Unconventional superconductivity without doping in infinite-layer nickelates under pressure

High-temperature unconventional superconductivity quite generically emerges from doping a strongly correlated parent compound, often (close to) an antiferromagnetic insulator. The recently developed dynamical vertex approximation is a state-of-the-art technique that has quantitatively predicted the superconducting dome of nickelates. Here, we apply it to study the effect of pressure in the infinite-layer nickelate SrxPr1−xNiO2. We reproduce the increase of the critical temperature (Tc) under pressure found in experiment up to 12 GPa. According to our results, Tc can be further increased with higher pressures. Even without Sr-doping the parent compound, PrNiO2, will become a high-temperature superconductor thanks to a strongly enhanced self-doping of the Ni \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${d}_{{x}^{2}-{y}^{2}}$$\end{document}dx2−y2 orbital under pressure. With a maximal Tc of 100 K around 100 GPa, nickelate superconductors can reach that of the best cuprates.


I. INTRODUCTION
Ever since the discovery of high-temperature superconductivity in LaBaCuO 4 [1], understanding or even predicting new unconventional (not electron-phonon-mediated) superconductors and identifying the pairing mechanism has been the object of an immense research effort.A new opportunity for a more thorough understanding arose with the discovery of superconductivity in several infinite-layer nickelates A 1−x B x NiO 2 [3,7,8,10,12,16], where A=La, Nd, Pr and B=Sr, Ca are different combinations of rare-earths and alkaline-earths.These nickelates are at the same time strikingly similar to cuprates (for this reason theory predicted nickelate superconductivity 20 years before experiment [8]) but also decidedly different.This constitutes an ideal combination to clarify the presumably common mechanism behind superconductivity in both systems.Superconductivity in nickelates was found to be quite independent of the rare earth A and dopant B [9] with a dome-like shape characteristic of unconventional superconductors.
Theoretical work has left little doubt that nickelates are, indeed, unconventional superconductors [10][11][12] and the similarity to the crystal and electronic structure of cuprates is striking [8,13].There are, however, subtle differences between cuprates and nickelates: Compared to Cu 2+ , the 3d bands of Ni 1+ are separated by a larger energy from the oxygen ones, hence hybridization is weaker and oxygen plays a less prominent role than in cuprates.On the other hand, the rare-earth A-derived bands cross the Fermi level in nickelates and form electron pockets.These electron pockets self-dope the Ni d x 2 −y 2 band with about 5% holes [2,12,[14][15][16][17][18][19], and prevent the parent compound from being an antiferromagnetic insulator.Given the inherent difficulty to incorporate the effect of strong electronic correlations, different theory groups have arrived at a variety of models for describing nickelates [2,16,19,[21][22][23][24][25][26].
Based on a minimal model consisting of a 1-orbital Hub- * simone.dicataldo@uniroma1.it bard model for the Ni d x 2 −y 2 band [12] plus largely decoupled electron pockets, that only act as electron reservoirs, Kitatani et al. [2] accurately predicted the superconducting dome in Sr-doped NdNiO 2 [2] prior to experiment [10,15].In particular, the agreement to more recent defect-free films [15] is excellent.This includes the quantitative value of T c , the doping region of the dome and the skewness of the superconducting dome, see Ref. [28].Also pentalayer nickelates [29] seamlessly fit the results of Ref. [28].In Refs.[2,30] some of us pointed out that larger T c 's should be possible if the ratio of interaction to hopping, U/t, is reduced.In a recent seminal paper, Wang et al. [13] reported a substantial increase of T c in Sr x Pr 1−x NiO 2 (x = 0.18) films on a SrTiO 3 (STO) substrate from 18 K to 31 K if a pressure of 12 GPa is applied in a diamond anvil cell.There are no indications of a saturation of the increase of T c with pressure yet.First calculations of the electronic structure, fixing the in-plane lattice constant to the ambient pressure value and relaxing (reducing) the out-of-plane c-axis have been presented [7].A large T c under pressure [33] or at least a resistivity drop [34], has also been reported in another nickelate: La 3 Ni 2 O 7 .With a 3d 7.5 electronic configuration and prevalent charge density wave fluctuations the mechanism in this compound is however clearly distinct from the (slightly doped) 3d 9 nickelates considered here.
In this work, we employ the same state-of-the-art scheme that was so successful in Ref. [2], which is based on density functional theory (DFT), dynamical mean-field theory (DMFT), and the dynamical vertex approximation (DΓA).We study the pressure dependence of the superconducting phase diagram of PrNiO 2 (PNO) with and without Sr doping.We find (i) a strong increase of the hopping t, almost by a factor of two, when going from 0 to 150 GPa, while the value of U, obtained through constrained random-phase approximation (cRPA), remains essentially unchanged as in cuprates [22].Importantly, pressure further results in (ii) deeper electron pockets, effectively increasing the hole doping δ of the Ni d x 2 −y 2 band with respect to half-filling.Altogether, this results in the phase diagrams shown in Fig. 1, the main result of our work.When going from (a) ambient pressure to (b) 50 GPa, T c increases by up to a factor of two and d-wave supercon- ductivity is observed in a much wider doping range -quite remarkably even without Sr doping.As a function of pressure, at doping fixed to x = 0.18 (c), the simulated phase diagram shows a very similar increase of T c from 0 to 12 GPa as in experiment [13].The figure further reveals that T c will continue to increase up to 49 K at 50 GPa, followed by a rapid decrease at higher pressures.For the parent compound, PrNiO 2 (x = 0, d), the enhanced self-doping alone is sufficient to turn it superconducting with a maximum predicted T c of close to 100 K at 100 GPa.

II. RESULTS
As the superconducting nickelate films are grown onto a STO substrate, particular care has to be taken when simulating the effect of isotropic pressure in the diamond anvil cell (cf.Supplemental Material (SM) [37] Section I A for a flowchart of the overall calculations and Section I C for further details of the pressure calculation).First, since the thickness of the film is 10-100 nm and thus negligible compared to that of the STO substrate, we calculate the STO equation of state in DFT and, from this, obtain the STO lattice parameters under pressure.Second, we fix the in-plane a (and b) lattice parameters to that of STO under pressure and find the lattice parameter c for the nickelate which minimizes the enthalpy at the given pressure.The resulting lattice constants are shown in Table I.This procedure better reflects the response of the system to the rather isotropic pressures realized in experiment and is more realistic than that used in Ref. [7] where the a-b lattice parameters had been fixed to that of unpressured STO [7].
With the crystal structure determined, we calculate the DFT electronic structure at pressures of 0, 12, 50, 100, and 150 GPa.Next, we perform a 10-and 1-orbital Wannierization around the Fermi energy, including all Pr-d plus Ni-d orbitals and only the Ni d x 2 −y 2 orbital, respectively.The DFT band structures and Wannier bands are shown in SM [37]    and as white lines in Fig. 2.
Following the method of Refs.[2,29], we supplement the Wannier Hamiltonian with a local intra-orbital Coulomb interaction of U = 4.4 eV (2.5 eV) and Hund's exchange J = 0.65 eV (0.25 eV) for Ni-3d (Pr-5d) as calculated in cRPA [24].For the thus derived 10-band model we perform DMFT calculations.The resulting DMFT spectral function of undoped PNO is shown in Fig. 2, for 0 and 50 GPa.We see that the Ni d x 2 −y 2 orbital crossing the Fermi energy is strongly quasiparticle-renormalized compared to the DFT result (white lines).In addition, there are pockets at the Γ and A momenta, which essentially follow the DFT band structure without renormalization.
Important for the following is that, with the overall increase of bandwidth under pressure, the size of the pockets grows dramatically under pressure in DFT and DMFT alike.The enlargement of the Γ pocket can also be seen from the Fermi surface Fig. 2 (d) vs. (b).The effect at higher pressures is shown in SM [37] Fig. S7-S8.
The results of the 10-band model show that the low-energy physics of the system boils down to one-strongly correlated Ni d x 2 −y 2 orbital plus weakly correlated electron pockets.Here, the A-pocket does not hybridize by symmetry with the Ni d x 2 −y 2 band, as is evident by the mere crossing of both between The above justifies a one-band minimal model for superconductivity in PNO [2,13,30], with the working hypothesis that superconductivity arises from the correlated Ni d x 2 −y 2 band only.However, the effective hole-doping δ of the Ni d x 2 −y 2 band (relative to half-filling) has to be calculated from the 10-band DFT+DMFT to properly account for the electrons in the pockets.In the following, it is thus imperative to always distinguish between the number of holes corresponding to Sr substitution of the Pr site (chemical doping x) and the holes in the Ni d x 2 −y 2 band compared to half filling (effective doping δ).
The electron pockets induce a nonlinear dependency of δ from x, and their growth with pressure P causes δ to increase by about 0.06 from 0 to 100 GPa, see SM [37] Fig. S6.
Using the effective doping δ of the Ni d x 2 −y 2 band, we perform a second DMFT calculation for the single Ni d x 2 −y 2 orbital, which we describe as a single-band Hubbard model with an interaction of U = 3.4 eV.This U is smaller than for the 10-band model due to additional screening, but it is notably insensitive to pressure (cf.SM [37] Tab.S2).The main effect of pressure is instead the increase of t as summarized in Table I and the already mentioned enhanced self-doping.
In Fig. 3, we show the spectral function A(k, ω) of the 1band model for PNO as a function of pressure and x = 0.The panels for 0 and 50 GPa can be compared to Fig. 2 (a,c) and show that the 1-band model reproduces the renormalization of the Ni d x 2 −y 2 orbital in the fully-fledged 10-band calculation.
Hubbard bands are visible at all pressures in Fig. 3, but become more spread out and less defined with increasing pressure.Simultaneously, the effective mass m * decreases, and the bandwidth widens.Similar results but for the experimentally investigated Sr-doping x = 0.18 can be found in SM [37] Fig. S11, and as a function of doping at 50 GPa in Fig. S10.
Next, we calculate the superconducting T c using DΓA [29,32].In Fig. 1, we follow four different paths in parameter space: as a function of doping at (a) 0 GPa and (b) 50 GPa as well as as a function of pressure with Sr-doping (c) x = 0.18, i.e., for the parent compound, and (d) x = 0.At ambient pressure in Fig. 1 (a) our results are in excellent agreement with our previous calculations for other nickelates [2,30], cf.SM [37] Fig. S14.The small differences are ascribable to the slightly different material, and are in good agreement with experiment [3,15].We find the effect of pressure to be significant: At 50 GPa, the maximum T c is enhanced by a factor of two compared to ambient pressure, while the maximum slightly shifts to lower doping (x = 0.10).Remarkably, even the undoped parent compound becomes superconducting at 50 GPa [x = 0 in Fig. 1 (b)], due to the increased self-doping from the electron pockets.
The experimental Sr-doping of x=0.18 [41] is close to optimal doping at 0 GPa.Increasing pressure in Fig. 1 (c), we observe an increase of T c by 0.81 K/GPa in excellent agreement with the experimental rate of 0.96 K/GPa [41], for pressures up to 12 GPa.The predicted T c of 30 K at 0 GPa is slightly higher in theory than the experimental 18 K [41], but still in good agreement.As pressure increases beyond 12 GPa, T c continues to grow and peaks at around 50 GPa with 49 K, before decreasing for higher pressures.
Most striking is the result for the undoped compound PNO in Fig. 1 (d).Here, superconductivity sets in below 50 GPa and peaks at almost 100 K around 100 GPa.Intrinsic doping Figure 4: The considered four paths in the U/t vs. effective hole doping δ parameter space: at a fixed pressure of (a) 0 GPa and (b) 50 GPa as a function of Sr-doping x; as a function of pressure for fixed Sr-doping (c) x = 0.18 and (d) x = 0.00.The gray color bar indicates the strength of superconductivity (superconducting eigenvalue λ at T = 0.01t; from [30]).The secondary y axis reports the pressure corresponding to the U/t values shown.from the electron pockets is sufficient to make the parent compound superconducting at high temperature.

III. DISCUSSION
To rationalize our results, we plot the four paths at fixed pressure, respectively, fixed Sr-doping x in Fig. 4, but now as a function of U/t and the effective hole doping δ of the Ni d x 2 −y 2 orbital.Superimposed is the DΓA superconducting eigenvalue λ [30], with the darker gray regions corresponding to a higher T c .
The application of an isotropic pressure on infinite-layer PrNiO 2 has two effects: First, it boosts the hopping t of the Ni d x 2 −y 2 orbital, which at 150 GPa becomes almost twice as large than at 0 GPa, see Table I.This increases the overall energy scale and thus enhances T c .Since U does not change significantly, the ratio U/t also decreases.This is preferable for superconductivity since at ambient pressure PNO exhibits a U/t slightly above the optimum of U/t ∼ 6 (above the darker gray region in Fig. 4).However, at high pressures of e.g. 100 GPa and 150 GPa curves (c) and (d) have passed the optimum in Fig. 4; T c in Fig. 1 decreases again.
Second, pressure enhances the effective hole doping δ even at fixed Sr-doping x, as the electron pockets become larger.For this reason, curves (c) and (d) in Fig. 4 deviate from a vertical line.For Sr-doping (c) x = 0.18, which is close to op-timum at 0 GPa, the curve moves away from optimum doping to the overdoped region when pressure is applied.This is a major driver for the decrease of T c above 50 GPa in Fig. 1 (c).In stark contrast, for the parent compound (x = 0; d) the effective hole doping δ goes from underdoping to optimal doping, and only at much larger pressures to overdoping and too small U/t.Consequently, PNO without doping hits the sweet spot for superconductivity in Fig. 4 at a pressure between 50 and 100 GPa.

IV. CONCLUSION
In short, our results strongly suggest that experiments for infinite-layer nickelates are still far from having achieved their maximum T c .Surprisingly, the maximum T c of almost 100 K is predicted to be found in undoped PrNiO 2 between 50 and 100 GPa.This places nickelates almost on par with cuprates in the Olympus of high-T c superconductors.The nickelate phase diagram under pressure will not only exhibit a significant increase in T c but also a wider dome.In particular, the maximum of this dome is shifted to lower Sr-doping x when a pressure of 50 -100 GPa is applied.
Such pressures can be achieved experimentally in diamond anvil cells.An alternative route to obtain the same in-plane lattice compression is using a substrate with smaller lattice parameters.For example, LaAlO 3 , YAlO 3 and LuAlO 3 have lattice parameters of 3.788 Å [42], 3.722 Å [43] and 3.690 Å [44], respectively, which are already close to the inplane lattice constants at 50 GPa.As this approach would not change the out-of-plane lattice parameter, the self-doping of the Ni d x 2 −y 2 band from the electron pockets should be less important.Hence we expect that with these substrates a higher T c might be achieved, but only with at least 10% doping.

V. METHODS
In this section, we summarize the computational methods employed.The interested reader can find additional information in the Supplementary Material [37], and data and input files for the whole set of calculations in the associated data repository [1].
Density functional theory calculations were performed using the Vienna ab-initio simulation package (VASP) [3,5] using projector-augmented wave pseudopotentials and Perdew-Burke-Ernzerhof exchange correlation functional adapted for solids (PBESol) [4,48], with a cutoff of 500 eV for the plane wave expansion.Integration over the Brillouin zone was performed over a grid with a uniform spacing of 0.25 Å −1 and a Gaussian smearing of 0.05 eV.Wannierization was performed using wannier90 [18].
The Hubbard U of the 1-orbital setup under pressure was computed from first principles using the constrained random phase approximation (cRPA) in the Wannier basis [19] for entangled band-structures [20], relying on a DFT electronic structure obtained from a full-potential linearized muffin-tin orbital method [21].
DMFT calculations were performed using w2dynamics [28], with values of U, J, and t as detailed in the main text, and in Tab.S1 and SM [37] Tab.S3.The 10-band calculations were performed at a temperature of 300 K, with a total of 30 iterations to converge the local Green's function, and a final step with higher sampling.The 1-band DMFT calculations were performed at variable temperature, with a total of 70 iterations, and a final step with higher sampling, which was increased at lower temperatures.
The calculation of the non-local quantities via ladder DΓA and the solution of the linearized Eliashberg equation was performed starting from the local vertex calculated with w2dynamics using our own implementation, available upon reasonable request.energy.Nevertheless, we checked extensively the quality of the VCA against Vergard's law, and on the structural and electronic properties of Pr 0.75 Sr 0.25 NiO 2 , compared with the results on a 2×2×2 supercell.As these checks are quite extensive and we do not want to interrupt the further discussion of the workflow here, we present in Section VII C, Figures S15 (Vergard's law) and S16 (comparison with 2×2×2 supercell).This aspect of Figure S4, which we present already in Section VI C for its information on how to obtain the c-axis parameter, is also discussed in Section VII C. All these tests confirmed that the VCA is consistent with the results obtained for supercells in the structure studied.

C. Structural Relaxation
In this section, we report the main results for the structural relaxation of the SrTiO 3 (STO) substrate.In Fig. S2 we show the equation of states along with the results from a fit with the Birch-Murnaghan equation for STO as a function of pressure.In Fig. S3 we show the corresponding lattice parameter.
The effect of isotropic pressure was computed in two steps, including the effect of the SrTiO 3 (STO) substrate on the in-plane lattice parameter, as well as the effect of pressure on the c axis.This strategy differs substantially from Ref. [S7], where the in-plane lattice constant is kept constant, as it is not suited to study the rather isotropic pressures that are realized in experiments using diamond anvil cells.
Our method of computing the crystal structure of Pr 1−x Sr x NiO 2 , on the other hand, is motivated by the consideration of how the actual crystal is grown in experiments.Infinite-layer nickelates are grown over a perovskite substrate.Often STO is used [S8-S12], this also includes the experiments by Wang et al. [S13] which motivated the present study.But, NdGaO 3 (NGO) [S14] and (LaAlO 3 ) 0.3 (Sr 2 TaAlO 6 ) 0.7 (LSAT) [S15] was employed as well.The nickelate layer is grown with a thickness between 10 and 100 nm [S8-S12, S14, S16] over a bulk substrate which can be regarded as infinite, and capped again with a few layers of the same substrate.Hence we work in the hypotheses that: 1.The nickelate is forced to assume the same in-plane lattice constant of the substrate on which it is grown.
2. In the xy plane the elastic response of the system is dominated by the substrate, due to the substrate being much thicker.
3. Along the z direction, the nickelate is not constrained by the substrate, hence its response is independent from it.
The above is then modelled in the two following steps: 1. We compute the equation of state for bulk STO in the cubic phase.At a given pressure, the equation of state is used to extract the in-plane lattice constants a = b (Supplementary Figure S2).
2. With a and b fixed to the value given by STO at the chosen pressure, we compute the enthalpy of the nickelate phase as a function of the c axis, see Fig. S4.The c value that minimizes the enthalpy corresponds to the equilibrium value.

D. Wannier Functions
The DFT bandstructure calculated in VASP is subsequently mapped onto a 10-and a 1-band Wannier basis using maximally localized Wannier orbitals and the wannier90 code [S18].Here, d orbitals centered on Pr and Ni were used for the initial projections for the 10-bands calculations, and a single d x 2 −y 2 orbital centered on Ni was used as initial projection for the 1-band calculation.The reader interested in the energy ranges for the disentanglement and frozen windows of each calculation can find the corresponding input files in [S1].The Hamiltonian H(k) was obtained as the Fourier transform of the wannier90 output.
Fig. S5 shows the direct comparison between Wannier and DFT bands.While there are some deviations of the 10-band model and the DFT band above the Fermi energy where additional bands -besides the 10-bands considered-cross, the agreement at low energies is excellent.And this is the relevant region for the subsequent DMFT calculation.From the wannierization, we can also extract the hopping amplitudes t, t ′ , and t ′′ for nearest, next-nearest, and next-nextnearest neighbour hopping for the effective 1-orbital model.These parameters for different pressures and dopings are reported in Table I and serve as a DMFT input for the 1-band calculation.In case of the fully-fledged 10-band DMFT calculation, we used the full H(k) of the Wannier bands, without restriction to shorter-range hoppings.

E. Constrained random phase approximation
This Wannier Hamiltonian needs to be supplemented by the Coulomb interaction.To this end, the static Hubbard interaction U was computed from first principles using the constrained random phase approximation (cRPA) in the Wannier basis [S19] for entangled band-structures [S20].Here, the underlying electronic structure was computed from DFT using a full-potential linearized muffin-tin orbital (fplmto) method [S21] and the local density approximation, applied to bulk LaNiO 2 using the relaxed tetragonal structures from Sec. VI C. The calculations use 10 3 reducible k-points, and a muffin-tin radius (RMT) for Ni of 1.97 Bohr radii, except for P = 100 GPa, where RMT=1.9.At P = 50 GPa the difference in RMT changes U by merely 0.4%.The cRPA results for the 1-band model are shown in Table II.Note that the Hubbard U can indeed display a non-trivial dependence on pressure [S22].E.g., in cuprates in-plane compression increases the local interaction in a d x 2 −y 2 setting [S23].
We find both the screened and the bare interaction-U and V-to be essentially insensitive to pressure: Both the localization of the d x 2 −y 2 -derived Wannier function and the screening remain constant.This justifies keeping U unchanged with pressure.To account for the frequency-dependence of U in cRPA a slightly larger value is needed, and we use a fixed U = 3.4 eV as is stated in Table I was already employed in [S2].
Given their weak pressure dependence, we also keep the interaction parameters fixed for the 10-band calculation.These have been calculated before in cRPA [S24] and are displayed in Table III a. 10-band case In the 10-band case, we employ a Kanamori Hamiltonian, and considered the Nd and Ni atoms as two different impurity sites, with interactions described in Supplementary Table III.The convergence of the local Green's function was achieved through a three-step process.The first two steps were performed with an increasing sampling of the quantum Monte-Carlo solver, for a total of 30 iterations, while a third, final step with a larger number of iterations was employed to better sample the Green's function.
b. 1-band case In the 1-band case we employed a Hamiltonian with only density-density interaction and the same two-step scheme of the 10-band case, with the addition of a fourth step with much larger sampling to obtain the local two-particle Green's function.

G. Dynamical vertex approximation
Based on this 1-band model, we perform ladder DΓA calculations to obtain the nonlocal magnetic and superconducting susceptibility starting from the local two-particle Green's function.As explained in more details in [S29], from the local twoparticle Green's function, first a local vertex Γ that is irreducible in the particle-hole channel is determined.From the local Γ in turn we calculate the DΓA lattice susceptibility using the Moriya-λ correction [S30-S32].In our case, the dominant susceptibility is the magnetic one.From this susceptibility in turn we extract the irreducible vertex in the particle-particle or Cooper channel Γ pp , cf. [S29, S33].Finally, from Γ pp and the bare susceptibility, we obtain the superconducting eigenvalue λ.Superconductivity is signalled by the leading eigenvalue (in our case this is in the d-wave channel) approaching one.
Following Ref. [S2], where low critical temperatures did not allow for a direct calculation below the critical temperature, the superconducting eigenvalue λ was fitted with the function λ(β) = A − B * np.log(β) to extrapolate the result to λ = 1.

VII. ADDITIONAL RESULTS
In this section, we report additional results from our study.

A. DFT+DMFT
In Fig. S6 we show the effective filling of the Ni d x 2 −y 2 band, obtained from the 10-band DMFT calculations as described in the main text.We computed the doping dependency at 0 and 50 GPa, and the pressure dependence at fixed Sr concentration x = 0.18.In general, the filling of the Ni d x 2 −y 2 band decreases linearly with increasing pressure and with larger Sr concentration.However, at 0 GPa the decrease flattens out for x ≥ 0.3, while it remains linear at 50 GPa.In addition to Fig. 2 of the main text, we show in Fig S7 the DMFT bandstructure of the parent compound, PrNiO−2, at further pressures, visualizing the evolution with pressure.This is supplemented by the Fermi surface displayed in Fig. S8 for the same pressure and doping x = 0. Note that at 150 GPa the electron pocket around Γ becomes so large that it touches the Ni d x 2 −y 2 band.At pressures higher than this point, the description of the correlated system in terms of 1 correlated orbital plus decoupled pockets likely needs to be refined.
In Fig. 3 of the main text, we showed the evolution of the 1-band DFT+DMFT spectrum of the parent compound (x = 0) as a function of pressure.This corresponds to path (c) in Fig. 1 and 4 of main text.Here, we also show the evolution with doping at 0 GPa [Fig.S9, path (a      Second in Fig. S14, we plot the change of T c as a function of pressure together with the change of t.This comparison clearly reveals that the difference between the parent compound (left) and 18% Sr-doping (right) originates from the parent compound moving to optimal doping at 100 GPa, whereas the doped sample moves from optimal doping to overdoped with pressure.

C. Tests of the Virtual Crystal Approximation
The Virtual Crystal Approximation should be approached carefully, especially when it is employed to interpolate between two atoms that are not adjacent on the periodic table.
We performed extensive checks of the quality of the Virtual Crystal Approximation (VCA) to simulate an average mixture of praseodymium (Pr) and strontium (Sr).In the following, we list the tests and their result, and briefly discuss their implication.
a. Vergard's Law The first simple test is to check that Vergard's Law is respected [S34], i.e. that the lattice parameter of the intermediate alloy is a weighted mean of the two isolate compounds.In Fig. S15 we report a comparison between the lattice parameter of a Pr-Sr mixture in a face-centered cubic structure at different concentrations, obtained with the VCA and in a 2×2×2 supercell.We note that not only the law is respected, but even the small deviation from the linear behavior are matched by the calculations in the supercell, i.e. they are a physical deviation, rather than an artifact of the VCA.In Fig. S4 we show the change in enthalpy of Pr 0.75 Sr 0.25 NiO 2 as a function of the c axis using the VCA and in five different supercell configurations.The equilibrium c value thus obtained is identical for all the cases considered, and close to the experimental value [S9], and deviations are only seen away from equilibrium.
c. Electronic structure The last test involves a direct comparison of the electronic band structure between the VCA case and the five inequivalent supercells at x = 0.25.We employed the relaxed structures describe in the previous paragraph, but we note that their lattice parameters are identical within DFT accuracy.In Fig. S16 we show the electronic band structure for the VCA and the five different supercells.Also in this case the two results are in excellent agreement, with only a few minute differences due to the symmetry breaking induced by Sr in the supercell, which splits a few bands.

Figure 2 :
Figure 2: Spectral function for undoped PNO.Panels (a) and (c) show the DMFT spectral function A(k, ω) (color scale) and the Wannier bands (white lines) along a path through the Brillouin zone at temperature T = 300K.Panels (b) and (d) show the same spectral function in the k z = 0 plane.A(k x , k y , k z = 0, ω = 0).The Γ point is at the center, i.e. (k x = 0, k y = 0).

Figure 3 :
Figure 3: DMFT spectral function (color bar) for undoped PNO for the 1-band model and as a function of pressure from 0 to 150 GPa at T = 0.0125t.The original DFT Wannier band and the same band renormalized with the effective mass are shown as solid white and dashed green lines, respectively.

Figure S1 :
Figure S1: Flowchart summarizing the whole workflow employed in the project.

Figure S2 :
Figure S2: Equations of state E(V) and P(V) for SrTiO 3 from 0 to 200 GPa.The blue dots and the orange line denote the values computed within Density Functional Theory and the fit, using the Birch-Murnaghan equation of state [S17].In addition, we report the results of the curve fit.

Figure S3 :
Figure S3: Lattice parameter of STO as a function of pressure from our DFT calculations.

Figure S4 :
Figure S4: Enthalpy versus value of the c axis for Pr 0.75 Sr 0.25 NiO 2 using the VCA and in five different supercells.The VCA value is shown as a black line, and individual points are shown as black dots.The results for the supercells are shown as colored lines.

Figure S5 :
Figure S5: DFT electronic band structure and 10-and 1-band wannierization as a function of pressure.The DFT bands, the 10 bands wannierization and the 1 band wannierization are shown as black solid, red dashed, and blue dashed lines, respectively.The figures are shown in order of ascending pressure.From top left to bottom right: 0, 12, 50, 100, and 150 GPa, respectively.

Figure S6 :
Figure S6: Left panel: Effective filling n d x 2 −y 2 of the Ni d x 2 −y 2 band as a function of Sr doping x, calculated in DMFT for the 10-band model.The results for 0 and 50 GPa are indicated as blue and orange lines, respectively.Right panel: n d x 2 −y 2 of the Ni d x 2 −y 2 band as a function of pressure at fixed doping x = 0.00 (blue curve) and x = 0.18 (orange curve).The hole doping of the main text is δ = 1 − n d x 2 −y 2 .

Figure S7 :
Figure S7: Evolution of the spectral function A(k, ω) of PrNiO 2 along a high-symmetry path in reciprocal space for the 10-bands DMFT calculations.(a), (b), (c), and (d) correspond to calculations at 12, 50, 100, and 150 GPa, respectively.The spectral function A(k, ω) is shown as a color scale.The DFT bands interpolated from Wannier functions are shown as white lines.

Figure S8 :
Figure S8: Evolution of the spectral function A(k, ω) of PrNiO 2 along the Fermi surface at k z = 0 for the undoped PrNiO 2 as a function of pressure.

Figure S9 :
Figure S9: Spectral function computed from the local self-energy at fixed pressure P = 0 GPa as a function of doping x, from 0.0 to 0.30, at β = 80t.The spectral function A(k, ω) is shown as a color gradient.

Figure S10 :
Figure S10: Spectral function computed from the local self-energy at fixed pressure P = 50 GPa as a function of doping x, from 0.0 to 0.30, at β = 80t.The spectral function A(k, ω) is shown as a color gradient.

Figure S11 :
Figure S11: Spectral function computed from the local self-energy at fixed doping x = 0.18 and as a function of pressure, from 0 to 150, at β = 80t.The spectral function A(k, ω) is shown as a color gradient.

Figure S12 :
Figure S12: Comparison of the superconducting dome at 0 GPa calculated for Sr x Pr 1−x NiO 2 (our work) and for Sr x Nd 1−x NiO 2 (Ref.[S2]).

Figure S13 :
Figure S13: Leading superconducting eigenvalue λ as a function of temperature at x = 0 and as a function of pressure.The critical temperature is selected as the value at which the curve (or its extrapolation) crosses the λ = 1 line.

Figure S14 :
Figure S14: Trend of the superconducting T c as a function of pressure, compared with the first-nearest-neighbor hopping t.

Figure S15 :
Figure S15: Vergard's Law for a Pr-Sr alloy in the face-centered cubic phase.Blue diamonds, green circles, and the orange line indicate the result obtained using the VCA, a 2×2×2 supercell, and the ideal law, respectively.

Figure S16 :
Figure S16: Electronic band structure for Pr 0.75 Sr 0.25 NiO 2 using the VCA and in five different supercells.The VCA value is shown as a black line, and the results for the supercells are shown as colored lines.

Table I :
Ab initio values for the lattice constants and the hoppings of the 1-orbital Hubbard model for PrNiO 2 under pressure.Here, t, t ′ , and t ′′ are the nearest, next-nearest, and next-next-nearest neighbour d x 2 −y 2 -hoppings.

Table I :
Calculated quantities for the single-band Hubbard model of the Ni d x 2 −y 2 band at the pressures and physical Sr-doping values x employed in this paper.The effective filling n = 1 − δ is reported.

Table II :
. Local Hubbard interaction U and bare (unscreened) Coulomb interaction V in the maximally localized Wannier function basis for the 1-band (Ni d x 2 −y 2 -band) model as calculated by cRPA.Pressure Temperature U Ni J Ni U' Ni U Nd J Nd U' Nd

Table III :
Intra-and inter-orbital Coulomb repulsion (U, U') and Hund coupling J for the Kanamori Hamiltonian employed in the 10-band DMFT calculations.