Shadowed triplet pairings in Hund’s metals with spin-orbit coupling

A key feature of superconductivity (SC) is the antisymmetric-wavefunction of the Cooper pair under the exchange of two electrons. This has limited the focus to even-parity spin-singlet and odd-parity spin-triplet pairings in a single Fermi surface (FS) system. In multiorbital systems, additional possibilities arise due to the orbital degree of freedom, such as even-parity spin-triplet, or odd-parity spin-singlet pairings. For example, it was shown earlier that Hund’s coupling in multiorbital systems allows for an even-parity spin-triplet pairing that is orbitally antisymmetric between two orbitals [1, 2, 3, 4, 5, 6, 7]. The Hund’s coupling, significant in transition metal systems, acts as an attractive pairing interaction [1]. Similar to the attractive Hubbard model, the Hund’s coupling allows a strong local Cooper pair to form between two orbitals with spin-triplet character.

However, when the electron motion is introduced, i.e., the kinetic term in the Hamiltonian, the pairing is drastically weakened, because the electrons comprising the Cooper pair with momenta ${\bf k}$ and $-{\bf k}$ in different orbitals have different energies, due to the different electronic dispersions of the orbitals. Thus, to stabilize such an orbital antisymmetric pairing, significant orbital degeneracy is required throughout momentum space near the FS [1, 2]. Furthermore, hybridization between different orbitals can further weaken interorbital pairings, by splitting the degeneracy where it occurs.

The story becomes more interesting when spin-orbit coupling (SOC) is introduced. Note that with SOC, purely spin-singlet and -triplet pairings are no longer well-defined, since the spin and orbital degrees of freedom are coupled. Thus one defines Cooper pairs in the total angular momentum (pseudospin) basis. When the SOC is strong, this forms the basis for SC in the heavy fermion superconductors, such as UPt₃, leading to odd-parity pseudospin-triplet SC [8, 9]. When the SOC is intermediate, i.e., comparable to the orbital degeneracy splitting terms, spin and orbital characters vary continuously, and one can define the pairing in either the orbital or band basis. Spin-orbital mixing then helps to stabilize the interorbital pairing. It was shown that SOC indeed enhances even-parity orbital-antisymmetric spin-triplet pairing [3]. The SOC not only supports the orbital-antisymmetric spin-triplet pairing, but it also transforms pure spin-triplet pairing into “both” pseudospin-singlet and -triplet pairings in the band basis.

Furthermore, the form of the SOC is not limited to atomic SOC, i.e., $\textbf{L}_{i}\cdot\textbf{S}_{i}$, where $i$ is the site index, and the precise form can determine the momentum dependence of the superconducting state. For example, $s$-wave SC proximate to a topological insulator or strong Rashba SOC with broken inversion symmetry leads to an effective $p+ip$ SC, which is odd-parity and a spinless triplet [10, 11, 12]. For materials with inversion symmetry, there is still the possibility of even-parity momentum-dependent SOC [13] (k-SOC), which has been discussed in the context of the unconventional superconductor, Sr₂RuO₄ [14, 15, 16, 17, 18, 19, 20]. Similar to a Rashba-SOC generated $p+ip$ SC, the inclusion of k-SOC in a microscopic model with $s$-wave pairing is reflected in an intraband pairing with the same momentum character as the SOC.

Here we study a microscopic route to k-SOC and how SOC transforms even-parity interorbital spin-triplet SC into pseudospin-singlet and -triplet SC in a Hund’s metal, where the multiorbital nature and strong Hund’s coupling are crucial. We also illustrate how pseudospin-triplet interband pairing remains, dubbed a “shadowed triplet” away from the Fermi energy. While this SC behaves like a singlet in response to low-energy excitations, its hidden identity shows up at finite magnetic fields, and it can be tested when the field strength reaches an appreciable percentage of the superconducting gap size [21]. Applying the concept to Sr₂RuO₄, we show that $s+id_{xy}$ pairing is stabilized in the $t_{2g}$ orbitals.

The paper is organized as follows. We first introduce the generic Hamiltonian that we will be concerned with throughout in Sec. II. We then consider a simple but general two-orbital model to show how even-parity spin-triplet pairing arises in Sec. III. This includes the stability conditions and how SOC transforms this pairing into a pseudospin-singlet and -triplet in the Bloch band basis. The SOC in the shadowed triplet not only plays an essential role in enhancing the pairing, but it also determines the pairing symmetry. In Sec. IV we investigate microscopic routes to several k-SOC terms with $d$-wave symmetry, which can lead to various $d$-wave pairing symmetries on the FS. In Sec. V, we apply the shadowed triplet pairing scenario to the prominent unconventional superconductor, Sr₂RuO₄ [14, 15, 16, 17], for which the SOC has been shown to be important [22, 23, 24, 3, 25, 26], and we discuss the leading instability towards $s+id_{xy}$ pairing within a three-orbital model for a range of SOC parameters.

We first introduce the generic Hamiltonian that we will be considering throughout. The Hamiltonian consists of three terms. The kinetic term, $H_{0}$, denotes a tight-binding (TB) model, for which we will discuss the precise form in the subsequent sections. The SOC Hamiltonian, $H_{SOC}$, refers to the atomic SOC written in the basis of $t_{2g}$ orbitals and various momentum-dependent terms, also specified later. The interaction, $H_{int}$, has the form of the Kanamori interactions,

	$\displaystyle H_{int}=$	$\displaystyle\frac{U}{2}\sum_{i,a,\sigma\neq\sigma^{\prime}}n_{a,i\sigma}n_{a,i\sigma^{\prime}}+\frac{U^{\prime}}{2}\sum_{i,a\neq b,\sigma\sigma^{\prime}}n_{a,i\sigma}n_{b,i\sigma^{\prime}}$		(1)
		$\displaystyle+\frac{J_{H}}{2}\sum_{i,a\neq b,\sigma\sigma^{\prime}}c^{\dagger}_{a,i\sigma}c^{\dagger}_{b,i\sigma^{\prime}}c_{a,i\sigma^{\prime}}c_{b,i\sigma}$
		$\displaystyle+\frac{J_{H}}{2}\sum_{i,a\neq b,\sigma\neq\sigma^{\prime}}c^{\dagger}_{a,i\sigma}c^{\dagger}_{a,i\sigma^{\prime}}c_{b,i\sigma^{\prime}}c_{b,i\sigma},$

where $U$ and $U^{\prime}$ are the intra- and interorbital Hubbard repulsions, $J_{H}$ is the Hund’s coupling and $c_{a,i\sigma}^{\dagger}$ is an electron operator creating an electron at site $i$ in orbital $a$ with spin $\sigma$. Decoupling these interaction terms into even-parity zero-momentum spin-singlet and spin-triplet order parameters[3, 7, 20, 21] gives,

		$\displaystyle H_{int}=\frac{4U}{N}\sum_{a,\textbf{k}\textbf{k}^{\prime}}\hat{\Delta}^{s\dagger}_{a,\textbf{k}}\hat{\Delta}^{s}_{a,\textbf{k}^{\prime}}$		(2)
		$\displaystyle+\frac{2(U^{\prime}-J_{H})}{N}\sum_{\{a\neq b\},\textbf{k}\textbf{k}^{\prime}}\hat{\mathbf{d}}_{a/b,\textbf{k}}^{\dagger}\cdot\hat{\mathbf{d}}_{a/b,\textbf{k}^{\prime}}$
		$\displaystyle+\frac{4J_{H}}{N}\sum_{a\neq b,\textbf{k}\textbf{k}^{\prime}}\hat{\Delta}^{s\dagger}_{a,\textbf{k}}\hat{\Delta}^{s}_{b,\textbf{k}^{\prime}}$
		$\displaystyle+\frac{2(U^{\prime}+J_{H})}{N}\sum_{a\neq b,\textbf{k}\textbf{k}^{\prime}}\hat{\Delta}^{s\dagger}_{a/b,\textbf{k}}\hat{\Delta}^{s}_{a/b,\textbf{k}^{\prime}},$

where $N$ is the number of sites, and the spin-triplet and -singlet order parameters are defined as

	$\displaystyle\hat{\textbf{d}}_{a/b,\textbf{k}}$	$\displaystyle=\frac{1}{4}\sum_{\sigma\sigma^{\prime}}[i\sigma^{y}\bm{\sigma}]_{\sigma\sigma^{\prime}}\bigl{(}c_{a,\textbf{k}\sigma}c_{b,-\textbf{k}\sigma^{\prime}}-c_{b,\textbf{k}\sigma}c_{a,-\textbf{k}\sigma^{\prime}}\bigr{)}$		(3)
	$\displaystyle\hat{\Delta}^{s}_{a/b,\textbf{k}}$	$\displaystyle=\frac{1}{4}\sum_{\sigma\sigma^{\prime}}[i\sigma^{y}]_{\sigma\sigma^{\prime}}\bigl{(}c_{a,\textbf{k}\sigma}c_{b,-\textbf{k}\sigma^{\prime}}+c_{b,\textbf{k}\sigma}c_{a,-\textbf{k}\sigma^{\prime}}\bigr{)}$
	$\displaystyle\hat{\Delta}^{s}_{a,\textbf{k}}$	$\displaystyle=\frac{1}{4}\sum_{\sigma\sigma^{\prime}}[i\sigma^{y}]_{\sigma\sigma^{\prime}}c_{a,\textbf{k}\sigma}c_{a,-\textbf{k}\sigma^{\prime}},$

and $\{a\neq b\}$ represents a sum over the unique pairs of orbital indices. An attractive interorbital spin-triplet, $\hat{\textbf{d}}_{a/b,\textbf{k}}$, channel is present when $J_{H}>U^{\prime}$, which will be our focus. While this corresponds to a Hund’s coupling larger than the typical value of $\sim 0.2U$ for Sr₂RuO₄ [26, 27], this type of pairing instability has also been found in several studies beyond mean field (MF) theory without such a requirement[4, 5, 6]. Furthermore, we note that the interorbital, $\hat{\Delta}^{s}_{a/b,\textbf{k}}$, and intraorbital, $\hat{\Delta}^{s}_{a,\textbf{k}}$, spin-singlet order parameters appear with repulsive interactions, but they can be induced by the spin-triplet order parameters through the SOC. Indeed, calculating the spin-singlet order parameters within a MF spin-triplet pairing state shows that they are generally one order of magnitude smaller than the spin-triplet order parameters.

To illustrate how the shadowed triplet pairing occurs, we first consider a two-orbital MF Hamiltonian, $H_{MF}$, consisting of a generic TB model with a specific SOC and an even-parity spin-triplet pairing term. The following model allows for a systematic study of the microscopic components relevant to such pairing in multiorbital systems and is generalized to systems with three orbitals. We use $\Psi_{\textbf{k}}^{\dagger}=(\psi_{\textbf{k}}^{\dagger},\mathcal{T}\psi_{\textbf{k}}^{T}\mathcal{T}^{-1})$, where $\mathcal{T}$ indicates time-reversal and $\psi_{\textbf{k}}^{\dagger}=(c^{\dagger}_{a,\textbf{k}\uparrow},c^{\dagger}_{b,\textbf{k}\uparrow},c^{\dagger}_{a,\textbf{k}\downarrow},c^{\dagger}_{b,\textbf{k}\downarrow})$ consists of electron operators creating an electron in one of the two orbitals $a,b$ with spin $\sigma=\uparrow,\downarrow$. We also introduce the Pauli matrices plus identity matrix, $\rho_{i},\sigma_{i},\tau_{i},(i=0,...3)$, in the particle-hole, spin and orbital spaces respectively, where the direct product between them is implied. $H_{MF}$ is given by,

		$\displaystyle H_{MF}=\sum_{\textbf{k}}\Psi_{\textbf{k}}^{\dagger}\bigl{(}H_{0}(\textbf{k})+H_{\text{SOC}}^{z}(\textbf{k})+H_{\text{pair}}(\textbf{k})\bigr{)}\Psi_{\textbf{k}}$		(4)
		$\displaystyle H_{0}(\textbf{k})=\rho_{3}\bigl{(}\frac{\xi_{\textbf{k}}^{+}}{2}\sigma_{0}\tau_{0}+\frac{\xi_{\textbf{k}}^{-}}{2}\sigma_{0}\tau_{3}+t_{\textbf{k}}\sigma_{0}\tau_{1}\bigr{)}$
		$\displaystyle H_{\text{SOC}}^{z}(\textbf{k})=-\lambda_{\textbf{k}}\rho_{3}\sigma_{3}\tau_{2}$
		$\displaystyle H_{\text{pair}}=-d^{z}_{a/b}\rho_{2}\sigma_{3}\tau_{2},$

where the TB contribution is given by the sum of the two orbital dispersions, $\xi_{\textbf{k}}^{+}$, and the difference between them, $\xi_{\textbf{k}}^{-}$, which are defined by $\displaystyle\xi_{\textbf{k}}^{\pm}=\xi_{\textbf{k}}^{a}\pm\xi_{\textbf{k}}^{b}$, as well as the orbital hybridization of the two orbitals, $t_{\textbf{k}}$. Here we consider $H_{SOC}^{z}$, with the form $\lambda_{\textbf{k}}L_{z}S_{z}$, to illustrate the effects of the SOC. Later, we derive several possible k-SOC from a microscopic perspective in Sec. IV, which differ from the $L_{z}S_{z}$ form included here. The pairing Hamiltonian is obtained through a MF decoupling of $H_{int}$ and favors the even-parity interorbital spin-triplet order parameter via $H_{SOC}^{z}$, with the $d$-vector aligned along the $z$-direction, and thus $d^{z}_{a/b}$ is given by

$$d^{z}_{a/b}\equiv(U^{\prime}-J_{H})\frac{1}{2N}\sum_{\textbf{k}}\langle{\hat{d}^{z}_{a/b,\textbf{k}}}\rangle.$$

(5)

To understand the stability of the SC state, we consider the relationship between the various components of our model and their effects on the quasi-particle (QP) dispersion. The gap on the FS is shown in Fig. 1 for various cases of $\xi_{\textbf{k}}^{-}$, $t_{\textbf{k}}$ and $\lambda_{\textbf{k}}$. Here, $\displaystyle\xi_{\textbf{k}}^{a}=-2t_{1}\cos{k_{y}}-2t_{2}\cos{k_{x}}-\mu$, and for $\xi_{\textbf{k}}^{b}$ we take $x\leftrightarrow y$. The orbitals are coupled through the SOC, for which we take two cases: the atomic SOC denoted by $\lambda_{0}$, and a $d$-wave SOC given by $\lambda_{\textbf{k}}=\lambda_{d}(\cos{k_{x}}-\cos{k_{y}})$, as well as the orbital hybridization, which we take as $\displaystyle t_{\textbf{k}}=-4t_{ab}\sin{k_{x}}\sin{k_{y}}$. With these dispersions, $\xi_{\textbf{k}}^{-}=2t^{-}(\cos{k_{x}}-\cos{k_{y}})$, where $t^{-}=t_{1}-t_{2}$ and all parameters are given in units of $2t_{1}=1$. The gap over the FS is shown for four cases: (a) $\lambda=0$, $\xi_{\textbf{k}}^{-}=0$, as $t_{ab}$ is increased from zero to $0.1$; (b) keeping $\lambda=0$, with $t_{ab}=0.1$ as $\xi_{\textbf{k}}^{-}$ is increased by tuning $t^{-}$ from zero to $0.13$; (c) both $t_{ab}$ and $t^{-}$ are kept the same as $\lambda_{0}$ is increased from zero to 0.48; and (d) the same as (c) but with the $d$-wave SOC, instead of $\lambda_{0}$, increased by tuning $\lambda_{d}$ from zero to 0.48.

When the orbital dispersions are completely degenerate, we see that the gap is non-zero everywhere over the two identical bands, as shown by the middle contour in Fig. 1(a). As the strength of the hybridization, $t_{\textbf{k}}$, is increased from zero, the energy separation of the two orbital dispersions increases wherever $t_{\textbf{k}}\neq 0$ and the gap arising from interband pairing disappears on the FS, except where $t_{\textbf{k}}$ vanishes along the $k_{x/y}=0$ lines, where the two orbital dispersions remain degenerate. Starting from there with zero interband pairing over most of the FS, Fig. 1(b) demonstrates the effect of $\xi_{\textbf{k}}^{-}$ in further reducing the interband pairing to zero everywhere on the FS, due to the absence of phase space for zero-momentum pairing. Keeping the same parameters used in (b), Fig. 1(c) reveals how the SOC revives the SC state by allowing for an intraband pairing on the FS. As the SOC is increased from zero, the intraband gap becomes non-zero over the entire FS. Additionally, the sign of the gap function is opposite on the two bands, matching the $\Delta^{s}(\textbf{k})\tilde{\tau}_{3}$ dependence, which we will show below in Eq. (10) and Eq. (11), but uniform on each band due to the lack of momentum dependence of the atomic SOC. In contrast, Fig. 1(d) displays the $d$-wave dependence of the gap arising from the $d$-wave SOC. Thus with the introduction of $t_{\textbf{k}}$, $\xi_{\textbf{k}}^{-}$ and $\lambda_{\textbf{k}}$, the pairing on the FS is transformed from an interband spin-triplet to a purely intraband pseudospin-singlet with the same momentum dependence as the associated SOC, while the pseudospin-triplet is active away from the Fermi energy.

The above results can also be understood via the commutation relations between the order parameter and other terms. The pair-breaking effects are revealed by the commuting behavior with the pairing term [28, 29, 30]. Conversely, the SOC anti-commutes with the pairing term and generally enhances the pairing state. The anti-commutators are given by

	$\displaystyle\{\frac{\xi_{\textbf{k}}^{-}}{2}\rho_{3}\sigma_{0}\tau_{3},H_{\text{pair}}\}$	$\displaystyle=\xi_{\textbf{k}}^{-}d^{z}_{a/b}\rho_{1}\sigma_{3}\tau_{1}$		(6)
	$\displaystyle\{t_{\textbf{k}}\rho_{3}\sigma_{0}\tau_{1},H_{\text{pair}}\}$	$\displaystyle=-2t_{\textbf{k}}d^{z}_{a/b}\rho_{1}\sigma_{3}\tau_{3}$
	$\displaystyle\{-\lambda_{\textbf{k}}\rho_{3}\sigma_{3}\tau_{2},H_{\text{pair}}\}$	$\displaystyle=0.$

These effects are reflected in the QP dispersion, given by

	$\displaystyle E_{\textbf{k}}$	$\displaystyle=\pm\frac{1}{2}\biggl{[}\xi_{\textbf{k}}^{-2}+\xi_{\textbf{k}}^{+2}+4\bigl{[}t_{\textbf{k}}^{2}+(d^{z}_{a/b})^{2}+\lambda_{\textbf{k}}^{2}\bigr{]}$		(7)
		$\displaystyle\pm 2\sqrt{\xi_{\textbf{k}}^{+2}\bigl{[}\xi_{\textbf{k}}^{-2}+4t_{\textbf{k}}^{2}+4\lambda_{\textbf{k}}^{2}\bigr{]}+4(d^{z}_{a/b})^{2}\bigl{[}\xi_{\textbf{k}}^{-2}+4t_{\textbf{k}}^{2}\bigr{]}}\biggr{]}^{\frac{1}{2}}.$

The general equation of the FS is $\displaystyle\xi_{\textbf{k}}^{-2}=\xi_{\textbf{k}}^{+2}-4(t_{\textbf{k}}^{2}+\lambda_{\textbf{k}}^{2})$, from which it can be seen that if $\lambda_{\textbf{k}}$ and $t_{\textbf{k}}=0$, and the orbitals are degenerate, i.e., $\xi_{\textbf{k}}^{-}=0$, we recover the conventional BCS result for the gap energy on the FS: $\pm d^{z}_{a/b}$. In the general case where these terms are non-zero, assuming $d^{z}_{a/b}$ is small, the QP gap is

$$\text{QP gap}_{\text{on FS}}\approx\pm\sqrt{\frac{(d^{z}_{a/b})^{4}+16\lambda_{\textbf{k}}^{2}(d^{z}_{a/b})^{2}}{4(\xi_{\textbf{k}}^{-2}+4(t_{\textbf{k}}^{2}+\lambda_{\textbf{k}}^{2}))}}.$$

(8)

From this, it is clear that increasing $\xi_{\textbf{k}}^{-}$ and $t_{\textbf{k}}$ decreases the overall gap size. The detrimental effects on the gap size, signified by the commuting nature of both $t_{\textbf{k}}$ and $\xi_{\textbf{k}}^{-}$ with the pairing Hamiltonian, are a result of shifting apart in energy the bands being paired, resulting in the gap moving away from the FS. However, turning on the SOC significantly enhances the gap size [3], as shown in Fig. 1(c), and as the SOC strength becomes larger than $\xi_{\textbf{k}}^{-}$ and $t_{\textbf{k}}$, the gap can be restored to the order of $d^{z}_{a/b}$. Furthermore, if $\lambda_{\textbf{k}}$ has a $d$-wave form factor such as ($\cos{k_{x}}-\cos{k_{y}})$, the gap reflects the exact same $d$-wave symmetry, as shown in Fig. 1(d). The enhancement of the SC state is accomplished by providing a non-zero intraband pseudospin-singlet pairing on the FS, as we show below in the band basis.

With the aim to further understand the nature of the SC state, we study how the pairing transforms to the Bloch band basis, labeled by band indices $\alpha,\beta$ and pseudospin $s=(+,-)$. The transformation is given by

$\displaystyle\begin{pmatrix}c_{a,\textbf{k}\sigma}\\[6.0pt] c_{b,\textbf{k}\sigma}\\ \end{pmatrix}=\begin{pmatrix}\frac{\eta_{\sigma}+1}{2}f_{\textbf{k}}-\frac{\eta_{\sigma}-1}{2}f_{\textbf{k}}^{*}&-g_{\textbf{k}}\\[6.0pt] g_{\textbf{k}}&\frac{\eta_{\sigma}+1}{2}f_{\textbf{k}}^{*}-\frac{\eta_{\sigma}-1}{2}f_{\textbf{k}}\end{pmatrix}\begin{pmatrix}\alpha_{\textbf{k},s}\\[6.0pt] \beta_{\textbf{k},s}\\ \end{pmatrix}$

(9)

where $\eta_{\sigma}=+1(-1)$ for $\sigma=\uparrow(\downarrow)$ and $s=+(-)$. The coefficients of the transformation are given by $f_{\textbf{k}}=-\frac{\gamma_{\textbf{k}}}{|\gamma_{\textbf{k}}|}\sqrt{\frac{1}{2}(1+\frac{\xi_{\textbf{k}}^{-}}{\sqrt{\xi_{\textbf{k}}^{-2}+4|\gamma_{\textbf{k}}|^{2}}})}$ and $g_{\textbf{k}}=-\sqrt{\frac{1}{2}(1-\frac{\xi_{\textbf{k}}^{-}}{\sqrt{\xi_{\textbf{k}}^{-2}+4|\gamma_{\textbf{k}}|^{2}}})}$, where $f_{\textbf{k}}$ is chosen to be complex with the same phase as $\gamma_{\textbf{k}}=t_{\textbf{k}}+i\lambda_{\textbf{k}}$, $g_{\textbf{k}}$ is real, and $|f_{\textbf{k}}|^{2}+g_{\textbf{k}}^{2}=1$. Applying this transformation on $H_{\text{pair}}$ results in the pairing in the band basis, where we have also defined the Pauli matrices $\tilde{\rho}_{i},\tilde{\sigma}_{i},\tilde{\tau}_{i}$ in the Nambu, pseudospin, and band spaces with the basis $\Phi_{\textbf{k}}^{\dagger}=(\phi_{\textbf{k}}^{\dagger},\mathcal{T}\phi_{\textbf{k}}^{T}\mathcal{T}^{-1})$ and $\phi_{\textbf{k}}^{\dagger}=(\alpha_{\textbf{k}+}^{\dagger},\beta_{\textbf{k}+}^{\dagger},\alpha_{\textbf{k}-}^{\dagger},\beta_{\textbf{k}-}^{\dagger})$. We obtain,

	$\displaystyle\tilde{H}_{\text{pair}}(\textbf{k})$	$\displaystyle=\tilde{\rho}_{2}\tilde{\sigma}_{0}\bigl{[}-\Delta^{s}(\textbf{k})\tilde{\tau}_{3}-\Delta^{s}_{\alpha/\beta}(\textbf{k})\tilde{\tau}_{1}\bigr{]}$		(10)
		$\displaystyle-d^{z}_{\alpha/\beta}(\textbf{k})\tilde{\rho}_{2}\tilde{\sigma}_{3}\tilde{\tau}_{2},$

where $\Delta^{s}(\textbf{k})$ and $\Delta^{s}_{\alpha/\beta}(\textbf{k})$ denote pseudospin-singlet intraband and interband pairings respectively, and $d^{z}_{\alpha/\beta}(\textbf{k})$ is a pseudospin-triplet interband pairing. The intraband and interband nature of these pairings becomes more apparent from the operator form,

	$\displaystyle\tilde{H}_{\text{pair}}(\textbf{k})=i\Delta^{s}(\textbf{k})\bigl{[}(\alpha_{\textbf{k},+}^{\dagger}\alpha_{-\textbf{k},-}^{\dagger}-\alpha_{\textbf{k},-}^{\dagger}\alpha_{-\textbf{k},+}^{\dagger})$	$\displaystyle-(\beta_{\textbf{k},+}^{\dagger}\beta_{-\textbf{k},-}^{\dagger}-\beta_{\textbf{k},-}^{\dagger}\beta_{-\textbf{k},+}^{\dagger})\bigr{]}$		(11)
	$\displaystyle+i\Delta^{s}_{\alpha/\beta}(\textbf{{k}})\bigl{[}(\alpha^{\dagger}_{\textbf{k},+}\beta^{\dagger}_{-\textbf{k},-}-\alpha^{\dagger}_{\textbf{k},-}\beta^{\dagger}_{-\textbf{k},+})$	$\displaystyle+(\beta^{\dagger}_{\textbf{k},+}\alpha^{\dagger}_{-\textbf{k},-}-\beta^{\dagger}_{\textbf{k},-}\alpha^{\dagger}_{-\textbf{k},+})\bigr{]}$
	$\displaystyle+d^{z}_{\alpha/\beta}(\textbf{{k}})\bigl{[}(\alpha^{\dagger}_{\textbf{k},+}\beta^{\dagger}_{-\textbf{k},-}+\alpha^{\dagger}_{\textbf{k},-}\beta^{\dagger}_{-\textbf{k},+})$	$\displaystyle-(\beta^{\dagger}_{\textbf{k},+}\alpha^{\dagger}_{-\textbf{k},-}+\beta^{\dagger}_{\textbf{k},-}\alpha^{\dagger}_{-\textbf{k},+})\bigr{]}.$

The above equation shows the sign change in the intraband pairing between the two bands, as displayed in Fig. 1(c) and 1(d). This relative sign between the bands is similar to the $s^{+-}$ gap structure [7], although it should be noted that here for simplicity we have ignored the SOC-induced intraorbital singlets [3, 7, 21, 31] that would add to the gap on each band. For small $(J_{H}-U^{\prime})$ they are small compared to $d^{z}_{a/b}$ and they do not affect the conclusions of this section. The pairings in the band basis have the following form:

	$\displaystyle\Delta^{s}(\textbf{k})$	$\displaystyle=-2d^{z}_{a/b}\text{Im}(f_{\textbf{k}})g_{\textbf{k}}=-\frac{2d^{z}_{a/b}\lambda_{\textbf{k}}}{\sqrt{\xi_{\textbf{k}}^{-2}+4(t_{\textbf{k}}^{2}+\lambda_{\textbf{k}}^{2})}}$		(12)
	$\displaystyle\Delta^{s}_{\alpha/\beta}(\textbf{k})$	$\displaystyle=-d^{z}_{a/b}\text{Im}(f_{\textbf{k}}^{2})=-2d^{z}_{a/b}|f_{\textbf{k}}|^{2}\frac{t_{\textbf{k}}\lambda_{\textbf{k}}}{t_{\textbf{k}}^{2}+\lambda_{\textbf{k}}^{2}}$
	$\displaystyle d^{z}_{\alpha/\beta}(\textbf{k})$	$\displaystyle=d^{z}_{a/b}[g_{\textbf{k}}^{2}+\text{Re}(f_{\textbf{k}}^{2})]=d^{z}_{a/b}\bigl{(}g_{\textbf{k}}^{2}+|f_{\textbf{k}}|^{2}\frac{t_{\textbf{k}}^{2}-\lambda_{\textbf{k}}^{2}}{t_{\textbf{k}}^{2}+\lambda_{\textbf{k}}^{2}}\bigr{)}.$

While the orbital order parameter is $s$-wave and contains no explicit momentum dependence, transforming to the band basis generates potentially complex momentum dependence from SOC and orbital hybridization. The orbital spin-triplet order parameter carries over to the interband pseudospin-triplet, which, in the limit of zero SOC, becomes equal to $d^{z}_{a/b}$, while both the intraband and interband pseudospin-singlets vanish. The interband pseudospin-singlet pairing also vanishes for $t_{\textbf{k}}=0$. However, an important feature for $\lambda_{\textbf{k}}\neq 0$ is the presence of the intraband pseudospin-singlet pairing, $\Delta^{s}(\textbf{k})$, which acquires the same symmetry dependence as a function of momentum as $\lambda_{\textbf{k}}$, as shown in Fig. 1(d). While the interband pseudospin-triplet is a signature of the fundamental interorbital spin-triplet order parameter, it is the intraband pseudospin-singlet that leads to a weak-coupling instability. This is because the interband pairing contribution to the gap will generally be negligible on the FS, such that the gap is given by $|\Delta^{s}|$. Considering the QP dispersion in terms of the band pairings,

		$\displaystyle E_{\textbf{k}}=\pm\biggl{[}\frac{(\xi_{\textbf{k}}^{\alpha})^{2}+(\xi_{\textbf{k}}^{\beta})^{2}}{2}+(\Delta^{s})^{2}+(\Delta^{s}_{\alpha/\beta})^{2}+(d^{z}_{\alpha/\beta})^{2}$		(13)
		$\displaystyle\pm\sqrt{\frac{\bigl{[}(\xi_{\textbf{k}}^{\alpha})^{2}-(\xi_{\textbf{k}}^{\beta})^{2}\bigr{]}^{2}}{4}+(\xi_{\textbf{k}}^{\alpha}-\xi_{\textbf{k}}^{\beta})^{2}\bigl{[}(\Delta^{s}_{\alpha/\beta})^{2}+(d^{z}_{\alpha/\beta})^{2}\bigr{]}}\biggr{]}^{\frac{1}{2}},$

and evaluating this on either the $\alpha$ or $\beta$ FS, for only $\Delta^{s}$ intraband pairing, we obtain the gap energy $\pm|\Delta^{s}|$. For either only $\Delta^{s}_{\alpha/\beta}$ or $d^{z}_{\alpha/\beta}$ interband pairing, a gap of $\pm\Delta^{s}_{\alpha/\beta}$ or $\pm d^{z}_{\alpha/\beta}$ forms where $\xi_{\textbf{k}}^{\alpha}=-\xi_{\textbf{k}}^{\beta}$. This corresponds to an energy gap on the FS only where $\xi_{\textbf{k}}^{\alpha}=\xi_{\textbf{k}}^{\beta}=0$, which is not a generic feature but rather requires fine-tuning to achieve, otherwise the gap formed will be away from the Fermi energy.

While the model introduced in this section is simple, its generality gives insight into the role of the orbital degeneracy, hybridization, and SOC for multiorbital systems in dictating the stability of the even-parity spin-triplet SC state. It is straightforward to extend to three-orbital descriptions. Furthermore, we have seen that in the band basis, the intraband pairing on the FS takes on the momentum dependence of the SOC, allowing for a rich collection of pairing symmetries unexpected from the original $s$-wave order parameter and in contrast to other forms of momentum-dependent SC that arise from nonlocal interactions. However, the possible pairing symmetries will depend on the forms of k-SOC that can be obtained from microscopic considerations. Therefore, we now turn to a study of how various forms of k-SOC can arise microscopically.

Here, we take as a specific microscopic example the layered perovskite Sr₂RuO₄, which has the tetragonal space group I4/mmm and point group $D_{4h}$, for which the Ru 4$d$ $t_{2g}$ orbitals are the relevant low-energy degrees of freedom. With this, we study how the various forms of k-SOC with different $d$-wave form factors such as (a) an in-plane $d_{xy}$ SOC in the B_2g representation, (b) in-plane $d_{x^{2}-y^{2}}$ SOC in the B_1g representation, and (c) interlayer $\{d_{xz},d_{yz}\}$ SOC in the E_g representation can arise microscopically, going beyond a purely symmetry-based analysis.