Nickelate superconductors—a renaissance of the one-band Hubbard model

The recently discovered nickelate superconductors appear, at first glance, to be even more complicated multi-orbital systems than cuprates. To identify the simplest model describing the nickelates, we analyse the multi-orbital system and find that it is instead the nickelates which can be described by a one-band Hubbard model, albeit with an additional electron reservoir and only around the superconducting regime. Our calculations of the critical temperature TC are in good agreement with experiment, and show that optimal doping is slightly below 20% Sr-doping. Even more promising than 3d nickelates are 4d palladates.


INTRODUCTION
Following the discovery of superconductivity in the cuprates 1 and the seminal work by Anderson 2 , the theoretical efforts to understand high-temperature superconductivity have been focusing to a large extent on a simple model: the one-band Hubbard model [3][4][5] . However, superconducting cuprates need to be doped, and the doped holes go into the oxygen orbitals [6][7][8] . This requires a more elaborate multi-band model such as the three-orbital Emery model 9 . If at all, the Hubbard model may mimic the physics of the Zhang-Rice singlet 10 between the moment on the copper sites and the oxygen holes.
The recent discovery of superconductivity in Sr 0.2 Nd 0.8 NiO 2 by Li et al. 11 marked the beginning of a new, a nickel age of superconductivity, ensuing a plethora of experimental and theoretical work; see, among others, refs  . Similar as for the cuprates, the basic structural elements are NiO 2 square lattice planes, and Ni has the same formal 3d 9 electronic configuration.
But at second glance, there are noteworthy differences, see Fig. 1a. For the parent compound NdNiO 2 , density functional theory (DFT) calculations show, besides the Ni d x 2 Ày 2 orbital, additional bands around the Fermi level E F 15,20,21,35 that are of predominant Nd-5d character and overlap with the former. Note that the Nd-5d bands in Fig. 1a extend below E F despite their centre of gravity being considerably above E F . Such bands or electron pockets have the intriguing effect that even for the parent compound NdNiO 2 , the Ni-d x 2 Ày 2 orbital is hole doped, with the missing electrons in the Nd-5d pockets. In other words, the parent compound NdNiO 2 already behaves as the doped cuprates. First calculations 16,19,27 for superconductivity in the nickelates that are valid at weak interaction strength hence started from a Fermi surface with both, the Nd-5d pockets and the Ni-d x 2 Ày 2 Fermi surface. Such a multi-orbital nature of superconductivity has also been advocated in refs 18,29 .
There is additionally the Nd-4f orbital which in DFT spuriously shows up just above the Fermi energy. But these 4f orbitals will be localised which can be mimicked by DFT + U or by putting them into the core, as has been done in Fig. 1a. Choi et al. 32 suggest a ferromagnetic, i.e., anti-Kondo coupling of these Nd-4f with the aforementioned Nd-5d states. A further striking difference is that the oxygen band is much further away from the Fermi level than for the cuprates, see Fig. 1a. Vice versa, the other Ni-3d orbitals are closer to the Fermi energy and slightly doped because of their hybridisation with the Nd-5d orbitals.
This raises the following issues: which orbitals are depopulated upon Sr doping NdNiO 2 ; the validity of a single-orbital description in the superconducting doping range; the doping range in which Sr x Nd 1−x NiO 2 is actually superconducting; and the upper superconducting transition temperature T C .
Addressing these points calls for a proper treatment of electronic correlations since e.g., the formation of Hubbard side bands has the potential to alter the orbitals that host the holes induced by Sr doping. In a material-specific, ab initio way this can be done using DFT + dynamical mean-field theory (DMFT) 36 and several groups have done such DFT + DMFT studies 25,26,28,33,34 . Most of these focus on the undoped parent compound and arrive at a picture that the main players are the Ni d x 2 Ày 2 orbital plus A (and Γ) pocket.
However, if we further Sr-dope the system this picture must break down. If we have e.g., two holes in the Ni orbitals as for LaNiO 2 H 28 , it is clear that because of Hund's rule coupling we need at least two Ni orbitals. Hence somewhere inbetween the undoped parent compound and the high Sr-doping regime, there must be a crossover from a Ni-d x 2 Ày 2 -orbital picture plus hole pockets to a multi-Ni-orbital picture. Whether this crossover happens before, within or after the superconducting Sr-doping regime is one of the key questions for identifying the minimal model of superconductivity in nickelates.
In this paper, we show that, if we properly include electronic correlations by DMFT, up to a Sr-doping x of about 30%, the holes only depopulate the Ni-3d x 2 Ày 2 and Nd-5d bands. Only for larger Sr-dopings, holes are doped into the other Ni-3d orbitals, necessitating a multi-orbital description. The hybridisation between the Ni-3d x 2 Ày 2 and Nd-5d orbitals as calculated from the DFT-derived Wannier Hamiltonian is vanishing. We hence conclude that up to a Sr-doping of around 30% marked as dark blue in Fig. 1b, a single Ni-3d x 2 Ày 2 band description as in the one-band Hubbard model is possible. However, because of the Nd-5d pocket(s), which acts like an electron reservoir and otherwise hardly interacts, only part of the Sr-doping [lower x-axis of Fig. 1b] goes into the Ni-3d x 2 Ày 2 -band [upper x-axis], cf. Supplementary Note 2 for the functional dependence. We also take small Sr-doping out of the blue-shaded region since for such small dopings there is, besides the Nd-5d A-pocket, the Γ-pocket which interacts with the Nd-4f moments ferromagentically 32 and might result in additional correlation effects. At larger doping and when including the Nd-interaction in DMFT the Γ-pocket is shifted above E F , see Fig. 2 below and e.g., refs 16,25,28,32 . Figure 1b further shows the superconducting critical temperature T C of the thus derived and doped Hubbard model, calculated by a method that is appropriate in the strong coupling regime: the dynamical vertex approximation (DΓA 37 ). The agreement between DΓA and experiment in Fig. 1b is reasonable given that the experimental T C can be expected to be lower because of e.g., impurity scattering and the theoretical one is somewhat overestimated (The back coupling of the particle-particle channel to the particle-hole and transversal particle-hole channel is not included. While a (tiny) hopping t z in the z-direction is necessary to overcome the Mermin-Wagner theorem, somewhat larger t z 's may suppress T C 38,39 ).

RESULTS AND DISCUSSION
DDFT + DMFT multi-orbital calculations Let us now discuss these results in more detail. We start with a DFT calculation [cf. Supplementary Note 1] which puts the Nd-4f orbitals just above E F . But since their hybridisation with the Ni-3 d x 2 Ày 2 orbital is weak 40 , jV x 2 Ày 2 ;4f j ¼ 25meV, see Supplementary Note 2, they will localise and not make a Kondo effect. This localisation can be described e.g., by the spin-splitting in DFT+U, or by including the Nd-4f states in the core. It leaves us with a well defined window with just five Nd-5d and five Ni-3d around the Fermi energy. For these remaining ten orbitals we do a Wannier function projection (see Supplementary Note 2) and subsequent  DMFT calculation with constrained random phase approximation (cRPA) calculated inter-orbital interaction U 0 ¼ 3:10eV (2.00 eV) and Hund's exchange J = 0.65 eV (0.25 eV) for Ni (Nd) 28 . Figure 2 presents the calculated DMFT spectral function for these ten bands. Let us first concentrate on Sr 0.2 Nd 0.8 NiO 2 for which also the k-resolved spectral function on the right hand side is shown. Clearly in DMFT there is a single, compared to the DFT strongly renormalized Ni-3d band of d x 2 Ày 2 character crossing E F = 0, see the zoom in Fig. 2e. Besides, there is also a pocket around the A-point of predominately Nd-5d xy character, but the Γ-pocket is shifted above E F , cf. Supplementary Notes 3 and 4 for other dopings. Hence, we have two bands of predominately Ni-3d x 2 Ày 2 and Nd-5d xy character. Their hybridisation is zero, see Supplementary  Table III, which can be inferred already from the lack of any splitting around the DFT crossing points in Fig. 2d, e. There is some hybridisation of the Nd-5d xy with the other Ni-bands, which results in minor spectral weight of Nd-5d xy character in the region of the other Ni-3d bands at −0.5 to −2.5 eV in Fig. 2a-c and viceversa of the Ni-3d z 2 orbital between 1 and 3 eV. This admixing is however so minor, that it can be described by properly admixed, effective orbitals that are away from the Fermi energy, without multi-orbital physics.
If we study the doping dependence in Fig. 2a-c, we see that with Sr-doping all bands move upwards. More involved and beyond a rigid-band picture, also the Ni-3d x 2 Ày 2 band becomes less and less correlated. The effective mass enhancement or inverse quasiparticle weight changes from m * /m = 1/Z = 4.4 for the undoped compound to m * /m = 2.8 at 20% Sr-doping to m * /m = 2.5 at 30% Sr-doping. The Hubbard bands gradually disappear.
At 30% Sr-doping we are in the situation that the other Ni-3d bands are now immediately below the Fermi energy. Hence around this doping it is no longer justified to employ a Nid x 2 Ày 2 -band plus Nd-d xy -pocket around A picture to describe the low-energy physics. The other Ni-3d orbitals become relevant. Because of the weak hybdization, the Nd-d xy -pocket only acts as an electron reservoir, which changes the doping of the Nid x 2 Ày 2 -band from the lower x-axis in Fig. 1b to the upper x-axis.
For this d x 2 Ày 2 -band we have done a separate Wannier function projection which results in the hopping parameters t = 395 meV, t 0 ¼ À95 meV, t″ = 47 meV between first-, second-, and third-nearest neighbors on a square lattice. We employed the onsite interaction U = 3.2 eV = 8t which is slightly larger than the cRPA zero frequency value [U = 2.6 eV 16,20 ] to mimic the frequency dependence, as well as beyond cRPA contributions 41 . In the Supplementary Note 5 we present further calculations within a realistic range of U values. The hopping parameters in the z-direction are negligibly small t z = 34 meV, leaving us with a-to a good approximation-two-dimensional one-band Hubbard model.

T C in DΓA
This two-dimensional one-band Hubbard model with the properly translated doping according to Fig. 1b can now be solved using more sophisticated, numerically expensive methods such as the DΓA 37 . DΓA starts from a local irreducible vertex, which depends on three frequencies and is calculated at DMFT convergence. From this, DΓA builds non-local vertices and self-energies through ladder or parquet diagrams. The types of diagrams are similar as in the random phase approximation (RPA) or fluctuation exchange (FLEX), but the important difference is the starting vertex which is non-perturbative. As a consequence all local DMFT correlations are included as well non-local spin-, charge-or superconducting fluctuations. This way, among others, (quantum) critical exponents, pseudogaps and superconductivity in the one-band Hubbard model can be calculated 37 . Applying ladder DΓA to cuprates yields quite reasonable T C 's 42 , giving us some confidence to now explore the largely unknown nickelates. As a matter of course effects beyond the Hubbard model, such as disorder and phonons or a strengthening of charge fluctuations 43 by non-local interactions which all are also considered to be of some relevance for superconductivity 44 are not included. Studying a (tetragonal) rotational symmetry breaking 45 would require a full parquet DΓA or an eigenvalue analysis like we do for d-wave superconductivity here. Figure 3 shows the thus obtained DΓA Fermi surfaces at different dopings and two different temperatures. For the superconducting Sr 0.2 Nd 0.8 NiO 2 , which corresponds to n d x 2 Ày 2 ¼ 0:822 electrons per site in the Ni-d x 2 Ày 2 -(Wannier)-band, we have a well defined hole-like Fermi surface, whereas for n d x 2 Ày 2 ¼ 0:9 and, in particular for n d x 2 Ày 2 ¼ 0:95, we see the development of Fermi arcs induced by strong antiferromagnetic spin fluctuations. The A = (π, π, π)-pocket is not visible in Fig. 3 because it is only included through the effective doping in the DΓA calculation. In any case it would be absent in the k z = 0 plane and only be visible around k z = π.
Whereas the Ni-3d x 2 Ày 2 -band is strongly correlated and has a Fermi surface that is prone to high-T C superconductivity, the Apocket is weakly correlated and hardly hybridises with the former. Nonetheless for some physical quantities, different from superconductivity, it will play a role. For example, the A-pocket will give an electron-like (negative) contribution to the Hall coefficient. Experimentally, the Hall coefficient 11 is large and electron-like (negative) for NdNiO 2 , whereas it is smaller and even changes its sign from negative (electron-like) to positive (hole-like) below 50 K for Sr 0.2 Nd 0.8 NiO 2 . This doping dependence at low temperatures is strikingly different to the cuprates and can be naturally explained from the balance between a 3d x 2 Ày 2 hole-contribution (like for cuprate) and an additional electron-like A-pocket contribution. In Fig. 3, we observed a pseudogap behaviour in the underdoped region. This should reduce the hole contribution somewhat as in underdoped cupates 46 . We expect hence that the electron-like A-pocket contribution wins in the low doping regime, but the 3 d x 2 Ày 2 hole-like contribution might well dominate for larger dopings, qualtitatively explaining the sign change of the Hall coefficient.
Moreover, in Fig. 3 we further see that for T = 92 K = 0.02t, we have a strong scattering at the antinodal point k = (π, 0), whereas at the lower temperature T = 46 K = 0.01t, we have a more well defined band throughout the Brillouin zone. This indicates that the 3d x 2 Ày 2 hole contribution is suppressed at higher temperatures, whereas the uncorrelated A-pocket contribution essentially remains the same. This possibly explains why the Hall coefficient changes sign 11 when increasing temperature for Sr 0.2 Nd 0.8 NiO 2 .
At the temperatures of Fig. 3, Sr x Nd 1−x NiO 2 is not yet superconducting. But we can determine T C from the divergence of the superconducting susceptibility χ, or alternatively the leading superconducting eigenvalue λ SC . These are related through, in matrix notation, χ = χ 0 /[1 − Γ pp χ 0 ]. Here χ 0 is the bare superconducting susceptibility and Γ pp the irreducible vertex in the particle-particle channel calculated by DΓA; λ SC is the leading eigenvalue of Γ pp χ 0 . If λ SC approaches 1, the superconducting susceptibility is diverging.
In Fig. 4 we plot this λ SC vs. temperature, and see that it approaches 1 at e.g., T = 36 K = 0.008t for n d x 2 Ày 2 ¼ 0:85. But outside a narrow doping regime between n d x 2 Ày 2 ¼ 0:9 and 0.8 it does not approach 1. There is no superconductivity. Let us also note that the phase transition is toward d-wave superconductivity which can be inferred from the leading eigenvector corresponding to λ SC .
Altogether, this leads to the superconducting dome of Fig. 1b. Most noteworthy Sr 0.2 Nd 0.8 NiO 2 which was found to be superconducting in experiment 11 is close to optimal doping n d x 2 Ày 2 ¼ 0:85 or Sr 0.16 Nd 0.84 NiO 2 . Our results call for a more thorough investigation of superconductivity in nickelates around this doping, which is quite challenging experimentally [12][13][14] .
Besides a slight increase of T C by optimising the doping, and further room of improvement by adjusting t 0 and t″, our results especially show that a larger bandwidth and a somewhat smaller interaction-to-bandwidth ratio may substantially enhance T C , see Supplementary Fig. 7, qualitatively similar as in fluctuation exchange (FLEX) calculations 16 . One way to achieve this is compressive strain, which enlarges the bandwidth while hardly affecting the interaction. Compressive strain can be realised by e.g., growing thin nickelate films on a LaAlO 3 substrate with or without SrTiO 3 capping layer, or by Ca-doping instead of Srdoping for the bulk or thick films. Another route is to substitute 3d Ni by 4d elements, e.g., in Nd(La)PdO 2 which has a similar Coulomb interaction and larger bandwidth 17,47 .
Note added: In an independent DFT+DMFT study Leonov et al. 48 also observe the shift of the Γ-pocket above E F with doping for LaNiO 2 and the occupation of further Ni 3d orbitals besides the 3d x 2 Ày 2 for large (e.g., 40%) doping.
Second note added: Given that our phase diagram Fig. 1b has been a prediction with only a single experimental data point given, the most recently experimentally determined phase diagram 49,50 turned out to be in very good agreement. In particular if one considers, as noted already above, that the theoretical calculation should overestimate T C , while the exerimentally observed T C is likely suppressed by extrinsic contributions such as disorder etc. Supplementary Note 6 shows a comparison, which corroborates the modelling and theoretical understanding achieved in the present paper.
Third note added: In recent single particle tunnelling measurements 51 , the authors observed the dominant d-wave and the additional s-wave component of the superconducting gap function, which is apparently consistent with our scenario: the dominant dwave single orbital system with additional Fermi pockets.

Density functional theory
We mainly employ the WIEN2K program package 52 using the PBE version of the generalised gradient approximation (GGA), a 13 × 13 × 15 momentum grid, and R MT K max = 7.0 with a muffin-tin radius R MT = 2.50, 1.95 and 1.68 a.u. for Nd, Ni and O, respectively. But we also double checked against VASP 53 , which was also used for structural relaxation (a = b = 3.86 Å, c = 3.24 Å) 40 , and FPLO 54 , which was used for Fig. 1a and Supplementary Fig.  1. The Nd-4f orbitals are treated as open core states if not stated otherwise.

Wannier function projection
The WIEN2K bandstructure around the Fermi energy is projected onto maximally localised Wannier functions 55 using WIEN2WANNIER 56 . For the DMFT we employ a projection onto five Ni-3d and five Nd-5d bands; for the parameterisation of the one-band Hubbard model we project onto the Ni-3 d x 2 Ày 2 orbital. For calculating the hybridisation with the Nd-4f, we further Wannier projected onto 17 bands, with the Nd-4f states now treated as valence bands in GGA.

Dynamical mean-field theory
We supplement the five Ni-3d plus five Nd-5d orbital Wannier Hamiltonian by a cRPA calculated Coulomb repulsions 28 U 0 ¼ 3:10eV (2.00 eV) and Hund's exchange J = 0.65 eV (0.25 eV) for Ni (Nd or La). The resulting Kanamori Hamiltonian is solved in DMFT 36 at room temperature (300 K) using continuous-time quantum Monte Carlo simulations in the hybridisation expansion 57 implemented in W2DYNAMICS 58 . The maximum entropy method 59 is employed for the analytic continuation of the spectra.

Dynamical vertex approximation
For calculating the superconducting T C , we employ the dynamical vertex approximation (DΓA), for a review see ref. 37 . We first calculate the particle-particle vertex with spin-fluctuations in the particle-hole and transversal-particle hole channel, and then the leading eigenvalues in the particle-particle channel, as done before in ref. 42 . This is like the first iteration for the particle-particle channel in a more complete parquet DΓA 37 .

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.