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0.0.0.1 Introduction. —

The discovery of high-temperature superconductivity (SC) in the bilayer nickelates La\(_3\)Ni\(_2\)O\(_7\) [1]–[6] has aroused a surge in the exploration of superconductors in the Ruddlesden-Popper (RP) phase nickelates [7]–[22], as well as investigation into their physical properties with respect to the crystal structure[23]–[29], electronic structure [26], [27], [29]–[55], [55]–[64], the competing orders [34]–[36], [48], [65]–[68], and the pairing nature [52], [53], [55], [56], [58], [60], [67]–[128], attracting a lot of interests. Currently, the nickel-based SC has become a new platform for exploration and study of HTSC other than the cuprates and the iron-based superconductors. Meanwhile, the pairing mechanism remains elusive.

It is now generally taken that the SC in the bilayer nickelates is driven by electron-electron interaction [2], [30]–[37], which is suggested by various experimental evidences including proximity of the SC to the spin-density-wave[37], the strong orbital-selective band renormalization[30]–[32], and the strange-metal transport behavior[2]. Very recently, the single-particle spectrum, including the angle-resolved-photoemission spectrum (ARPES) and the scanning-tunneling-microscope (STM) has been obtained. While the ARPES reveals a nearly isotropic nodeless gap function[69], the STM spectrum takes a prominent U-shape[71], [72], which unambiguously reveals an s-wave pairing symmetry. These clear features of the single-particle spectrum set strong constraint on theories of the pairing mechanism.

Existing theories on the pairing mechanism of La\(_3\)Ni\(_2\)O\(_7\) include strong-coupling theories and weak-coupling ones. Typical strong-coupling theories are roughly divided into two classes. The first class adopts molecular orbital constructed via combining the atom orbitals from different layers into bonding or antibonding state as basis [119]–[123], and proposes that intralayer superexchange interaction between these molecular orbitals drive intralayer d-wave pairing, which can hardly understand the APRES/STM observations. The second class holds that the ultimate driving force of the pairing mechanism originates from the interlayer antiferromagnetic (AFM) superexchange interaction between the nearly half-filled \(d_{z^2}\) orbitals [67], [68], [92]–[118]. To settle the difficulty that the low hole density and low mobility of the \(d_{z^2}\) orbital strongly suppress the phase coherence, this class is further divided into two types. In the first type, the \(d_{z^2}\)-orbital electrons can gain coherence through hybridization with the \(d_{x^2-y^2}\) orbital and dominates the SC [92]–[96], [98], [99]; in the second type, the \(d_{x^2-y^2}\)-orbital electrons gain an effective interlayer superexchange interaction which is transferred from the \(d_{z^2}\) orbital via the Hund’s rule and dominate the SC [60], [67], [68], [100]–[118]. Both types yield interlayer s-wave pairings. Weak-coupling type of theories, e.g. the random-phase-approximation (RPA)[52], [53], [60], [78]–[80], [82]–[84], the fluctuation-exchange (FLEX) [55], [85], [87], [88] and the functional renormalization group (FRG)[56], [89], [90], yield that spin fluctuations mediate s\(^\pm\)-wave pairing. It is eager now to find useful ways to single out among the three types of theories which yield s-wave pairing the one most relevant to La\(_3\)Ni\(_2\)O\(_7\).

In this paper, we study the single-particle spectrum of La\(_3\)Ni\(_2\)O\(_7\) revealed by ARPES and STM, and relate it to the pairing nature and pairing mechanism. We propose that the pairing gap along the BZ diagonal can serve as a useful probe to detect the orbital-selective pairing nature which provides insight for the pairing mechanism. Due to different symmetries between \(d_{z^2}\)- and \(d_{x^2-y^2}\)- orbitals, their hybridization vanishes along the Brillouin zone (BZ) diagonal, so that the pairings on the \(\gamma\)- and \(\alpha/\beta\)- pockets reflect the intrinsic pairing strengths of the \(d_{z^2}\) and \(d_{x^2-y^2}\) orbitals respectively. Under the \(d_{z^2}\)-orbital dominated orbital-hybridization driven pairing mechanism, the pairing gaps along the BZ diagonal on the \(\alpha\)- and \(\beta\)- pockets vanish, leading to a V-shaped STM curve, inconsistent with experiments. Under the \(d_{x^2-y^2}\)-orbital dominated Hund’s rule driven pairing mechanism, instead, the pairing gap is almost isotropic on each pocket, leading to a fully-gaped s\(^\pm\)-wave pairing consistent with ARPES and STM. For the weak-coupling theory, our RPA calculations also yield gap nodes near the BZ diagonal which conflicts with experiments because, due to large density of states (DOS) on the \(\gamma\)-pocket, the \(d_{z^2}\)-orbital dominates the pairing. Our analysis establishes the \(d_{x^2-y^2}\)-orbital dominated Hund’s rule driven pairing mechanism as the most relevant one in La\(_3\)Ni\(_2\)O\(_7\).

0.0.0.2 Orbital-Selective Pairing Nature. —

As the low-energy degree of freedom in La\(_3\)Ni\(_2\)O\(_7\) is dominated by the \(d_{z^2}\)- and \(d_{x^2-y^2}\)- orbitals, we adopt the two-orbital tight-binding (TB) model from Ref. [54] for the thin film, \[\begin{align} \label{H0} H_{0} &=& H_0^x + H_0^z + H_0^{xz}=\sum_{\mathbf{k}} H_0(\mathbf{k})\nonumber\\&=&\sum_{\mathbf{k}} \left[ H_0^x(\mathbf{k}) + H_0^z(\mathbf{k}) + H_0^{xz}(\mathbf{k})\right], \end{align}\tag{1}\] where \(H_0^x\) (\(H_0^z\)) denotes the intra-\(d_{x^2-y^2}\)- (\(d_{z^2}\)-) orbital component and \(H_0^{xz}\) indicates the orbital-hybridization, \[\require{upgreek} \begin{align} H_{0}^{xz} &= \sum_{i,\updelta,\mu,\nu}t_{\mu\nu}^{xz}(\updelta)c_{i\mu x}^\dagger c_{i+\updelta,\nu z} + \text{h.c.} \\ &= \sum_{\mathbf{k}\mu\nu} h_{\mu\nu}^{xz}(\mathbf{k})c_{\mathbf{k}\mu x}^\dagger c_{\mathbf{k}\nu z}, \end{align}\] where \(\mu,\nu=\{\text{top},\text{bottom}\}\) labels layer and \(h_{\mu\nu}^{xz}(\mathbf{k})\) is the Fourier transformation of the hopping integral \(\require{upgreek} t_{\mu\nu}^{xz}(\updelta)\).

Consider the mirror reflection about the in-plane diagonal \(\hat{M}\). Under this operation, while the \(d_{z^2}\)-orbital wave function remains unchanged, the \(d_{x^2-y^2}\)-orbital one changes sign. Then the invariance of the system under this symmetry operation requires \(\require{upgreek} t^{xz}_{\mu\nu}(\hat{M}\updelta) = -t^{xz}_{\mu\nu}(\updelta)\), and hence \(h_{\mu\nu}^{xz}(\hat{M}\mathbf{k})= -h_{\mu\nu}^{xz}(\mathbf{k})\). For the momenta along the BZ diagonal \(\mathbf{k}_{\text{diag}}\), since \(\mathbf{k}_{\text{diag}}=\hat{M}\mathbf{k}_{\text{diag}}\), we have \(h_{\mu\nu}^{xz}(\mathbf{k}_{\text{diag}}) = 0\), and hence \[\begin{align} \label{H095diag} H_{0}(\mathbf{k}_{\text{diag}}) =H_0^x(\mathbf{k}_{\text{diag}}) + H_0^z(\mathbf{k}_{\text{diag}}). \end{align}\tag{2}\] Eq. (2 ) indicates that the TB Hamiltonian decouples into the \(d_{x^2-y^2}\)- component and the \(d_{z^2}\)- one along the BZ diagonal. Specifically, along the BZ diagonal, the \(\alpha/\beta\) pocket corresponds to purely \(d_{x^2-y^2}\)- component, and the \(\gamma\) pocket correspond to purely \(d_{z^2}\) component, as clearly displayed in the band structure shown in Fig. 1 (a) and the Fermi surface (FS) shown in Fig. 1 (b).

The orbital decoupling along the BZ diagonal brings convenience for investigating the orbital-selective pairing nature. In most proposed pairing mechanism for La\(_3\)Ni\(_2\)O\(_7\), the intraorbital pairing is dominant [52], [53], [55], [56], [78], [87], [89], [107], [127]. Therefore we only consider intraorbital pairing here, under which the pairing state can be described by the mean-field (MF) Hamiltonian \[\begin{align} \label{eq:H95MF} H_{\rm MF} &=& H_0+H^x_{\text{pair}}+H^z_{\text{pair}}=\sum_{\mathbf{k}} H_{\rm MF}(\mathbf{k})\nonumber\\ &=&\sum_{\mathbf{k}} \left[ H_0(\mathbf{k}) + H^x_{\text{pair}}(\mathbf{k})+H^z_{\text{pair}}(\mathbf{k})\right]. \end{align}\tag{3}\] As along the BZ diagonal, \(H_0^{xz}(\mathbf{k}_{\rm diag})=0\), we have \[\begin{align} \label{eq:H95MF95diag} H_{\rm MF}(\mathbf{k_{\rm diag}}) &=& \left[ H_0^x(\mathbf{k_{\rm diag}}) + H^x_{\text{pair}}(\mathbf{k_{\rm diag}})\right]\nonumber\\&+&\left[H_0^z(\mathbf{k_{\rm diag}}) + H^z_{\text{pair}}(\mathbf{k_{\rm diag}})\right]. \end{align}\tag{4}\] As the \(\alpha/\beta\) (\(\gamma\)) pocket is purely contributed by \(d_{x^2-y^2}\) (\(d_{z^2}\)) orbital along the BZ diagonal, the gap there is solely determined by \(H^x_{\text{pair}}\) (\(H^z_{\text{pair}}\)), as suggested by Eq. (4 ).

This orbital-selective pairing nature along the BZ diagonal provides insight for the pairing mechanism. In the following, we compare the pairing states obtained from the above introduced three types of pairing mechanism with ARPES and STM data. In the comparison, we adopt the intraband pairing approximation and project the gap onto the FS, see SM for details.

0.0.0.3 Orbital-Hybridization Driven Pairing Mechanism. —

This strong-coupling type of theories take that the interlayer AFM superexchange drives interlayer s-wave pairing between \(d_{z^2}\)- electrons, which further gain coherence through hybridization with \(d_{x^2-y^2}\) orbital and dominate the SC [92]–[96], [98], [99]. In these theories, although the \(d_{x^2-y^2}\)-orbital electrons also become SC through proximity (i.e. hybridization) with the \(d_{z^2}\)-orbital electrons, their intrinsic pairing is extremely weak due to quarter-filling (i.e. heavily overdoping in an analogy to the cuprates) of this orbital and can be ignored. The obtained pairing state can be described by the MF Hamiltonian, \[\begin{align} H_{\rm MF} &=& H_0^x + H_0^z + H_0^{xz} + H^z_{\text{pair}}\nonumber\\ H^z_{\text{pair}} &=& \Delta^z_{\perp} \sum_{i}\left( c_{izt\uparrow}^\dagger c_{izb\downarrow}^\dagger - c_{izt\downarrow}^\dagger c_{izb\uparrow}^\dagger + \mathrm{H.c.} \right). \label{eq:H95MF95z} \end{align}\tag{5}\] Note that the strong correlation in La\(_3\)Ni\(_2\)O\(_7\) causes orbital-selective band renormalization [30]–[32], which can be effectively treated as bringing about different renormalization factors to the \(H_0^x\), \(H_0^z\) and \(H_0^{xz}\) terms. However, the basis formula of Eq. (5 ) and particularly the above symmetry argument maintain.

As along the BZ diagonal, \(H_0^{xz}(\mathbf{k}_{\rm diag})=0\), we have \[\begin{align} \label{eq:H95MF95diag951} H_{\rm MF}(\mathbf{k_{\rm diag}})=H_0^x(\mathbf{k_{\rm diag}})+H_0^z(\mathbf{k_{\rm diag}}) + H^z_{\text{pair}}(\mathbf{k_{\rm diag}}). \end{align}\tag{6}\] As the \(\alpha/\beta\) pocket there is purely contributed by \(d_{x^2-y^2}\) orbital, the Hamiltonian there only contains the pure TB term \(H_0^x(\mathbf{k_{\rm diag}})\), and thus the pairing gap vanishes there, which is inconsistent with ARPES/STM.

Using Eq. (5 ), projection of the gap onto the FS leads to the distribution shown in Fig. 2 (a), which is compared with ARPES in Fig. 2 (b). We adopt \(\Delta^z_{\perp} = 13\) meV to well fit the ARPES gap on the \(\gamma\)- pocket, to find that the calculated gap distribution on the \(\beta\) pocket distinct from that of ARPES: Besides the distinct gap amplitude, the pronounced difference lies in that the calculated gaps are strongly anisotropic and form nodes at the BZ diagonal while the ARPES reveals a full gap with low anisotropy. Due to presence of the gap nodes, the calculated STM curve shown in Fig. 2 (c) exhibits a V- shape, distinct from the U-shaped experimental curve.

Despite absence of intrinsic pairing between the \(d_{x^2-y^2}\)- electrons, hybridization with \(d_{z^2}\) orbital induces nonzero expectation value of their interlayer pairing operator \(\langle \Delta_x \rangle = 1/\sqrt{2}\langle c_{ixt\uparrow}^\dagger c_{ixb\downarrow}^\dagger - c_{ixt\downarrow}^\dagger c_{ixb\uparrow}^\dagger \rangle\), as shown in Fig. 2 (d). Such a “hierarchical structure” of pairing in La\(_3\)Ni\(_2\)O\(_7\) has been reported in recent studies [96], [99]. However, this nonzero value only originates from the average over the whole BZ. Focusing on the BZ diagonal, since the orbital hybridization vanishes there, gap nodes are inevitable on the \(\alpha/\beta\)-pockets. Therefore, an intrinsic pairing gap in the \(d_{x^2-y^2}\) orbital is required to understand the full pairing gap observed experimentally.

0.0.0.4 Hund’s Rule Driven Pairing Mechanism. —

This strong-coupling type of theories hold that, although the interlayer AFM superexchange between the \(d_{z^2}\)- orbitals can induce intrinsic interlayer pairing between the \(d_{z^2}\)- electrons, the low hole density and low mobility of the \(d_{z^2}\)- orbital prevent it from dominating the SC. In contrast, the \(d_{x^2-y^2}\)- electrons can form a dominant interlayer pairing induced by the effective interlayer superexchange transferred from the \(d_{z^2}\)- orbital through the Hund’s rule [60], [67], [68], [100]–[118]. The resultant pairing state can be described by the MF Hamitonian, \[\begin{align} \label{eq:H95MF95x} H_{\rm MF} &= H_0 + \Delta^x_{\perp} \sum_{i,\sigma}\left( c_{ixt\uparrow}^\dagger c_{ixb\downarrow}^\dagger - c_{ixt\downarrow}^\dagger c_{ixb\uparrow}^\dagger + \mathrm{H.c.} \right) \nonumber\\ &\quad + \Delta^z_{\perp} \sum_{i,\sigma}\left( c_{izt\uparrow}^\dagger c_{izb\downarrow}^\dagger - c_{izt\downarrow}^\dagger c_{izb\uparrow}^\dagger + \mathrm{H.c.} \right), \end{align}\tag{7}\] where \(\Delta^x_{\perp}\) and \(\Delta^z_{\perp}\) denote the intrinsic interlayer pairing gaps in the \(d_{x^2-y^2}\) and \(d_{z^2}\) orbitals, respectively. Here the finite value of \(\Delta^z_{\perp}\) relies on the presence of the \(\gamma\)- pocket which provides finite hole density of the \(d_{z^2}\) orbital. Generally we have \(\Delta^x_{\perp}>\Delta^z_{\perp}\).

As claimed above, the \(\Delta^x_{\perp}\) directly opens the gap on the \(\alpha\)- and \(\beta\)- pockets along the BZ diagonal. Setting \(\Delta^x_{\perp} = 20\) meV and \(\Delta^z_{\perp} = 13\) meV as an example, we display the pairing gap on the FS in Figs. 3 (a, b). Clearly, we obtain full gaps on all pockets, which qualitatively fit the ARPES/STM data. The fitting can be further improved through introducing a small interlayer nearest-neighbor (NN) pairing \(\Delta^{x1}_{\perp}\) between the \(d_{x^2-y^2}\) electrons, whose physical origin is schematically illustrated in Fig. 3 (c): The Hund’s rule transfers the interlayer AFM superexchange between the \(d_{z^2}\)- electrons \(J^z_\perp\) to the \(d_{x^2-y^2}\)- electrons as \(J^x_\perp\), and the intralayer hopping of the \(d_{x^2-y^2}\)- electrons \(t^x_{\parallel}\) further transfers \(J^x_\perp\) to between the interlayer NN sites as \(J^{x1}_\perp\), which induces \(\Delta^{x1}_{\perp}\). Setting \(\Delta^{x}_{\perp} = 27\) meV, \(\Delta^z_{\perp} = 13\) meV, and \(\Delta^{x1}_{\perp} = -6\) meV, the gap distribution shown in Figs. 3 (d, e) perfectly fits the ARPES data. Furthermore, using the same set of \(\Delta^{x}_{\perp}\) and \(\Delta^{x1}_{\perp}\) but slightly different \(\Delta^z_{\perp} = 10\) meV, the calculated STM spectrum also well fits the U-shaped experiment data, as shown in Fig. 3 (f). The slight variation of \(\Delta^z_{\perp}\) may be caused by sample difference which influences the oxygen content and hence the size of the \(\gamma\)-pocket, while the \(\Delta^{x}_{\perp}\) and \(\Delta^{x1}_{\perp}\) are not sensitive to the \(\gamma\)-pocket.

0.0.0.5 Weak-coupling Pairing Mechanism. —.

The pairing mechanism of bilayer nickelate has been extensively studied within various weak-coupling approaches. These weak-coupling studies consistently yield gap nodes or near-nodes on the \(\beta\) pocket near the BZ diagonal regime [52], [53], [55], [56], [60], [78]–[80], [82], [84], [85], [87]–[90], whose physical origin however has not been clarified. For illustration, we perform an RPA calculation adopting the Hubbard-Kanamori model with choosing typical interaction parameters provided in Appendix B, and the results are shown in Figs. 4 (a-b). Clearly on the \(\beta\) pocket, the gap near the BZ diagonal regime is much weaker than that in other regimes, which conflicts with the nearly isotropic gap function observed experimentally.

To clarify the physical origin of the gap nodes or near-nodes along the BZ diagonal, we calculated the real-space pairing pattern through a Fourier transformation of the obtained \(\mathbf{k}\)- space pairing gap, with the results displayed in Appendix B. Two features are prominent for these data: First, the interorbital pairing is very weak and can be ignored. Second, the \(d_{z^2}\)- orbital pairing dominants the \(d_{x^2-y^2}\)- orbital pairing. The first feature justifies the validity of Eq. (3 ) and Eq. (4 ) and subsequent analysis, based on which the second feature further causes the near-nodes around the BZ diagonal regime on the \(\beta\)-pocket through orbital-hybridization. The dominance of the \(d_{z^2}\)- orbital in the pairing is common in weak-coupling theories [52], [56], [78], [87], [89], [90], which originates from the \(\gamma\)- pocket formed by the flat top of the bonding \(d_{z^2}\)- band that hosts a large DOS and hence contributes significantly to the pairing.

Therefore, the weak-coupling theories and the orbital-hybridization driven strong-coupling theories share the same feature that the \(d_{z^2}\) orbital dominates the pairing, which inevitably leads to near-nodes along the BZ diagonal on the \(\alpha/\beta\)- pocket, which however conflicts with experiments. The limitation of these theories lies in overestimation of \(d_{z^2}\) orbital in the pairing. Physically, due to near half-filling of the \(d_{z^2}\) orbital, the Hubbard repulsion will strongly suppress the phase coherence and hence strongly suppress the SC order by a Gutzwiller renormalization factor [129]. However, the weak-coupling approach treats the \(d_{z^2}\) electrons as purely itinerant and ignore such renormalization factor, which inevitably overestimates the significance of the \(d_{z^2}\)-electrons. The orbital-hybridization driven strong-coupling theories ignore the intrinsic pairing between the \(d_{x^2-y^2}\) orbital, which also overestimates the role of the \(d_{z^2}\)- orbital. In contrast, in the Hund’s-rule driven pairing mechanism, the \(d_{x^2-y^2}\) orbital dominantly carries the SC, which properly accounts for the strong-correlation effect and consistently explains the fully gapped spectrum.

0.0.0.6 Conclusions. —.

In conclusion, based on symmetry analysis, we propose that vanish of orbital hybridization along the BZ diagonal causes orbital-selective pairing nature on different Fermi pockets, which provides a probe of pairing mechanism in La\(_3\)Ni\(_2\)O\(_7\). Analysis based on this proposal suggests that, in \(d_{z^2}\)- orbital dominated pairing mechanism such as the orbital-hybridization driven strong-coupling theories and the weak-coupling theories, gap nodes or near-nodes are inevitable along the BZ diagonal on the \(\alpha/\beta\) pockets, which conflicts with the ARPES/STM spectrum. In contrast, in \(d_{x^2-y^2}\)-orbital dominated mechanism such as the Hund’s rule driven strong-coupling theories, the calculated gap function can well fit the ARPES and STM data. This analysis clarifies the dominant role of \(d_{x^2-y^2}\) orbital in the pairing and establishes the Hund’s rule driven pairing mechanism as the one most relevant to La\(_3\)Ni\(_2\)O\(_7\).

Based on the interlayer \(d_{x^2-y^2}\)-orbital pairing nature, we made further prediction on the ARPES. Along the BZ diagonal, the bonding and antibonding states of the \(d_{x^2-y^2}\) orbital reside on the \(\alpha\)- and \(\beta\)- pockets, respectively. As derived in Appendix C, we predict that the gaps on the two pockets have equal amplitude and opposite signs. This explains why the presence of two pockets does not lead to fluctuation of the observed pairing amplitude near the BZ diagonal. We have further studied the gap distribution in the case when the \(\gamma\) pocket is absent, as some ARPES experiments suggest. Without the \(\gamma\) pocket, \(\Delta^z_{\perp}\) is suppressed due to poor coherence of the \(d_{z^2}\) orbital. In this case, we predict that the gap on the \(\alpha\) and \(\beta\)- pockets reaches its maximum along the BZ diagonal because the weight of the \(d_{x^2-y^2}\) orbital is maximized there. See Appendix C for details.

1 Appendix A: Gap Function and STM spectrum↩︎

Here we provide the derivation details relevant to the calculation results shown in the main text, including the gap distribution on FS and the STM spectrum.

The distribution of gap on FS.  We start from the mean-field Hamiltonian \[\begin{align} H_{\rm MF} &= H_0 + \sum_{m,i}\Delta_{\perp}^{m} \left( c_{imt\uparrow}^\dagger c_{imb\downarrow}^\dagger - c_{imt\downarrow}^\dagger c_{imb\uparrow}^\dagger + \mathrm{H.c.} \right) \\ &=\sum_{\mathbf{k},\sigma}\sum_{mm'\mu\nu}h_{\mu\nu}^{mm'}(\mathbf{k})c_{\mathbf{k}m\mu\sigma}^\dagger c_{\mathbf{k}m'\nu\sigma}+\mathrm{H.c.} \\ &+ \sum_{m\mathbf{k}}\Delta_{\perp}^{m} \left( c_{\mathbf{k}mt\uparrow}^\dagger c_{-\mathbf{k}mb\downarrow}^\dagger - c_{\mathbf{k}mt\downarrow}^\dagger c_{-\mathbf{k}mb\uparrow}^\dagger + \mathrm{H.c.} \right)\notag, \end{align}\] where \(m,m'\in\{x,z\}\) labeling the orbitals and \(\mu,\nu\in\{t,b\}\) labeling the layers. Diagonalizing the matrix \(h(\mathbf{k})\) yields the band eigenstates \(|\mathbf{k},l\rangle\) with eigenvector components \(\xi_{l,m\mu}(\mathbf{k})\) and eigenvalues \(\epsilon_l(\mathbf{k})\).

We reasonably adopt the intraband pairing approximation, in which only the intraband Cooper pairing of the form \(c_{\mathbf{k}l\sigma}^\dagger c_{-\mathbf{k}l\sigma'}^\dagger\) is retained. The band-basis mean-field Hamiltonian then reads \[H_{\rm MF} = \sum_{\mathbf{k},l,\sigma} \epsilon_l(\mathbf{k}) c_{\mathbf{k}l\sigma}^\dagger c_{\mathbf{k}l\sigma} + \sum_{\mathbf{k},l} \left[ \Delta_{l}(\mathbf{k}) c_{\mathbf{k}l\uparrow}^\dagger c_{-\mathbf{k}l\downarrow}^\dagger + \mathrm{H.c.} \right]\notag,\] with \[\Delta_{l}(\mathbf{k}) = 2\Delta_{\perp}^{x}\, \xi_{l,xt}^*(\mathbf{k}) \, \xi_{l,xb}^*(-\mathbf{k}) + 2\Delta_{\perp}^{z} \, \xi_{l,zt}^*(\mathbf{k}) \, \xi_{l,zb}^*(-\mathbf{k})\notag\]

When the NN interlayer pairing \(\Delta_{\perp}^{x1}\) is included, we need to add a term \(4\Delta_{\perp}^{x1}(\cos k_x + \cos k_y) \xi_{l,xt}^*(\mathbf{k}) \xi_{l,xb}^*(-\mathbf{k})\) into \(\Delta_l(\mathbf{k})\).

STM spectrum. At finite temperature, the differential conductance is obtained by convolving the BCS density of states with the thermal kernel \(K_T(E)=-\partial f/\partial E\), \[\frac{dI}{dV}(V) \propto \sum_{l,\mathbf{k}} \left[ u_{l\mathbf{k}}^2 K_T\bigl(E_{l\mathbf{k}}-eV\bigr) + v_{l\mathbf{k}}^2 K_T\bigl(E_{l\mathbf{k}}+eV\bigr) \right]\notag,\] where \(u_{l\mathbf{k}}\) and \(v_{l\mathbf{k}}\) are the standard BCS coherence factors, \(E_{l\mathbf{k}}\) is the quasiparticle dispersion, and \[K(E) = \frac{1}{4k_{\rm B}T}\,\mathrm{sech}^2\!\left(\frac{E}{2k_{\rm B}T}\right)\notag.\]

We take \(k_{\rm B}T = 0.362\) meV in calculation, corresponding to the experimental measurement temperature.

2 Appendix B: Random Phase Approximation Study↩︎

In RPA framework, SC is driven by the spin or charge fluctuation with the propagator given by the spin or charge susceptibility \(\chi^{(s/c)}\) renormalized up to the RPA level. When the interaction strength \(U\) is smaller than the critical value \(U_c\), both \(\chi^{(s)}\) and \(\chi^{(c)}\) do not diverge and the spin and charge fluctuation mediates SC.

We consider the multi-orbital Hubbard-Kanamori Hamiltonian and set the onsite intraorbital Coulomb repulsion strength \(U=1\text{ eV}\) and the strength of Hund’s coupling \(J_H=U/6\), and the onsite interorbital Coulomb repulsion strength \(V\) is obtained from the relationship \(U=V+2J_H\) [130]. Then we apply the standard RPA approach [131]–[137] to obtain the \(\mathbf{k}\)-space gap function in the full BZ. After getting the \(\mathbf{k}\)-space gap function \(\Delta_{l}(\mathbf{k})\) in the full BZ, we obtain the real-space pairing components by Fourier transformation.

The ratios between some dominant pairing components and the interlayer pairing of the \(d_{z^2}\) orbital \(\Delta^{z,\perp}_{(0,0)}\) are shown in Table 1. For each case in \(d_{z^2}\) intralayer pairing, \(d_{z^2}\) interlayer pairing, \(d_{x^2-y^2}\) intralayer pairing, \(d_{x^2-y^2}\) interlayer pairing and interorbital pairing, only the relative large values are shown.

3 Appendix C: The prediction of the gap distribution↩︎

Here, we perform two experimentally testable predictions, namely, the gap on the \(\alpha\) pocket and the gap evolution when the \(\gamma\) pocket is absent.

First, as shown in Fig. 5 (a), we predict that the gap on the \(\alpha\) pocket should be essentially isotropic, with a magnitude comparable to the gap on the \(\beta\) pocket along the diagonal direction (\(\sim 16\) meV).

This prediction follows directly from the orbital structure along the diagonal direction. There the \(\alpha\) and \(\beta\) states are the bonding and antibonding combinations of the two layer \(d_{x^2-y^2}\) orbitals, \[\begin{align} |\mathbf{k}_{\rm diag},\alpha/\beta\rangle &= \frac{1}{\sqrt{2}}\left(|\mathbf{k}_{\rm diag},tx\rangle \pm |\mathbf{k}_{\rm diag},bx\rangle\right),\\ |\mathbf{k}_{\rm diag},t/b,x\rangle &= \frac{1}{\sqrt{2}}\left(|\mathbf{k}_{\rm diag},\alpha\rangle \pm |\mathbf{k}_{\rm diag},\beta\rangle\right). \end{align}\]

On the \(\beta\) and \(\alpha\) FSs, only the interlayer pairing \(\Delta^x_{\perp}\) can project onto the momentum \(\mathbf{k}_{\rm diag}\). The pairing term reads \[\begin{align} H_{\rm pair}(\mathbf{k}_{\rm diag}) &= \Delta_{\perp}^{x} \left( c_{\mathbf{k}tx\uparrow}^\dagger c_{-\mathbf{k}bx\downarrow}^\dagger - c_{\mathbf{k}tx\downarrow}^\dagger c_{-\mathbf{k}bx\uparrow}^\dagger + \mathrm{H.c.}\right)\\ &=\Delta_{\perp}^{x} \left[(c_{\mathbf{k}\alpha\uparrow}^\dagger+ c_{\mathbf{k}\beta\uparrow}^\dagger)(c_{-\mathbf{k}\alpha\downarrow}^\dagger -c_{-\mathbf{k}\beta\downarrow}^\dagger)+\mathrm{H.c.}\right]\\ &=\Delta_{\perp}^{x} \left(c_{\mathbf{k}\alpha\uparrow}^\dagger c_{-\mathbf{k}\alpha\downarrow}^\dagger-c_{\mathbf{k}\beta\uparrow}^\dagger c_{-\mathbf{k}\beta\downarrow}^\dagger\right)+\mathrm{H.c.}\notag, \end{align}\] It follows that \(\Delta_{\alpha}(\mathbf{k}_{\rm diag})=-\Delta_{\beta}(\mathbf{k}_{\rm diag})=\Delta_{\perp}^{x}\). The gaps on the two pockets have equal magnitude and opposite sign along the diagonal. Meanwhile, the smaller size of the \(\alpha\) pocket limits angular variation of the gap, leading to the nearly isotropic gap seen in Fig. 5 (a).

Second, we consider the case without the \(\gamma\) pocket, as suggested by some ARPES experiments. In this case, the \(d_{z^2}\) orbital is much closer to half-filling, so the \(d_{z^2}\) pairing gap \(\Delta^z\) is strongly suppressed. With \(\Delta^z\) negligible, the \(\beta\)-pocket gap is dominated by the \(d_{x^2-y^2}\) interlayer pairing. We predict that the gap remains fully opened, but its angular dependence changes, with the gap now reaching a maximum along the diagonal direction, as shown in Fig. 5 (b).

References↩︎