Introduction.– The (A=K, Rb, Cs) kagome superconductors are an exciting class of compounds which exhibit various symmetry-broken states [1]–[33], including charge orders and superconductivity (SC). While the presence of these broken symmetry states are well established, the nature of the mechanism driving the charge order and the SC states is still under active investigation [34]–[67].

A plethora of experiments support the formation of charge bond-order [5]–[7], [68]–[71], with charge density modulations on the vanadium (V) bonds. Further, time-reversal (TR) symmetry breaking has been reported in the charge-ordered [6], [9], [10], [12], [13], [69], [72]–[75] state, despite the absence of local-moment spin order [3], [76], thereby suggesting loop-current order as a possible origin of the TR-breaking. However, a number of other studies [8], [68], [74], [77] have been unable to detect a stable long-range loop current order, leaving its status contested. These observations, however, do not preclude the existence of fluctuating charge loop-currents, or other fluctuating modes in the spin channel. Further, applying pressure [29], [78]–[81] or doping [14], [82], [83] suppresses and eventually destroys the charge order. Hence, in this disordered state, various fluctuating bond-orders (BO) and loop-currents (LC) all coexist, and the effect of their competing fluctuations on SC in the presence of a realistic band structure is yet to be explored.

In this letter, we present a microscopic study of the pairing interaction in the kagome metals mediated by fluctuations of BO and LCs in the charge and spin channels, all of which originate from nearest-neighbor Coulomb interactions. Inspired by [62], we include BO and LC order parameters between V and in-plane Sb (Sb-ip) sites as well as those between nearest neighbor V sites. First, we construct an effective microscopic model that captures the van-Hove singularities (VHS) and a Fermi surface (FS) near the zone center with appropriate orbital contents. Using this model, we compute the RPA-corrected susceptibilities in the disordered state and identify various fluctuating BO & LC channels close to the P-type VHS which are strongly enhanced near the \(M\) point in the Brillouin zone. Finally, by tuning the relative strengths of the various fluctuating channels incorporated in the RPA-corrected Cooper vertex, we show that the different fluctuations compete with each other to mediate distinct types of SC. The properties of the resulting superconducting states are discussed in light of recent experiments.

Electronic structure.– The electronic band structure of the kagome metals features multiple VHSs at the \(M\) points close to the Fermi energy [1], [2]. The VHS mostly come from the V \(3\)-d orbitals [3], [4], [45], [84]–[91], but a strong hybridization between the out-of-plane Sb (Sb–op) and V is responsible for forming the upper M-type VHS [45], [90], [92]–[94]. Close to van-Hove fillings, the approximate nesting wave vector \(\mathbf{Q}_{ab}\) (Fig. 1 (b)) enhances various susceptibilities at the charge-ordering wave vector \(M_{c}\), which has been proposed to be the driver of the charge BO formation [49], [95], [96]. It has been shown that the putative LC states may primarily emerge from the mirror-odd \(\sigma_h=-1\) orbitals [45], [54] and distinct mirror symmetries of the wavefunction at the P- and M-type VHSs help in stabilizing the LC ordered state [45], [54]. In the disordered state, this would likely enhance the LC fluctuations. In this study, we construct an 8-band tight-binding model, with orbitals in the \(\sigma_h=-1\) sub-sector. This model, based on the DFT band structure from Ref. [45], includes one effective V-\(3d\) orbital per V site, Sb-\(5p_z\) from the Sb-ip, and four linear combinations of the Sb-op \(5p\) orbitals (see Supplement [97] for details of the model). The band structure of this model in Fig. 1 (c), features the P- and M-type VHSs with the correct mirror symmetries as in [54]. It also has a circular Fermi \(\Gamma\)-pocket which is primarily composed of the Sb-ip \(p_z\) orbitals (Fig. 1 (d)). Removal of this \(\Gamma\)-pocket through a Lifshitz transition at 7.5 GPa coincides with the disappearance of superconductivity in low pressure regime of [1], [82], [98], [99], hinting towards its importance in the low-pressure SC state. Also, quasi-particle interference spectroscopy has observed a large gap on this Sb-ip dominated \(\Gamma\)-pocket in [30]. This crucial \(\Gamma\)-pocket, and the two VHSs with the correct mirror symmetries, are built into our 8-band model.

Interaction Hamiltonian.- We choose the chemical potential \(\mu\) to be close to a P-type VHS (see Figs. 1 (c, d)) so that effects of the on-site Hubbard-U are suppressed due to sub-lattice interference [95], [100]. We therefore focus on two kinds of nearest-neighbor (NN) Coulomb repulsions: \(\mathcal{V}_{\text{vv}}\) between the V (v) sites and \(\mathcal{V}_{\text{sv}}\) between the V and Sb-ip (s) sites (for details, see Supplement [97]), \[\begin{gather} H_{\text{int}} =\mathcal{V}_{\text{vv}} \sum_{\mathcal{\mathbf{r}}_j,(a,b,c)} \left(n^{\text{v}}_{\mathcal{\mathbf{r}}_j,a} n^{\text{v}}_{\mathcal{\mathbf{r}}_j,b}+n^{\text{v}}_{\mathcal{\mathbf{r}}_j,a} n^{\text{v}}_{\mathcal{\mathbf{r}}_j-\mathbf{x}_{c},b}\right)\nonumber\\ + \mathcal{V}_{\text{sv}} \sum_{\mathcal{\mathbf{r}}_j,(a,b,c)} n^{\text{s}}_{\mathcal{\mathbf{r}}_j+\mathbf{\tilde{x}}_{a}} \left(n^{\text{v}}_{\mathcal{\mathbf{r}}_j,b}+ n^{\text{v}}_{\mathcal{\mathbf{r}}_j,c} \right) \label{eq:NN95Coulomb95interacting95ham}, \end{gather}\tag{1}\] where \((a,b,c)\) denotes cyclic permutations of the \((A,B,C)\) and \((\mathbf{{\tilde{x}}}_A, \mathbf{{\tilde{x}}}_B, \mathbf{{\tilde{x}}}_C) = (0, -\mathbf{x}_C, \mathbf{x}_B)\). \(\mathbf{x}_A, \mathbf{x}_B, \mathbf{x}_C\) are defined in Fig.1 caption. In the unit cell \(\mathcal{\mathbf{r}}_j\), \(n^{\text{v}}_{\mathcal{\mathbf{r}}_j,a}\) is the local density of V at sub-lattice \(a\), and \(n^{\text{s}}_{\mathcal{\mathbf{r}}_j}\) is the local density of the Sb-ip. We can reorganize Eq. 1 as an interaction between different types of bond-order and loop-current channels. In this decomposition, in addition to the conventionally studied charge bond-order (cBO) and charge loop-current (cLC) channels [34], [45], [49], [54], spin bond-order (sBO) and spin loop-current (sLC) channels are also present. We define four distinct types of order parameters: charge bond-orders: \(\mathcal{\hat{B}}^{\text{c,vv}}_{ \mathbf{q}}\), \(\mathcal{\hat{B}}^{\text{c,sv}}_{ \mathbf{q}}\); charge loop-currents: \(\mathcal{\hat{J}}^{\text{c,vv}}_{ \mathbf{q}}\), \(\mathcal{\hat{J}}^{\text{c,sv}}_{ \mathbf{q}}\); spin bond-orders: \(\mathcal{\hat{B}^{\text{s,vv}}_{ \mathbf{q}}}\), \(\mathcal{\hat{B}^{\text{s,sv}}_{ \mathbf{q}}}\); and spin loop-currents: \(\mathcal{\hat{J}^{\text{s,vv}}_{ \mathbf{q}}}\), \(\mathcal{\hat{J}^{\text{s,sv}}_{ \mathbf{q}}}\), with each type allowed between either V-V (vv) or V-Sb-ip (sv). Each order parameter is composed of three components, e.g. \(\mathcal{\hat{B}}^{\text{c,vv}}_{ \mathbf{q}}=\left(\mathcal{\hat{B}}^{\text{c,vv}}_{\mathbf{q},A},\mathcal{\hat{B}}^{\text{c,vv}}_{\mathbf{q},B},\mathcal{\hat{B}}^{\text{c,vv}}_{\mathbf{q},C}\right)\), along the three bond directions (\(\hat{\text{w}}\)) shown in Fig. 1 (a). For each component, condensation or ordering may occur for a different momentum \(\mathbf{q}=\mathbf{q}_A, \mathbf{q}_B, \mathbf{q}_C\). The V-V BO & LC order-parameter along the \(\hat{\text{w}}_C\) direction in Fig. 1 (a) is, \[\begin{align} \mathcal{\hat{B}}^{\texttt{i}\text{,vv}}_{ \mathbf{q},C}=\sum_{\mathbf{k}\sigma\sigma'}\left[X^{A\sigma,B\sigma'}_{\mathbf{k},\mathbf{k+q}}f^{\text{vv}}_{C,\mathbf{k+q}}+X^{B\sigma,A\sigma'}_{\mathbf{k},\mathbf{k+q}}f^{\text{vv}}_{C,\mathbf{-k}} \right]\frac{\Gamma^\texttt{i}_{\sigma\sigma'}}{2}\tag{2}\\ \mathcal{\hat{J}}^{\texttt{i}\text{,vv}}_{\mathbf{q},C}=i\sum_{\mathbf{k}\sigma\sigma'} \left[ X^{B\sigma,A\sigma'}_{\mathbf{k},\mathbf{k+q}}f^{\text{vv}}_{C,\mathbf{-k}} -X^{A\sigma,B\sigma'}_{\mathbf{k},\mathbf{k+q}}f^{\text{vv}}_{C,\mathbf{k+q}}\right]\frac{\Gamma^\texttt{i}_{\sigma\sigma'}}{2}\tag{3} \end{align}\] where, \(X^{A\sigma,B\sigma'}_{\mathbf{\mathbf{p},q}}=c^{\dagger}_{\mathbf{p},{\cal O}(A),\sigma}c_{\mathbf{q}, {\cal O}(B),\sigma'}\) with \({\cal{O}}(A),{\cal{O}}(B)\) referring to the V orbitals at the sublattice sites \(A, B\) (one effective \(d\)-orbital per V sub-lattice site) and spin indices \(\sigma,\sigma'\). \(f^{\text{vv}}_{C,\mathbf{k}}=1-e^{-i\mathbf{k}\cdot \mathbf{x}_C}\) is a form factor associated with the order parameter along the \(\hat{\text{w}}_C\) direction (see Fig. 1 (a)). In Eqs. 2 , 3 , \(\texttt{i}\in \{c,s\}\) specify the charge (\(\mathbb{1}\)) and spin (\(\Vec{\boldsymbol{\sigma}}\)) channels respectively, i.e. \((\Gamma^{\text{c}}, \Gamma^{\text{s}}) = (\mathbb{1}, \Vec{\boldsymbol{\sigma}})\). The order parameters along the other bond directions, or in V-Sb-ip channels (sv) can be defined similarly (see Supplement [97]). The bare interaction Hamiltonian can be rewritten as \[\begin{align} H_{\text{int}}=-&\frac{\mathcal{V}_{\text{vv}}}{2N}\sum_{\mathbf{q},\texttt{i}\in \{c,s\}} \left(\mathcal{\hat{B}}^{\texttt{i},\text{vv}}_{\mathbf{q}}\cdot\mathcal{\hat{B}}^{\texttt{i},\text{vv}}_{\mathbf{-q}}+ \mathcal{\hat{J}}^{\texttt{i},\text{vv}}_{\mathbf{q}}\cdot\mathcal{\hat{J}}^{\texttt{i},\text{vv}}_{\mathbf{-q}} \right)\nonumber\\ -&\frac{\mathcal{V}_{\text{sv}}}{2N}\sum_{\mathbf{q},\texttt{i}\in \{c,s\}}\left( \mathcal{\hat{B}}^{\texttt{i},\text{sv}}_{\mathbf{q}}\cdot\mathcal{\hat{B}}^{\texttt{i},\text{sv}}_{\mathbf{-q}}+ \mathcal{\hat{J}}^{\texttt{i},\text{sv}}_{\mathbf{q}}\cdot\mathcal{\hat{J}}^{\texttt{i},\text{sv}}_{\mathbf{-q}} \right).\label{eq:final95interaction95ham95split} \end{align}\tag{4}\] In the subsequent sections, we obtain the RPA-corrected effective interaction, which incorporates FS features into the vertex.

Susceptibilities.– We compute the static (\(\omega=0\)) RPA-corrected susceptibilities for bond-orders \(\chi^{\texttt{i}}_{\mathcal{BB}}(\mathbf{q})=\left \langle \mathcal{\hat{B}}_{\mathbf{q}}^{\texttt{i}} \mathcal{\hat{B}}_{\mathbf{-q}}^{\texttt{i}} \right\rangle\) and loop-currents \(\chi^{\texttt{i}}_{\mathcal{JJ}}(\mathbf{q})=\left \langle \mathcal{\hat{J}}_{\mathbf{q}}^{\texttt{i}} \mathcal{\hat{J}}_{\mathbf{-q}}^{\texttt{i}} \right\rangle\), for both the V-V and V-Sb-ip channels in the disordered state (see Fig. 2 (j)). To gain more insight, we have parameterized the charge-spin sector of the vertex by, \(V_{\text{charge-spin}}(\alpha)=(1-\alpha)\Gamma^{\text{c}}\Gamma^{\text{c}}+\alpha\Gamma^{\text{s}}\cdot \Gamma^{\text{s}}\), where \(\alpha\) tunes the relative strengths of the charge and spin channels. While the NN-Coulomb interaction fixes \(\alpha=\frac{1}{2}\), electron–phonon couplings [101]–[104] can potentially renormalize \(\alpha\). The orbital-sector of the vertex remains unchanged.

In Figs. 2 (a,c,e,g), we show the BO & LC susceptibilities, i.e. \(\chi_{\mathcal{BB}}\) and \(\chi_{\mathcal{JJ}}\) respectively in both V-V (vv) and the V-Sbip (sv) channels for \(\mu=3.88\) near the P-type VHS. As seen from the columns in Fig. 2, tuning the interaction vertex \(\alpha\) between the charge and spin channels respectively enhances the charge (\(\mathtt{i}=c\)) and spin (\(\mathtt{i}=s\)) susceptibilities (shown with different color scales in each panel in Fig. 2). Additionally, as a general trend we see that both the BO & LC susceptibilities in all the channels become most strongly peaked close to all the three \(M_c\)-points (see Fig. 1 (b), where \(c\in \{A,B,C\}\)) with increasing interaction strengths. Further, the strengths of the fluctuations are roughly comparable as evident from the peak heights. For V-V charge orders (see Fig. 2 (a1)), this \(M_{c}\)-ordering tendency is in agreement with experiments [6], [7], [9] which detect the formation of a \(2\times 2\) charge order on the V kagome net. Further, in Figs. 1 (a,c) we find that the V-V BO susceptibilities \(\chi^{\texttt{i},\text{vv}}_{\mathcal{BB}}(\mathbf{q})\) are larger than the V-V LC susceptibility \(\chi^{\texttt{i},\text{vv}}_{\mathcal{JJ}}(\mathbf{q})\), in agreement with previous studies which find BO phase close to the P-type VHS [42], [49], [54]. We also find that moving \(\mu\) closer to the M-type VHS results in an increase in the ratio of the LC to BO susceptibility or \(\chi^{\texttt{i},\text{vv}}_{\mathcal{JJ}}(\mathbf{q})\)/\(\chi^{\texttt{i},\text{vv}}_{\mathcal{BB}}(\mathbf{q})\) at \(\mathbf{q}=M_{c}\) which is caused by enhanced LC fluctuations. This is expected as Refs. [45], [54] found the mean-field LC-order to be stabilized close to the M-type VHS. However, on the V-Sb-ip bonds, the \(M_{c}\)-peaked V-Sb-ip LC susceptibility \(\chi_{\mathcal{JJ}}^{\texttt{i},\text{sv}}(\mathbf{q})\) is stronger than the V-Sb-ip BO susceptibility \(\chi_{\mathcal{BB}}^{\texttt{i},\text{sv}}(\mathbf{q})\) for the same interaction strength (see Figs. 2 (e,g)). We show in Figs.  2 (b, d, f, h), the pattern of the different long-range ordered states if they were allowed to condense at all the three \(\mathbf{q}=M_{c}\), resulting in a \(3\mathbf{Q}\) ordered state (i.e. for the three components: \(\mathbf{q}_A=M_A\), \(\mathbf{q}_B=M_B\), and \(\mathbf{q}_C=M_C\) respectively). The cBOs have modulations of bond charge densities, and the sBOs have alternating bonds of increased spin-up and spin-down occupation [95], [105]. The LCs (in Figs. 2 (d, h)) preserve Kirchhoff’s law and also the Bloch’s global-current constraint [62], [106], [107]. Additionally, the cLCs conserve the local density \(n_{\mathcal{\mathbf{r}}_j}\), and the sLCs conserve the local spin \(S^a_{\mathcal{\mathbf{r}}_j}\) (see Fig. 2 (i)). In Fig. 2 (a,c,e,g) , the spin and charge susceptibilities are equal for \(\alpha=0.5\). However, inclusion of beyond RPA-bubble diagrams are expected to comparatively strengthen the charge channels [41]. This would be equivalent to reducing \(\alpha\), while leaving the other results unchanged.

Since each kind of fluctuation can act as a pairing glue to potentially distinct kind of SC [62], [106], we henceforth study the phase diagram as we tune between different fluctuating channels, by adjusting their relative strengths using \(\{\alpha,\beta, \gamma\}\). As defined earlier, \(\alpha\) tunes between the charge and spin channels. The relative strength of the bond-order and loop-current channel is tuned by \(\beta\): \(V_{\text{BO+LC}}(\beta)=(1-\beta)V_{\text{BO}}+\beta V_{\text{LC}}\). We do not consider mixing among the BO and LC channels in the RPA susceptibility [97]. \(\gamma\) tunes between the V-V and V-Sb-ip interactions: \(\mathcal{V}_{\text{vv}}=(1-\gamma)\xi\) and \(\mathcal{V}_{\text{sv}}=\gamma\xi\) in Eq. 4 , here \(\xi\) fixes the overall interaction scale.

SC phase diagram.– The leading SC instability is found by solving a linearized gap equation using the RPA-corrected Cooper vertex incorporating the different fluctuating modes. Since we only focus on the competition between different channels, we set the overall interaction scale \(\xi=4\). See End Matter for more details. The various SC instabilities at \(\mu=3.88\), as a function of the three parameters \(\{\alpha, \beta, \gamma\}\) are shown in Fig. 3 (a). Most of the parameter space is dominated by singlet \(A_{1g}\) (\(s\)-wave) and \(E_{2g}\) (\(d\)-wave) channels. This is in agreement with experiments which report evidence of singlet pairing [108], although the actual pairing symmetry is still under debate [1], [2].

First, we discuss the SC driven by fluctuations produced by V-V interactions, i.e. the \(\gamma=0\) plane in Fig. 3 (a). We see that fluctuations of V-V cBO or sLC lead to \(s\)-wave (\(A_{1g}\)) pairing with a large amplitude on the outer two FSs, but a small amplitude on the \(\Gamma\)-pocket, as shown in Fig. 3 (d). This is due to a small projection of the V-V channel interactions on the \(\Gamma\)-pocket. In Fig. 3 (e) we see a qualitatively similar distribution of gaps across the FSs for the \(d\)-wave (\(E_{2g}\)) pairing driven by V-V cLC or sBO (see Figs. 3 (a)). Upon including non-linear corrections to the gap equation, the two-fold degenerate \(E_{2g}\) state can break time-reversal symmetry to become a chiral \(d_{x^2-y^2}\pm i d_{xy}\) [109] to maximize condensation energy.

On the other hand, as seen in Fig. 3 (a), the V-Sb-ip channel fluctuations always mediate \(s\)-wave (\(A_{1g}\)) pairing. We find in Figs. 3 (f,g) that fluctuating V-Sb-ip channels (close to \(\gamma\approx 1\)) induce a large gap across the \(\Gamma\)-pocket, in addition to comparably large gaps on the outer two FSs. This may be consistent with experiments that suggest sizable gaps on the \(\Gamma\)-pocket [30]. We can further sub-classify the \(A_{1g}\) region in Fig. 3 (b), by the relative sign of the average pairing gap across the inner \(\Gamma\)-pocket (in) and the outer two FS (out). We quantify this by the ratio of angular summed gaps on the inner and outer FSs, i.e. \(\tilde{\Delta}_{\text{in}}/\tilde{\Delta}_{\text{out}}=\oint _{\text{in}}\Delta_{\text{in}}(k) dk/\oint_{\text{out}} \Delta_{\text{out}}(k) dk\), and it is shown in Fig. 3 (b). Fluctuations of V-Sb-ip cLC or sBO channels create a gap with different signs across the outer two FS and inner \(\Gamma\)-pocket as shown in Fig. 3 (g), resulting in \(\tilde{\Delta}_{\text{in}}/\tilde{\Delta}_{\text{out}}<0\) in Fig. 3 (b). This is due to a large repulsion in these channels between the outer V-dominated and inner-Sb-ip dominated FSs, leading to a \(s_{+-}\) pairing. The opposite behavior is seen for the V-Sb-ip cBO or sLC channels which is always attractive, leading to an \(s_{++}\) pairing as shown in Fig. 3 (f), or \(\tilde{\Delta}_{\text{in}}/\tilde{\Delta}_{\text{out}}>0\) in Fig. 3 (b).

All the \(s\)-wave states in Fig. 3 (a) have an anisotropic gap, whose angular distribution is shown in Fig. 4 (a). Further, in Figs. 3 (a,c) we see a change in the pairing symmetry occurs on tuning \(\alpha\) for a given mode (BOs or LCs), between \(\alpha\sim 0.25-0.3\). This is because the spin vertex, with three components, contributes more than the charge vertex (details in Supplement [97]). In Fig. 3 (a), we also see a small region of a triplet \(p\)-wave (\(E_{1u}\)) for \(\beta\approx 0.5\). However, this weak triplet phase, with small \(\lambda_{\text{max}}\) values compared to the nearby singlet states, disappears on increasing the interaction scale \(\xi\) as shown in Fig. 4 (b). Additionally, for each value of \(\gamma\) we find that cLC and sBO fluctuations favor the same pairing symmetry (see Fig. 3 (a)), as do cBO and sLC fluctuations.

Time-reversal parity of fluctuating mode and SC.- The leading singlet pairing instability is closely tied to the time-reversal parity of the fluctuating modes [110]. TRS even modes lead to conventional \(s\)-wave pairing, and TRS odd modes favor unconventional \(s_{+-}\) or chiral \(d+id\) pairing with sign-changing gaps. Since the BO & LC channels each have definite TR parities, the pairing symmetry of the singlet states in Figs. 3 (a,b) can be understood qualitatively based on the TR parity of the strongest fluctuating mode. See End Matter for a more detailed discussion.

Discussions.– In summary, by considering an electron model which incorporates nearest neighbor V-V and V-planar Sb interactions and the appropriate band structure effects, we have shown that \(s_{++}\), \(s_{+-}\), or \(d+id\) superconductivity can arise from the competition between bond-order and loop-current fluctuations. All of these are consistent with experimental observations of a full gap in the SC state [12], [14], [111]–[114] at low pressures.

Recent experiments probing the SC gap structure using electron-irradiation techniques [112], ARPES [114], penetration depth measurements [111], and point-contact Andreev-reflection spectroscopy (PCARS) [113] have all reported the presence of highly anisotropic gaps. Further, the ARPES [114] observed the gap on the Sb-ip dominated \(\Gamma\)-pocket to be isotropic, with most of the anisotropy residing on the outer FSs. A similar coexistence of isotropic and anisotropic gaps was reported in the PCARS study [113]. In the absence of TRS breaking, both the \(s_{++}\) or \(s_{+-}\) gaps in Figs. 4 (a2,a3) can naturally explain this scenario. These states also have a large gap on the \(\Gamma\)-pocket (in the presence of a sufficiently large V-planar Sb interaction), consistent with quasi-particle interference spectroscopy measurements [30]. Distinguishing \(s_{+-}\) and \(s_{++}\) would require a carefully designed phase sensitive measurement. There have been various reports of time-reversal symmetry breaking [12], [14], [26], which may favor \(d+id\) state. However, this issue is still under debate as the impact of the extrinsic effects such as impurities and defects have yet to be clarified.

Recently, Schultz et. al [62] investigated the pairing mediated by charge-LC fluctuations between V-V and V-Sb-ip by incorporating a phenomenological LC propagator in the Cooper vertex, which introduced \(M_{c}\)-fluctuations without specifying its origin. Our results are consistent with these findings when we restrict our computations to the cLC sector.

Acknowledgements.– DJS and YBK thank G. Palle, R.M. Fernandes, J. Schmalian for earlier collaboration on a related subject and helpful discussions. This work was supported by the Natural Science and Engineering Council of Canada (NSERC) and the Center for Quantum Materials at the University of Toronto. Computations were performed on the Cedar cluster hosted by WestGrid and SciNet in partnership with the Digital Research Alliance of Canada. This work was further supported by the German Research Foundation (DFG) through CRC TRR 288 “Elasto-Q-Mat,” project A07 (D.J.S.).

1 END MATTER↩︎

1.1 Superconducting Instabilities↩︎

To include the effect of the different fluctuating modes (present across all \(\mathbf{q}\)) in the interaction vertex, we compute the RPA-corrected vertex: \(V=V_0+V_{\text{RPA}}\), by summing over all bubble-diagrams (more details in [97]). \[\begin{gather} \left[V_{\text{RPA}}(\mathbf{k},\mathbf{k'})\right]^{\tilde{\mu}_1\tilde{\mu}_2}_{\tilde{\mu}_3\tilde{\mu}_4}=-\left[ V_0 (1+\chi_0 V_0)^{-1} \chi_0 V_0 (\mathbf{k},\mathbf{k'})\right]^{\tilde{\mu}_1\tilde{\mu}_2}_{\tilde{\mu}_3\tilde{\mu}_4},\label{eq:RPA95correction95to95V} \end{gather}\tag{5}\] where \(\tilde{\mu}=(\mu,s)\) is a combined orbital-spin index. Eq. 5 includes a static orbital-resolved susceptibility \(\chi_0(\mathbf{q},\omega=0)\) (defined in [97]) and no dynamical effects to the pairing are considered. Eq. 5 is then projected onto the Cooper channel and anti-symmetrized. This results in an effective pairing interaction of the form, \[\begin{gather} H_{\text{eff}}=\frac{1}{2N}\sum_{\mathbf{k},\mathbf{k'},\{\tilde{\mu}\}}V(\mathbf{k},\mathbf{k'})^{\tilde{\mu}_1\tilde{\mu}_2}_{\tilde{\mu}_3\tilde{\mu}_4}c_{\mathbf{k}\tilde{\mu}_1}^{\dagger}c_{\mathbf{-k}\tilde{\mu}_3}^{\dagger} c_{-\mathbf{k'}\tilde{\mu}_2}c_{\mathbf{k'}\tilde{\mu}_4}.\label{eq:Cooper95195mt} \end{gather}\tag{6}\] We find the leading and sub-leading SC instabilities, with intra-band pairing, by solving a linearized gap equation for the Cooper vertex in the singlet \((\eta=0)\) and triplet (\(\eta=\{x,y,z\}\)) channels, \[\begin{gather} \lambda_{\eta}\Delta_{\eta}(n_1\mathbf{k})= - \sum_{n_2}\oint_{\mathbf{k'}\in \text{FS}_{n_2}} \frac{\tilde{V}_{\eta}(n_1\mathbf{k},n_2\mathbf{k'}) \Delta_{\eta} (n_2\mathbf{k'}) d\mathbf{k'}}{\left|\nabla_{\mathbf{k'}}\xi_{\mathbf{k'}n_2}\right| \left(2\pi \right)^2}. \label{eq:LGE} \end{gather}\tag{7}\] Here, \(n\mathbf{k}\) denotes a FS momenta \(\mathbf{k}\) of band \(n\). The triplet channels are degenerate because of unbroken spin-rotation symmetry in the disordered state and the absence of spin-orbit coupling. The eigenvector \(\Delta_{\text{max}}(n\mathbf{k})\) of the leading SC instability with the largest eigenvalue \(\lambda_{\text{max}}\), encodes its gap structure across the FS. We classify the gap functions based on lattice harmonics and irreps of the \(D_{6h}\) point group. Since in this study we primarily focus on the competition between different fluctuating channels, we set the overall interaction scale \(\xi=4\). For reference, the average band-width of the V-\(d\) bands near the VHS is \(W_{\text{band}}\sim 12\) (here, we set \(t_{\text{V-V}}/2=1\), see [97]). Small changes to \(\xi\) do not qualitatively change the leading SC instability (see Fig. 3 (c)). All our results are obtained at \(T/t_{\text{V-V}}\sim 0.005\), representative of the low-temperature limit.

1.2 Time-reversal parity of fluctuating mode and SC↩︎

The singlet-channel Cooper vertex for a fluctuating mode of TR parity \(P_T\) can be expressed as \[\begin{gather} \tilde{V}_{\eta=0}(n_1\mathbf{k},n_2\mathbf{k'})=\frac{-g^2P_T}{2}\left( \left|\mathcal{O}^{n_1,n_2}_{\mathbf{k},\mathbf{k'}}\right|^2 +\left|\mathcal{O}^{n_1,n_2}_{\mathbf{k},\mathbf{-k'}}\right|^2 \right)\label{eq:main95singlet95parity}, \end{gather}\tag{8}\] where \(\mathcal{O}^{n_1,n_2}_{\mathbf{k},\mathbf{k'}}\) is the fluctuating mode in the band-basis (a simple proof presented in [97]). Eq. 8 shows that the \(P_T = 1\) singlet channels are attractive (\(\tilde{V}_{\eta=0}(\mathbf{k},\mathbf{k}') < 0\)) across the entire FS, leading to conventional \(s\)-wave pairing. \(P_T = -1\) modes are repulsive (\(\tilde{V}_{\eta=0}(\mathbf{k},\mathbf{k}') > 0\)), therefore favoring unconventional sign-changing gaps. The BO & LC channels each have definite TR parities and therefore the pairing symmetry of the singlet states in Figs. 3 (a,b) can be understood qualitatively based on the TR parity of the strongest fluctuating mode. For example, focusing on the charge sector (\(\alpha=0\)) in Fig. 3 (a), dominant TRS even cBO fluctuations (at large \(\beta\)) drive \(s_{++}\)-wave pairing. On the other hand, dominant cLC fluctuations (at small \(\beta\)) drive unconventional chiral \(d+id\) or \(s_{+-}\) pairing. These conclusions remain valid after incorporating RPA corrections to Eq. 8 . However, this argument alone does not specify whether the leading pairing state is singlet or triplet.

References↩︎