Orbital-Nematic and Two-Fluid Superconductivity in Hole-Doped NdNiO₂

Abstract

Based on DFT + DMFT, we investigate the orbital-nematic and s-wave superconducting states of a hole-doped ${\mathrm{NdNiO}}_2$ superconductor. We emphasize the role played by the interorbital proximity effect in determining the orbital-selective electronic state both in the normal and superconducting phases. Specifically, we show how orbital-nematic plus s-wave pairing symmetry acting on the $xz$ orbital might have pronounced effects on proximitized non-superconducting Ni-$3d$ orbitals due to many-particle electron–electron interactions. This work represents a step forward in understanding the emergence of two-fluid superconductivity (with superconducting $xz$ and non-superconducting $xy,yz,x^2-y^2,3z^2-r^2$ channels) in hole-doped ${\mathrm{NdNiO}}_2$ superconductors.

1. Introduction

There is a complex interplay of spin, charge and orbital degrees of freedom in strongly correlated electron systems [1], such as layered cuprates [2] and iron superconductors [3], or in d-band transition metals on general grounds [4]. In these correlated many-particle systems, a variety of theoretical and experimental studies have revealed the presence of competing symmetry-broken orders in the normal state [5,6]. Although the pairing-glue mechanism for unconventional (non-BCS) superconductivity is under debate [7,8,9], it has been recognized that intertwined orders might play a fundamental role in determining the exotic nature of the superconducting phase.

Finding superconductivity in Sr-doped ${\mathrm{NdNiO}}_2$ thin films [10] has reinforced the search for analogies and differences between distinct superconducting families, shedding new light on the emergence of complex quantum phase instabilities. An important open question related to these layered systems is whether superconductivity is two-dimensional like in cuprates [11] or more three-dimensional like in the multi-orbital (MO) iron superconductors [12,13]. Understanding the interplay of dimensionality and many-particle electronic correlations is an important step in the identification of spontaneous lattice symmetry breaking in normal [14] and superconducting [15] states. Extant theoretical studies indicate that orbital degrees of freedom [16,17,18,19,20] are relevant in infinite-layer nickelates, giving rise the question of whether orbital selectivity plays an important role in establishing an emergent superconducting state which might coexist with magnetism driven by Mottness [21,22].

Experiments investigating the critical currents and the angular dependence of magnetoresistance [15,23,24,25] reveal a reduction in the rotational symmetry of doped M${\mathrm{NiO}}_2$ (M = La, Nd, Pr) systems from four-fold $\left(C_4\right)$ to two-fold $\left(C_2\right)$ symmetry that might be a fingerprint of nematic phase instability, leading to $b\ne a$ [26], where $a,b$ are planar unit cell distances. Importantly, the coexistence of different competing orders [5,6] suggests an underlying physics with similarities akin to some Fe-superconductors [3,15,27,28], prompting us to question to what extent the correlated electronic properties of infinite-layer nickelates and Fe-superconductors are similar [13,29]. Here, we shed light on this problem, showing the role played by the dynamic MO proximity effect in the orbital-nematic [30,31,32] and s-wave superconducting [33,34,35] states of a hole-doped ${\mathrm{NdNiO}}_2$ superconductor. We explore the effect of MO electron–electron interactions [16,17,18,19,20] and orbital nematicity [27,30,31,32,36] in the Ni-$3d$ shell of ${\mathrm{NdNiO}}_2$.

In Fe-based superconductors, the antiferromagnetic order is preceded by, or in some cases concurrent with, a tetragonal–orthorhombic $\left(\mathcal{T}-\mathcal{O}\right)$ lattice distortion. In the unconventional FeSe superconductor, for example, a transition from a $\mathcal{T}$ to a weakly distorted $\mathcal{O}$ phase occurs at around 90 K, i.e., before the emergence of superconductivity below 8 K. The $\mathcal{O}$ phase is also referred to as the nematic phase because experimental measurements detect significant asymmetry along the a and b directions. It has been argued in Refs. [27,30] that orbital degrees of freedom can drive the nematic ordering in the form of a ferro-orbital (FOO) order [37,38]. This orbital ordering, which manifests itself as an unequal occupation of $xz,yz$ orbitals [36], breaks the in-plane local symmetry and generates two non-equivalent $a,b$ directions. Thus, as in systems with a stripe-type antiferromagnetic order, the nematic order breaks the rotational four-fold lattice symmetry of a high-T phase [39], inducing a $\mathcal{T}-\mathcal{O}$ structural transition. However, unlike hole-doped ${\mathrm{NdNiO}}_2$ superconductors, where rotational symmetry breaking appears below $T_c$ [15,24], nematic instability is a characteristic feature of the normal state, and thus, nematicity is considered to be a precursor to superconductivity.

These caveats, together with experimental observations [15,23,24,25], call for an investigation into the microscopic consequences of nematic-induced FOO within the $xz,yz$-orbital sector of hole-doped ${\mathrm{NdNiO}}_2$. Here, we address this problem using DFT + DMFT [40]. Our aim is to explore the emergent electronic state which arises due to nematic-induced residual interactions present in the many-particle fluid of infinite-layer ${\mathrm{NdNiO}}_2$. Thereupon, we extend our earlier DFT + DMFT study [41] towards an orbital-nematic state, and we show how unusual localization arises as direct instability of a correlated non-Fermi liquid metal near to orbital-selective Mottness. However, since the mechanism at the origin of the observed rotational symmetry breaking in superconducting nickelates is not yet fully understood [15,24], we simulate nematic asymmetry by enhancing the FOO induced by orthorhombic distortion. When the FOO (or orbital nematicity) is adjusted above a critical value, the correlated electronic DFT + DMFT state shows the emergence of an orbital-selective phase transition with coexisting classical Mott [42] and Mott–Kondo [43,44,45] localized electrons. As pointed out in earlier DFT + DMFT studies [20,41], within the $d^9$ (${\mathrm{Ni}}^{1+}$) ionic configuration, ${\mathrm{NdNiO}}_2$ is close to Mott localization [21,22], and electronic correlations promote interesting physical effects on hole doping. MO electron–electron interactions naturally induce changes in orbital polarization [36,46,47] yielding co-existent orbital-selective metallic, insulating and bad-metallic states. In this work, we employ the MO DFT + DMFT [40] treatment to microscopically unveil the effect of FOO on the electronic structure reconstruction in a five-orbital problem within the $d^{8.5}$ electronic configuration, a total electron band filling which is close to that reported in Ref. [48] for ${\mathrm{NdNiO}}_2$ and analogs.

However, before presenting our results for the orbital-nematic and s-wave superconducting states of a hole-doped ${\mathrm{NdNiO}}_2$ superconductor, we shall mention here that the DFT + DMFT results for ${\mathrm{NdNiO}}_2$ and analogs are sensitive not only to multi-orbital electron–electron interactions, but also to bare DFT inputs, particularly near the Mott metal–insulator transition, where small changes in orbital and band fillings derived using different computation schemes, together with the transfer of spectral weight induced by MO correlations, determines the phase boundary between the metallic and the Mott insulating states. After the publication of a plethora of studies, it is now recognized that complete many-particle treatment of the all-electron problem is not currently feasible [49]. For real systems of interest, some progress has been made by reducing the actual all-electron problem to a smaller and therefore more tractable many-body problem, as defined in the electronic subspace near the Fermi level $\left(E_F\right)$. In our earlier publications in this field of research [29], we have studied correlated electronic structure reconstruction within the minimal model approach consisting of $x^2-y^2$ and $3z^2-r^2$ orbitals for infinite-layer ${\mathrm{NdNiO}}_2$. However, to provide a more realistic description of its correlated electronic structure in a subsequent study [41], we considered both the $e_g$ and the $t_{2}g$ orbitals. Similarly to other five-orbital studies, our theory and the results below must be seen as a natural extension of or a step beyond the minimal two-band models discussed in the literature, which involves the identification of a more complete subset of single-particle states that are then used to construct the complete Ni $3d$-orbital space, where the correlated many-body problem is to be solved. As shown in the five-orbital problem for ${\mathrm{NdNiO}}_2$, and similarly to the DFT + DMFT results of Ref. [19], for example, the spectral functions of the low lying orbitals undergo an increase in energy, while the $x^2-y^2$ channel is dynamically shifted to low energies and becomes more populated to account for the total Ni-$3d$ band filling. A particularly interesting feature in our DFT + DMFT treatment for the five-orbital model of ${\mathrm{NdNiO}}_2$ is the emergence of low energy Kondo-quasiparticle resonances near $E_F$ in the $3z^2-r^2,xz,yz$ and $xy$ orbitals [41] as a result of the strong inter-orbital proximity effect induced by hidden MO correlations. Importantly, the emergence of quasiparticle peaks (or peak–dip–hump lineshapes) in these nearly fully polarized orbitals has also been derived for ${\mathrm{Nd}}_3{\mathrm{Ni}}_2{\mathrm{O}}_6$ and ${\mathrm{Nd}}_6{\mathrm{Ni}}_5{\mathrm{O}}_{12}$ systems [19], and for pure and doped ${\mathrm{NdNiO}}_2$ [16,50] and ${\mathrm{SrNiO}}_2$ [20], providing support to the orbital-selective electronic structure reconstruction in pore and doped nickelate superconductors in spite of different model parameter values, impurity solvers and one-band inputs to DFT + DMFT. Taken together, these earlier studies reflect the important role played by MO correlations in determining the orbital-selective electronic state in the infinite-layer nickelates or in MO systems in more general groups.

2. Theory and Results

Similarly to Ref. [27], the free-electron Hamiltonian considered here for ${\mathrm{NdNiO}}_2$ is $H_0={\sum }_{\mathbf{k},a,\sigma }{ϵ}_a\left(\mathbf{k}\right)c_{\mathbf{k},a,\sigma }^{†}{c}_{\mathbf{k},a,\sigma }+{\sum }_i\Delta (n_{i,xz}-n_{i,yz})$, where $a=xy,xz,yz,x^2-y^2,3z^2-r^2$ represent the Ni-$3d$ orbitals and ${ϵ}_a\left(\mathbf{k}\right)$ is the corresponding band dispersion of the tetragonal structural phase [51]. The last term of $H_0$ describes the orthorombicity (or the FOO) acting within the $xz,yz$-orbital sector [27,30,31,32]. These five Ni orbitals with lifted $xz,yz$-orbital degeneracy at finite $\Delta$ are the microscopic single-particle inputs for MO DFT + DMFT, which generates a strongly reshaped electronic structure, as shown below. The local MO interactions in ${\mathrm{NdNiO}}_2$ [41] considered here for $H_{int}$ are the intra- and inter-orbital Coulomb interaction terms, U and $U^{\prime }(=U-2J_H)$, and the Hund’s coupling $J_H$ [20]. We compute the one-particle Green’s functions $G_a\left(\omega \right)={\displaystyle \frac{1}{N}}{\sum }_{\mathbf{k}}{\displaystyle \frac{1}{\omega -{\sum }_a\left(\omega \right)-{ϵ}_a\left(\mathbf{k}\right)}}$ of the five-orbital Hamiltonian $H=H_0+H_{int}$ using the MO iterated perturbation theory (MO-IPT) as an impurity solver [52], which gives real frequency results in accordance with continuous-time quantum Monte Carlo simulations for real MO quantum systems [53].

Herein, we use the DFT density of states (DOS) computed in Ref. [41] as the input for the MO DFT + DMFT calculations. Figure 1 shows the orbitally projected DOS derived in Ref. [41]: based on experimental crystal data [51], DFT calculations for ${\mathrm{NdNiO}}_2$ in Ref. [41] are performed with the Quantum Espresso 7.2 package. As seen, the $x^2-y^2$ orbital is characterized by an asymmetric two-dimension-like electronic structure around the Fermi level, $E_F=\omega =0$. Consistent with earlier studies [54,55], the tetragonal symmetry of ${\mathrm{NdNiO}}_2$ lifts the Ni-$3d$-orbital degeneracy, with the $3z^2-r^2$ orbital being the ground-state orbital followed by the $yz/xz,xy,x^2-y^2$ orbitals. As mentioned above, and in analogy with the Fe-superconductor, the two-fold degeneracy is set within the $yz,xz$ orbital sector as a result of $D_{4h}$ splitting. However, unlike $3d^6$ superconducting Fe-containing materials, pronouced orbital polarization emerges in the $3d^9$ valence configuration of $\mathcal{T}$ Ni-superconductors. As is visible in Figure 1, apart from the half-filled $x^2-y^2$ orbital, all other $3d$ orbitals are nearly fully filled, with the top of their valence bands sitting slightly below $E_F$. How these orbital-selective DFT spectral functions of hole-doped ${\mathrm{NdNiO}}_2$ are reshaped by the interplay between dynamic correlations and orbital nematicity is discussed below.

Before discussing this problem, let us first discuss our DFT + DMFT results, computed using fixed $U=4.5$ eV (and $J_H=0.7$ eV) and the total Ni-$3d$ band filling $n=8.5$ of hole-doped ${\mathrm{NdNiO}}_2$, an occupation value consistent with those considered in Ref. [48]. As already pointed out in Ref. [41], the shape of the $x^2-y^2$ DFT + DMFT DOS displays emergent lower and upper Hubbard bands at approximately −2.5 and 1.8 eV. Interestingly, and similarly to Ref. [19], the spectral functions of the low-laying orbitals undergo an increase in energy, while the $x^2-y^2$ orbital is dynamically shifted to lower energies, increasing its band filling. Also noteworthy in our results for $\Delta =0$ in Figure 1 is the emergence of Kondo-quasiparticles near $E_F$ in the $3z^2-r^2,xz,yz,xy$ orbitals due to inter-orbital correlation effects induced by $U^{\prime }$ and $J_H$.

In Figure 1, we show the DFT + DMFT DOS for the Ni-$3d$ orbitals computed using $U=4.5$ eV and $\Delta =0.1$ eV. A closer examination reveals small but interesting changes in the DFT + DMFT spectral functions. The main role of the finite $\Delta$ is to remove the $xz,yz$-orbtial degeneracy of the $\mathcal{T}$ phase, inducing FOO. With $U=4.5$ eV ($U^{\prime }=3.1$ eV), MO dynamic correlations encoded in the MO-IPT self-energies (${\sum }_a\left(\omega \right)$) drive the noticeable transfer of spectral weight over energies in the order of 4.0 eV, particularly at high binding energies below $E_F$ in response to the renormalization induced by the orbital field $\Delta$. This feature, stemming from the $U^{\prime }$-driven interorbital proximity effect, is relevant to understanding strongly correlated quantum matter. Future observations of orbital fluctuations associated with nearly degenerate $xz,yz$-orbital states and sizeable spectral weight transfer across the $\mathcal{T}-\mathcal{O}$ transition should provide support to our proposal.

Although the degree of rotational symmetry breaking between the a-axis and b-axis is currently under debate [15,24], in Figure 2, we show the electronic structure evolution with an increasing orbital field $\Delta$. Similarly to Ref. [56], the enhanced nematicity induces spectral weight redistribution in the Ni-$3d_{xy,yz}$ orbitals with distinct pseudogap features at low emergence for $\Delta \le 0.3$ eV. Moreover, at $\Delta$ between 0.3 and 0.4 eV, a first-order transition occurs from a pseudogap to an orbital-selective electron-localized state, with distinct degrees of localization and band-gap features near $E_F$. However, since the $xz$ orbital is pushed below its position in the $\mathcal{T}$ phase, it implies enhanced orbital nematicity $\langle N\rangle \equiv {\displaystyle \frac{\langle n_{xz}\rangle -\langle n_{yz}\rangle }{\langle n_{xz}\rangle +\langle n_{yz}\rangle }}$ [27], i.e., the FOO order parameter [30] (here, $n_{xz,yz}$ corresponds to the occupation of the $xz,yz$ orbitals). As already discussed in Ref. [27], $U,U^{\prime }$ and $J_H$ will have a larger localizing effects on the more populated $xz$ states. Thus, the observed rotational symmetry breaking [15,24] might be linked to changes in orbital polarization and selective orbital incoherence, the latter already noticeable in the $\mathcal{T}$ crystal structure, as shown in Figure 1. Sizable MO correlations and orbital-selective Mottness computed using DFT + DMFT in Figure 2 are expected to provide qualitative reconciliation of the features seen in the nematic (or FOO) phase of infinite-layer nickelates and those observed in Fe-based superconductors [27]. We predict that similar orbital-selective many-particle physics induced by sizable $\langle N\rangle$ could be seen in strained infinite-layer nickelate thin films, since unlike pressure [57], strain sharpens the nematic order by increasing $xz,yz$ splitting.

To provide additional information regarding changes in the normal electronic state of the nematic ${\mathrm{NdNiO}}_2$ superconductor, in Figure 3, we display the frequency dependence of the self-energy $\left[{\sum }_a\left(\omega \right)\right]$ imaginary parts. As seen in this figure, and similarly to a nematic FeSe superconductor [56,57], the $xz$ orbital remains the most strongly correlated compared to the $yz$ orbital, and it might provide an important source for superconducting glue in orbital nematic superconductors [56]. Moreover, examination of the self-energy imaginary parts when approaching a first-order transition at ${\Delta }_c=0.4$ eV reveals orbital-selective behavior with coexistent Mott $(3z^2-r^2,x^2-y^2,xz)$ and Mott-assisted Kondo $(yz,xy)$ insulators. Meanwhile, the Mott-localized orbitals are characterized by a pole in $Im{\sum }_a\left(\omega \right)$ near $E_F$ [42] in the Mott-assisted Kondo insulator $Im{\sum }_a\left(\omega \right)=0$ in the gap region. Apart from a strong interorbital proximity effect, here, one might be tempted to link this Mott–Kondo behavior [43,44,45] to a reconstructed correlated insulator due to the dynamic transfer of the spectral weight of the DFT spectra induced by the interplay of many-particle electron–electron interactions and enhanced FOO in the normal state. However, the central conclusion to be drawn from Figure 2 and Figure 3 is that while the transfer of spectral weight from lower to higher energies is obtained on all Ni-$3d$ orbitals, the conduction and valence band spectral lineshapes of the nearly half-filled $x^2-y^2$ orbital remain close to those of the tetragonal phase as a manifestation of its intrinsic strongly correlated nature. Future studies, as a function of uniaxial or tensile stress [58,59], should show whether this is obtained in nematic ${\mathrm{NdNiO}}_2$ superconductors and related infinite-layer nickelate superconductors. If confirmed, it may have implications for the instabilities of a selective quantum critical regime when faced with competing orders. This problem is left to future theoretical and experimental studies.

Although the microscopic origin of the distinct paring symmetries of superconductivity is under debate [9,33,34,35,60], we now concentrate on the emergent electronic state induced by s-wave superconductivity [33,34,35] within the weakly nematic regime (see our results in Figure 1) of hole-doped ${\mathrm{NdNiO}}_2$. Since quasiparticles are steadfast excitations in good Fermi liquid metals, the s-wave (or BCS-like) pairing acting on these quasiparticles is forceful, particularly within the $xz$ orbital, which shows clear Kondo-quasiparticle excitations at low emergence for $\Delta$ = 0.1 eV (see Figure 1). Following the philosophy used earlier [61,62], we focus on the $xz$ channel considering a singlet superconducting s-wave gap [33,34,35,60]. Within this approximation, we model the two-fluid [63] orbital-nematic superconducting state in infinite-layer nickelates using the mean-field Hamiltonian $H_{SC}={\Delta }_{sw}{\sum }_k(c_{xz,k,\uparrow }^{†}{c}_{xz,-k,\downarrow }^{†}+H.c.)$, with ${\Delta }_{sw}$ being the induced s-wave pairing potential [33,34,35,60,64]. As shown below, the two-fluid [63] assumption made here, with superconducting $xz$ and non-superconducting $xy,yz,x^2-y^2,3z^2-r^2$ channels, has significant effects on the one-particle spectrum at lower energies for the orbital-nematic plus s-wave regime induced by the strong interorbital proximity effect hidden in the many-body problem of hole-doped ${\mathrm{NdNiO}}_2$.

To describe the excitation spectrum of hole-doped infinite-layer ${\mathrm{NdNiO}}_2$, which is mediated by the superconducting transition, we revisited our nematic normal state calculation for $\Delta$ = 0.1 eV in Figure 1 to treat $H_{pair}$ above using DFT + DMFT for the superconducting state [61,62]. Based on our conjecture for the s-wave pair-field ${\Delta }_{sw}$, the DFT + DMFT equations for the $xz$ orbital can be implemented to describe the superconducting phase. As in Ref. [65], the one-particle Green’s function is self-consistently computed by extending the orbital-nematic DFT + DMFT solution while taking into account the pairing potential term. After considering the ${\Delta }_{sw}$ term, the DFT + DMFT propagator for the $xz$ orbital reads [61,62]

$$\begin{array}{c}\hfill G_{xz,\sigma }(\omega ,\mathbf{k})={\displaystyle \frac{1}{\omega -{\Sigma }_{xz,\sigma }\left(\omega \right)-{ϵ}_{xz}\left(\mathbf{k}\right)-\frac{{\Delta }_{sw}^2}{\omega +{\sum }_{xz,\overline{\sigma }}^{*}\left(\omega \right)+{ϵ}_{xz}\left(\mathbf{k}\right)}}},\end{array}$$

However, since this equation couples the non-superconducting channels via interorbitial electron–electron interactions, the opening up of a superconducting gap in the $xz$ orbital will, as shown below, induce secondary MO effects in all Ni-$3d$ orbitals.

We shall now discuss our results for the orbital-nematic s-wave state of hole-doped ${\mathrm{NdNiO}}_2$. Considering the non-superconducting DFT + DMFT solution for $\Delta =0.1$ eV, we show, in the lower-right panel of Figure 4, the changes driven by superconductivity in the $xz$-orbital DOS across the superconducting transition. As expected, the Ni-$3d_{xz}$ channel is clearly reshaped by the pairing mechanism, inducing the emergence of a superconducting gap with a nearly V-shape form [33] at low energies as a result of the superconducting phase instability [9]. Also interesting is the absence of Bogoliubov quasiparticles [33], which, according to our results, are smeared (see our discussion below) by strong electron–electron interactions. It is worth mentioning that the appearance of sharp singularities, so-called Bogoliubov quasiparticles, at low energies, as shown in top-right panel of Figure 4 for the uncorrelated $U=0$ regime, are canonical fingerprints of conventional BCS superconductivity [64]. Thus, the presence of both full-gap-type and V-shaped tunneling spectra in the extant experiment at the surface of ${\mathrm{NdNiO}}_2$ thin films [33] can be taken as a manifestation on the degree of electronic correlations due to screening and disorder (or thermalization) effects hidden in rough surfaces [66].

Finally, since Bogoliubov quasiparticles exist only in a free electron or in a weakly correlated superconducting regime [60], our results for the $xz$ orbital in the lower-right panel of Figure 4 suggest that the Bogoliubov quasiparticle peaks are unstable in the presence of strong electron correlations [67] and thermalization [68]. We recall here that inner thermalization generally arises from scatterings due to impurities or disorder [69] and electron interactions, which induces lifetime broadening in the one-particle spectral functions [62]. A direct comparison between the uncorrelated and correlated (see Figure 4) $xz$ spectral functions reveals that the latter are thermally smeared, shrinking the superconducting gap upon decreasing the lifetime of the Bogoliubov quasiparticles. In this gap-filling [70] landscape, the Bogoliubov quasiparticles lose their coherence, and an incoherent spectral weight appears in the s-wave gap region at low energies, giving rise to a nearly gapless superconducting phase [64], as reported, for example, in Ref. [33]. Also interesting is the spectral weight transfer induced by the interorbital proximity effect on the non-superconducting orbitals. While weak spectral weight transfer at low energies above $E_F$ occurs in all channels, the valance band states are clearly reshaped, particularly at high binding energies, as visible in Figure 4, with increasing ${\Delta }_{sw}$. Future tunneling spectroscopy [33] measurements are called for to corroborate our results for the two-fluid superconducting scenario and the changes in the one-particle spectral functions across the s-wave superconducting phase transition in orbital-nematic ${\mathrm{NdNiO}}_2$.

3. Conclusions

In conclusion, we performed DFT + DMFT calculations for a five-orbital Hubbard model to show the role played by the interorbital proximity effect in the orbital-nematic electronic state of nickelate superconductors. Importantly, to determine whether ${\mathrm{NdNiO}}_2$ is an appropriate system within the Ni-age of high-$T_c$ superconductivity [71], we analyzed its orbital-selective electronic behavior, unraveling its electronic reconstruction in a two-fluid regime [63] where the $xz$ orbital undergoes an s-wave superconduting phase transition [33,34,35] while the $xy,yz,x^2-y^2,3z^2-r^2$ orbitals remain in the normal state. Within our DFT + DMFT calculations, we predicted the emergence of strong orbital selectivity when orbital nematicity is enhanced above a critical value, which could be tested in future studies on strained thin films.