)NiO₂. As mentioned, the ODS tendency was seen fifteen years ago in LaNiO₂, which is commonly used as a stand-in for NdNiO₂ in correlated electron calculations. It should be mentioned that others have not reported obtaining this singlet tendency in LaNiO₂. However, in BNOAS we obtain it with two different codes and GGA+U implementations. We obtain the magnetic states others report but they have higher energies (for physical values of $U$). There are still things to be learned about the peculiar behavior of the $d_{z^{2}}$ orbital in infinite layer nickelates. It is of considerable interest to try to electron-dope BNOAS toward the $d^{9-\delta}$ regime where superconductivity emerges in (Nd,Sr)NiO₂.

Another related system is Ba₂CuO_3.2, which superconducts at 73Kpnas after having been synthesized at 18 GPa and 1,000C. Although not described as such, this compound is an infinite layer cuprate that is hole-doped with 0.2 apical oxygen atoms per copper, making it formally somewhat overdoped. The (average) apical Cu-O distance is small at 1.86Å, whereas the in-plane Cu-O separation is 2.00Å, larger than other superconducting cuprates. The interpretation of spectral data suggested that the crystal field has exchanged the positions of the $d_{z^{2}}$ and $d_{x^{2}-y^{2}}$ levels, which would provide a new paradigm for cuprate superconductivity. The relationship of this material to BNOAS suggests the usefulness of further study of both systems.

While BNOAS is not simple to synthesize due to the need of high pressure, it is nevertheless available (after synthesis) to experimental probes that are not available to MnO at 100 GPa pressure. We encourage further synthesis and study of BNOAS by thermodynamic and spectroscopic (electron, neutron, and muon) probes to illuminate the peculiar magnetic behavior observed in this infinite layer nickelate.