Peierls distortion driven multi-orbital origin of charge density waves in the undoped infinite-layer nickelate

We have systematically performed calculations on a number of 1D stripe phases as a function of charge periodicity. We begin by considering two exemplar stripe phases that are schematized in Fig. 1. We assume that the stripe phases have the same ferromagnetic inter-layer exchange couplings as in the $C$-AFM phase. Firstly, we study a bond-centered stripe phase with charge periodicity $P_{c}=5$ and magnetic periodicity $P_{m}=5$, see Fig. 1 (a). The calculated magnetic moments of Ni atoms are around 1.06 $\mu_{B}$ with small modulations of $\sim$0.02$\,\mu_{B}$, leading to a slightly enhanced magnetic density around the antiphase boundaries (APBs). This phenomenon is discussed in further detail in the following section. Secondly, we study a site-centered stripe phase with charge and magnetic periodicities of $P_{c}=4$ and $P_{m}=8$, see Fig. 1 (b). Notably, we find the self-consistent magnetic moment of each Ni atom along the APBs to be 0.00 $\mu_{B}$, whereas the Ni atoms away from the APB exhibit a 0.99 $\mu_{B}$ moment in site-centered cases. This difference between bond- and site-centered cases persists for stripe phases within the more comprehensive set of stripe phases that we have studied. This set contains both bond- and site-centered cases for $P_{c}=3$, 4, 5, 6, 7, 9, 11, 13, and 15. Computations for larger values of $P_{c}$ become impractical. Next, we discuss these results in more detail.

Figure 2 (a) shows the total energies of each site and the bond-centered stripe state for undoped LaNiO₂ relative to the $C$-AFM phase Zhang2021 . Our total energy calculations show the bond-centered stripe is more stable than the site-centered stripe, consistent with results on cuprates Zhang2020 . Moreover, we find that the bond-centered stripes become more stable as the charge periodicity increases, consistent with pristine YBa₂Cu₃O₆ (YBCO₆) Zhang2020 . Figure 2 (b) compares the total energy of the various magnetic phases with the average Ni magnetization. Here, we include all of the 1D stripe phases up to $P_{c}=15$ and the $C$-AFM phase as the reference, along with a variety of uniform magnetic phases that have been previously discussed in Ref.  Zhang2021 . Overall, we find that a larger average Ni magnetic moment results in a more stable magnetic phase and that several high-magnetic-moment phases are nearly degenerate in energy. Two of the bond-centered stripe phases ($P_{c}=11$ and $P_{c}=13$) are found to be more stable than the $C$-AFM phase. In particular, multiple phases with strong local magnetic moments are found to be nearly degenerate in energy, with quite small energy differences of less than 2 meV. The bond-centered stripe phase with $P_{c}=13$ is found to be the ground state. These results are in remarkable agreement with our recent study of the yttrium-based cuprates Zhang2020 , which revealed that “intertwined orders” with larger magnetic moments are crucial for understanding the properties of correlated materials. Notably, we rule out any significant role of the non-magnetic (NM) phase in the ground state since it is not energetically competitive ($\sim$400 meV above the $C$-AFM phase). We emphasize that, similar to the doped cuprates, Fig. 2 clearly provides evidence for the existence of competing low-energy states in LaNiO₂.

The computational cost becomes prohibitive for larger $P_{c}$ values using DFT calculations. To explore the full manifold of possible low-energy stripe phases, we perform a random phase approximation (RPA) based instability analysis of the NM state [Section 1 of Supplementary Information(SI)]. The RPA results are consistent with DFT, showing that the total energy of the nickelate system is lowered by hosting a number of nearly degenerate stripe phases.

Based on the DFT total energy calculations, we find one significant difference between the cuprates and nickelates: in the undoped cuprates, the stripe phases are always metastable, with excess energy $\sim$$1/P_{c}$, leading to a well-defined AFM ground state Zhang2020 , whereas in the nickelates a stripe phase is the lowest energy state similar to YBCO₇ Zhang2020 .

This suggests that the nickelates may be closer to the Slater regime, where Fermi surface nesting can lower the stripe energy. This enhanced stripe-AFM competition may explain why LaNiO₂, like YBCO₇, has no long-range magnetic order OrtizPRR2022 ; Li2019a . Indeed, the similarity of Fig. 2(b) to Fig. 3 of ref. Zhang2020 , with both displaying an accumulation point of phases near the ground state, is striking, and it provides a plausible basis for the intertwined orders found in many correlated materials.

To understand the stabilization of these stripe phases, we plot selected charge differences between the bond-centered stripes and $C$-AFM phases in Figs. 3 (a) and S2 in section 2 of SI. For $P_{c}\leq 9$ phases, strong charge redistribution is found around the APBs, which minimizes the effect of parallel neighboring magnetic configurations on the boundaries. However, for $P_{c}\geq 11$, we find that the charges are deconfined from the APBs, predominantly redistributing in the magnetic domains, which leads to $P_{c}=11$ and $13$ showing lower energies than the $C$-AFM phase. Taking $P_{c}=13$ as an example, in Fig. 3 (a) we plot the electron density difference – the $P_{c}=13$ charge density minus the $C$-AFM charge density stretched to have the same lattice constant. We find a strong charge polarization along the $x$ direction, but not along the $z$ direction, where the electron clouds around each Ni or O atom are polarized to point away from a depletion region centered half way between the two neighboring antiphase boundaries. Consistent with this, all of the Ni and O atoms are displaced away from the depletion layer, Fig. 3 (b), bottom, the accompanying Peierls distortion, leading to a more stable stripe phase than $C$-AFM. [Note, displacements are enhanced by a factor of 10 for ease of viewing.] Due to unequal atomic displacements, the Ni atoms are shifted away from the midpoint between the two neighboring O atoms, Fig. 3 (c), leading to additional polarization effects. To relate this to the observed CDW, in the top part of Fig. 3 (b) we sketch a CDW of the correct periodicity. We note that the shape of the central lobe closely matches the profile of the depletion layer, as seen from either the electron polarization (a) or the atomic displacements (b), while the side lobes are not in evidence in the data.

In Fig. 3 (d), we show the spin density of the bond-centered stripe phase with $P_{c}=13$. From the shape of the Ni orbitals, one can clearly see the local magnetic moments of Ni sites are mainly composed of Ni $3d_{x^{2}-y^{2}}$ and $3d_{z^{2}}$ orbitals. We also note that O atoms in the magnetic domains show small spin polarization, while O atoms around the APBs show a dumbbell spin density, which implies that they have a small magnetic moment, 0.037$\mu_{B}$. These findings strongly indicate that nickelates are not a charge-transfer system like the cuprates Zhang2020 .

Figure 3 (e) shows the 1D Fourier transform (FT) of the charge density of the bond-centered stripe phase with $P_{c}=P_{m}=13$ along the $x$ direction. We identify three peaks in the first Brillouin Zone (B.Z.). Note that although the first peak located at 0.15 has the strongest amplitude, its wavelength may overlap with the tail of the Q=0 peak in experiments. Importantly, the wave vector of our second peak is 4/13 = 0.31, which is very close to the experimental values of 0.33 2021arXiv211202484R ; Krieger2021 ; Tam2021 . In addition, a weaker peak around 6/13 = 0.47 is noted in both our calculations and experiment 2021arXiv211202484R . While the charge transfer between the sites is small [Fig. S3], there is a significant charge polarization associated with the Peierls distortion [Figs. 3 (a) and (b)], which presumably drives CDW formation.

To gain insight into the electronic properties of the ground state, we compare the electronic band structures of the ground state $P_{c}=P_{m}=13$ bond-centered stripe and the uniform $C$-AFM phase. Figure 4 presents the unfolded electronic band structures of LaNiO₂ in the $C$-AFM (a) and $P_{c}=P_{m}=13$ bond-centered stripe (c) phases, respectively. As pointed out in Ref. Zhang2021 , the magnetic state appears as a Hund’s phase, where the magnetic moments primarily consist of $d_{x^{2}-y^{2}}$ character (0.75$\mu_{B}$), with a smaller admixture from $d_{z^{2}}$ (0.25$\mu_{B}$). This is reflected in the band structure. The Ni-$d_{x^{2}-y^{2}}$ band (red) is fully gapped out, consistent with recent experiment Tam2021 , and the $d_{z^{2}}$ is spin-split (blue), similar to the cuprates ChrisLCO2018 . Due to the fractional filling of the upper Ni-$d_{z^{2}}$ band, it is pinned to the Fermi level producing a flat band in the $k_{z}=\pi$ plane, consistent with previous studies Zhang2021 ; ChoiPRR2020 . Flat bands dominated by Ni-$d_{z^{2}}$ have also been predicted in multilayer nickelates using DFT+DMFT calculations LechermannPRB2022 . Similar results are found for the $P_{c}=P_{m}=13$ stripe (0.78 $\mu_{B}$ from $d_{x^{2}-y^{2}}$, 0.26$\mu_{B}$ from $d_{z^{2}}$). Remarkably, this pinning is absent in the NM phase, and arises from a strong reorganization of the $d_{z^{2}}$ dispersion when a magnetic gap opens on the $d_{x^{2}-y^{2}}$ band. This flat band gives rise to a Van Hove singularity (VHS) in the DOS, which can enhance the many-body interaction and drive more exotic correlated phenomena Robert2021 . This flat band has also been suggested to couple electronic and lattice instabilities Zhang2021 ; ChoiPRR2020 . Note that the VHS in LaNiO₂, involving the Ni $3d_{z^{2}}$ orbital, is stronger than the $3d_{x^{2}-y^{2}}$ VHS found in cuprates. However, it is extremely rare to find a VHS exactly at the Fermi level, without driving a transition to a new phase.

Upon unfolding the unit cell to capture the $P_{c}=P_{m}=13$ stripe, we find the resulting band dispersions [Figs. 4(c)] are only slightly modified from those of the $C$-AFM phase. Specifically, the degeneracy of bands in the stripe phase gets lowered due to the reduction in symmetry. Despite the lower symmetry, the 3$d_{z^{2}}$ flat band remains pinned to the Fermi level, as is also found in other stripe phase, as we show in SI, section 4. Away from the Fermi level, minigaps form in the Ni-$3d_{z^{2}}$/La-$d$ hybridized band due to the supermodulation of the crystal potential, typical of stripe-like structures MarkiewiczPRB2000 . This is also consistent with cuprates, where the stripes were identified as topological defects of an underlying AFM order. From Fig. 4(d), we find clear evidence that the stripe stabilization energy is associated with minigaps induced by lattice distortion in the $3d_{z^{2}}$ and $3d_{xz/yz}$ bands, which open a pseudogap in the $3d_{z^{2}}$ DOS and a small gap in the $3d_{xz/yz}$ dispersion.

In contrast to the cuprates, LaNiO₂ differs from the insulating cuprate parent compounds in that the former is metal, with localized Ni 3$d_{z^{2}}$, itinerant 3$d_{xz/yz}$, and La 5$d$ hybridized bands, indicating that a multiorbital model is necessary to properly describe the role of valence fluctuations between Ni²⁺ ($d^{8}$) and Ni¹⁺ ($d^{9}$). The behavior of the magnetic state is similar to the cuprates, except for a somewhat larger energy scale. That is, the magnetic gap in the Ni-$d_{x^{2}-y^{2}}$ band is approximately double that in the cuprates owing to the larger magnetic moments. Consequently, the energetic ordering scale of magnetic and stripe states in Fig. 2(b) is a factor of 5-7 larger than those in YBCO₇ Zhang2020 .