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Due to unique crystal structure of Kagome compounds AV₃Sb₅, the electronic structures of normal-state AV₃Sb₅ are rather complicated. The band structures of the pristine phase of CsV₃Sb₅Tan et al. (2021), calculated by the density functional theory (DFT), predicted that seven bands cross the Fermi energy E_F and are composed of V 3$d$ orbitals and Sb 5$p$ orbitals, which shows typical multi-orbital character. Since the unit cell of Kagome lattice contains three V sites in the pristine phase, at the same time, the metallic ground state in AV₃Sb₅ and partial filled V 3$d$ orbitals also imply the multi-orbital character in the low-energy model. These inevitably bring difficulty in constructing an effective multiorbital model for superconducting AV₃Sb₅. At present the microscopic origin of the charge density wave is generally attributed to the van Hove singularity, whereas, to understand the essential superconductive nature, a proper effective model correctly describing the low-energy processes plays key roles and is highly expected.

So-far experimental data strongly support unconventional superconductive pairing mechanism in AV₃Sb₅: conventional e-ph coupling mechanism is not enough to account for the superconductive microscopic origin, neither the transition temperatures estimated by the McMillan’s formula for RbV₃Sb₅ and CsV₃Sb₅ considerably deviate from the experimental T_C Tan et al. (2021), nor the experimental T_C violates the expectation of the conventional e-ph mechanism when A varies from K, Rb to Cs. AV₃Sb₅ superconductors also display many features of unconventional superconductivity: with the increase of hydrostatic pressure, the P-T phase diagram demonstrates two superconductive regions Chen et al. (2021b); Zhu et al. (2021), similar to the two-dome superconductive phase diagrams in iron-based superconductorsSun et al. (2012); Shahi et al. (2018). Though the local magnetic moment was argued to be absent Kenney et al. (2021), the observation of the anomalous Hall effect in CsV₃Sb₅ Yang et al. (2020) strongly favors of the presence of local magnetic field, hence the local spins. The first-principles electronic structure calculations also suggested there exists local magnetic moments in AV₃Sb₅ Zhang et al. (2021b); Wu et al. (2021a), implying that the spin-orbital fluctuations mechanism is the possible pairing origin, since not only the system posses local magnetic moments and multiorbital character, but also Kagome lattice structure favors strong spin frustrations and fluctuations. These eager further deep theoretical investigations.

On the other hand, the details of the superconductive nature in AV₃Sb₅ are not clear. A very recent studyWu et al. (2021b) suggested that the superconducting electrons in CsV₃Sb₅ are f-wave triplet pairing: i.e. the superconductive Cooper pairs are spatial f-wave, and spin parallel or triplet. Such a prediction was inconsistent with the temperature-dependence Knight shift experimentMu et al. (2021). One also finds that in similar sixfold-symmetric superconductor Na_xCoO₂.yH₂O, an earlier theoretical investigation suggested to be f-wave pairing symmetry Mazin and Johannes (2005), however, the experiments supported that the pairing symmetry of superconducting cobaltates is singlet d-wave Matano et al. (2007). This arises the puzzle whether the Cooper pairs with high angular momentum could stably exist in AV₃Sb₅. In this Letter, based on our calculated band structures for CsV₃Sb₅ in the pristine phase, we fit an effective low-energy 6-band model; utilizing the random phase approximation (RPA) on the effective minimal model, we show that the superconducting pairing strengths increase with the increase of Coulomb correlation, and the superconductive pairing symmetry is singlet d_xy-wave-like. The present theoretical results favor our understand on the unconventional superconductivity in Kagome compounds AV₃Sb₅.

Our band structures and Fermi surfaces are in agreement with available literature LaBollita and Botana (2021); Luo et al. (2021). There are three distinct Fermi surface sheets at k_z=0 : (i) a central ring sheet around $\Gamma$ contributed by Sb-$p_{z}$ orbitals, (ii) a hexagonal sheet composed of V-$d_{xy}$, V-$d_{x^{2}-y^{2}}$, and V-$d_{z^{2}}$, and (iii) two additional sheets composed of V-$d_{xz}/d_{yz}$ orbitalsLaBollita and Botana (2021); Luo et al. (2021). In fitting the band structures with an effective low-energy tight-binding model, we consider the large hexagonal Fermi pocket (ii) and one of the Fermi surface sheets (iii), to which the corresponding energy bands are hybridized weakly with the $p$ orbitals of SbLaBollita and Botana (2021), and then construct a six-band tight-binding (TB) model with two local orbitals on each V site. By constructing localized Wannier functions, the hopping integrals between different V orbitals $\alpha$ and $\beta$ ($\alpha=xz,yz$, $\beta=xy,3z^{2}-r^{2},x^{2}-y^{2}$) are zeros, see the TABLE S1 in Supplementary Materials, indicating that interorbital hybridization between the $\alpha$ and $\beta$ orbits of vanadiums is vanishing small, so we neglect the interorbital hopping terms in the tight-binding model. The fitted tight-binding bands and the full electronic structures of CsV₃Sb₅ are the red and black lines in Fig. 2(a), respectively.

The fitted six pack-band tight-binding model for CsV₃Sb₅ can be written as

where the inequivalent V sites $i,j=A,B,C$, the orbital indices $\alpha,\beta=1,2$, and the hopping matrix elements $\xi_{ij,\alpha}(k)=\left[(\varepsilon_{\alpha}+t^{\prime\prime}_{\alpha}\phi^{3}(k)-\mu)\delta_{ij}+t_{\alpha}\phi^{1}_{ij}(k)+t^{\prime}_{\alpha}\phi^{2}_{ij}(k)\right]$, where $\phi^{n}$ denotes the lattice structure factor defined in Supplementary Materials. The $\varepsilon_{\alpha}$ denotes the energy level of the orbital $\alpha$ and $\mu$ is the chemical potential. The operator $c^{\dagger}_{k,i,\alpha\sigma}$ ($c_{k,i,\alpha\sigma}$) creates (annihilates) an electron with momentum $k$ and spin $\sigma$ of sublattice $i$ in orbital $\alpha$. The $t_{\alpha}$, $t^{\prime}_{\alpha}$ and $t^{\prime\prime}_{\alpha}$ denote intra-orbital hopping integrals for the first, second and third nearest neighbors respectively. The hopping parameters are given in Supplementary Materials, see Table S3. Within the present six-band model, the Fermi surface is plotted in Fig. 2(b). The corresponding electron filling number is $n=5.45$ to match the chemical potential of CsV₃Sb₅ .

To investigate the spin fluctuations and superconducting pairing properties in CsV₃Sb₅, we consider the on-site Coulomb interaction as following

here $l$ denote the $l-$th unit cell. The on-site intra- and interorbital Coulomb repulsions are denoted by $U$ and $U^{\prime}$. $J_{H}$ and $J_{P}$ are the Hund’s rule exchange and pair-hopping term, respectively. Throughout this paper, we set the Couloub and Hund’s interaction parameters $U^{\prime}=U-2J_{H}$ and $J_{H}=J_{P}=U/8$, which satisfy spin rotational invariance. The fact that the DFT band structures are in good agreement with the angle-resolved photoemission spectroscopy (ARPES) dataOrtiz et al. (2020), demonstrates that the electronic correlation in CsV₃Sb₅ is not very strong.

Within the random phase approximation (RPA), the singlet (s) and triplet (t) pairing vertices arising from spin and charge fluctuations for the multiorbital case Takimoto et al. (2002, 2003); Kubo (2007) are given by

here $U^{s}$ and $U^{c}$ represent the $12\times 12$ matrices in the V orbital space, and $\chi^{RPA}_{S}$ and $\chi^{RPA}_{O}$ are the ones RPA spin and orbital (charge) susceptibility matrix given in the Sec. III of Supplementary Materials.

By projecting the zero frequency $(\omega=0)$ pairing vertex into the band space Graser et al. (2009); Scalapino (2012), we obtain

The $\Gamma^{s}_{i,j}(k,k^{\prime})$ (or $\Gamma^{t}_{i,j}(k,k^{\prime})$) determines the scatterings of two electrons of opposite (or same) spin from the state $(k,-k)$ on the Fermi surface sheet $C_{i}$ to the state $(k^{\prime},-k^{\prime})$ on the Fermi surface sheet $C_{j}$. By means of a mean-field decouping of the interaction, the gap equation in singlet and triplet channels can be obtained and expressed as

with

It is worthy noting that $\Delta^{i,s}_{k}$ is an even function with respect to k and $\Delta^{i,t}_{k}$ an odd function. The superconducting instability appears at temperature. When the temperature just below T_c the gap function $\Delta^{i,s/t}_{k}$ are expected to be very small, and hence we study the superconducting instability by solving the linearized gap equations Graser et al. (2009); Maier et al. (2019)

Here the eigenvalues $\lambda_{\alpha}$ are the superconducting pairing strengths and $g_{i}^{\alpha}(k)$ the corresponding eigenfunctions. $V_{G}$ is the area of a Brillouin zone, $i$ and $j$ are the band indices of Fermi surface vectors $k$, $k^{\prime}$, respectively, $|v_{F_{j}}(k^{\prime})|$ is the magnitude of the Fermi velocity. The largest pairing strength $\lambda_{\alpha}$ determines the highest transition temperature and the corresponding eigenfunction $g_{i}^{\alpha}(k)$ shows the symmetry of the gap function. Throughout this paper we perform the calculations at temperature of $k_{B}T=0.002$ eV.

Note that the e-ph coupling mechanism could not account for the origin of the superconductivity in CsV₃Sb₅, the spin-fluctuation mechanism seems a plausible candidate. In the theory of spin-fluctuation mediated superconductivity, the singlet pair scattering $\Gamma^{s}_{i,j}(k,k^{\prime})$ of a Cooper pair at k to k^′ is proportional to the RPA spin susceptibilityKreisel et al. (2017); Scalapino (2012), one expects that the gap function on different parts Fermi surface connected by ${\bf q}_{1}\approx\frac{1}{4}\Gamma K$ and other sixfold-symmetric wavevectors has opposite signs, according to the linearized gap equations.

The largest singlet pairing gap functions g(k) at $U=0.5$ eV and 0.8 eV are plotted in Fig. 5. These gap functions show sign-change at the high symmetry k points on the Fermi surface. Further analysis in Fig. 6 shows when $U=0.5$ eV, the gap function exhibiting the antisymmetry with respect to the $x$ axis and the $y$ axis, i.e., d_xy-wave-like symmetry, on the outer Fermi sheet and weak g-wave symmetry on the inner Fermi sheet. The gap function on the inner Fermi sheet is about smaller than that on the outer Fermi sheet by two orders in magnitude. When $U=0.8$ eV, the gap functions slightly change, while still are d_xy-wave-like. These suggest that the d_xy-wave-like paring is dominant in CsV₃Sb₅. The d_xy-wave-like pairing is in consistent with the results of finite linear term of thermal conductivity at ultra low temperatureZhao et al. (2021b) and the V-shaped gapChen et al. (2021d), which suggest the nodal superconductivity in CsV₃Sb₅. However, the study of impurity effectsXu et al. (2021) on superconductivity by STM and the appearance of Hebel-Slichter peakMu et al. (2021) in $1/T_{1}T$ just below T_C proposes S-wave pairing in CsV₃Sb₅. To uncover the structure of the superconducting gap in CsV₃Sb₅, further experiments on superconducting state, such as angular-resolved photoemission spectroscopy (ARPES), Bogoliubov quasiparticle interference, and phase-sensitive tests, are crucial and highly expected.

In fitting the present effective model (1), we elaborately consider the major bands contributed from the 3d orbits of V ions and ignore the bands contributed from Sb 5p orbits. The latter consists of the central Fermi surface around the $\Gamma$ point which arises from Sb 5p orbits and a few of Fermi surface fragments arising from the hybridization between dominant Sb 5p orbits and partial V 3d orbit. One notices an experimental fact that upon applying pressure, the central Fermi surface and Sb-related 5$p$ bands of CsV₃Sb₅ almost do not vary with the disappearance of the charge-density-wave order Liu et al. (2021), while the superconducting properties change considerably with increasing pressure, suggesting that the present fitted effective minimal model is responsible for the unconventional superconductivity in CsV₃Sb₅.

Contrast to the present singlet d_xy-like superconductive pairing symmetry in our effective minimal model for CsV₃Sb₅, very recently Wu et al. fitted another effective 6-orbital model Wu et al. (2021b) based on the electronic structures of KV₃Sb₅, they found that within the RPA their model favors the triplet f-wave pairing symmetry. This indicates that the superconductive pairing symmetry is sensitive to the details of the band structures in AV₃Sb₅. Our present singlet pairing results agree with recent experiments, suggesting the plausible of the present effective minimal model for the superconductivity in AV₃Sb₅.

In summary, we have obtained an effective low-energy six-band model for CsV₃Sb₅ by fitting the the first-principles electronic structures. Within the random phase approximation and on the basis of the effective minimal model, we have found that the superconducting pairing strength increases with the lift of Coulomb correlation and decline of temperature, and superconductive pairing symmetry is singlet, most probably with the d_xy-wave-like symmetry. These results highlight the essence of the unconventional superconductivity in Kagome compounds AV₃Sb₅. Also, one may recall that the full effective model should include the central Fermi surface and a few of Fermi surface fragments ignored in the present study, the role of these dominant Sb 5p bands on the nature of the superconductivity in Kagome compounds AV₃Sb₅ is worthy of further investigations.