Effective minimal model and unconventional spin-singlet pairing in Kagome superconductor CsV3Sb5

Recently synthesized Kagome compounds AV$_3$Sb$_5$ attract great attention due to the unusual coexistence of the topology, charge density wave and superconductivity. In this {\it Letter}, based on the band structures for CsV$_3$Sb$_5$ in pristine phase, we fit an effective 6-band model for the low-energy processes; utilizing the random phase approximation (RPA) on the effective minimal model, we obtain the momentum-resolved static spin susceptibility attributing the spin-fluctuation pairing mechanism, we find that the superconducting pairing strengths increase with the lift of the Coulomb correlation, and the superconductive pairing symmetry is singlet, the gap functions are antisymmetric with respect to the x-axis and the y-axis in the intermediate to strong Coulomb correlated regime,indicating the unconventional superconductivity in Kagome compounds AV$_3$Sb$_5$.

So-far experimental data strongly support unconventional superconductive pairing mechanism in AV 3 Sb 5 : 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 RbV 3 Sb 5 and CsV 3 Sb 5 considerably deviate from the experimental T C 22 , nor the experimental T C violates the expectation of the conventional e-ph mechanism when A varies from K, Rb to Cs. AV 3 Sb 5 superconductors also display many features of unconventional superconductivity: with the increase of hydrostatic pressure, the P-T phase diagram demonstrates two superconductive regions 7,10 , similar to the two-dome superconductive phase diagrams in iron-based superconductors 34,35 . Though the local magnetic moment was argued to be absent 36 , the observation of the anomalous Hall effect in CsV 3 Sb 5 37 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 AV 3 Sb 5 23,24 , 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 AV 3 Sb 5 are not clear. A very recent study 33 suggested that the superconducting electrons in CsV 3 Sb 5 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 experiment 20 . One also finds that in similar sixfold-symmetric superconductor Na x CoO 2 .yH 2 O, an earlier theoretical investigation suggested to be f-wave pairing symmetry 38 , however, the experiments supported that the pairing symmetry of superconducting cobaltates is singlet d-wave 39 . This arises the puzzle whether the Cooper pairs with high angular momentum could stably exist in AV 3 Sb 5 . In this Letter, based on our calculated band structures for CsV 3 Sb 5 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 AV 3 Sb 5 .  Fig. 1(a) shows the crystal structure of the layered CsV 3 Sb 5 . Recent resistivity measurement on CsV 3 Sb 5 4 showed remarkable anisotropy of the ab-plane to the c-axis, indicating that the interlayer V-V hybridization is rather small. Throughout this paper we confine our study on the V-Sb Kagome plane, as seen in Fig. 1 Our band structures and Fermi surfaces are in agreement with available literature 25,26 . There are three distinct Fermi surface sheets at k z =0 : (i) a central ring sheet around Γ 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 orbitals 25,26 . 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 Sb 25 , 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 α and β (α = xz, yz, β = xy, 3z 2 − r 2 , x 2 − y 2 ) are zeros, see the TABLE S1 in Supplementary Materials, indicating that interorbital hybridization between the α and β 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 CsV 3 Sb 5 are the red and black lines in Fig. 2(a), respectively.
The fitted six pack-band tight-binding model for CsV 3 Sb 5 can be written as where the inequivalent V sites i, j = A, B, C, the orbital indices α, β = 1, 2, and the hopping matrix elements where φ n denotes the lattice structure factor defined in Supplementary Materials. The ε α denotes the energy level of the orbital α and µ is the chemical potential. The operator c † k,i,ασ (c k,i,ασ ) creates (annihilates) an electron with momentum k and spin σ of sublattice i in orbital α. The t α , t ′ α and t ′′ α 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 CsV 3 Sb 5 .
To investigate the spin fluctuations and superconducting pairing properties in CsV 3 Sb 5 , 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 ′ . 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 ′ = 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) data 4 , demonstrates that the electronic correlation in CsV 3 Sb 5 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 [40][41][42] are given by here U s and U c represent the 12 × 12 matrices in the V orbital space, and χ RPA The Γ s i, j (k, k ′ ) (or Γ t i, j (k, k ′ )) 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 ′ , −k ′ ) 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 ∆ i,s k is an even function with respect to k and ∆ i,t k an odd function. The superconducting instability appears at temperature. When the temperature just below T c the gap function ∆ i,s/t k are expected to be very small, and hence we study the superconducting instability by solving the linearized gap equations 43,45 Here the eigenvalues λ α are the superconducting pairing strengths and g α i (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 ′ , respectively, |v F j (k ′ )| is the magnitude of the Fermi velocity. The largest pairing strength λ α determines the highest transition temperature and the corresponding eigenfunction g α i (k) shows the symmetry of the gap function. Throughout this paper we perform the calculations at temperature of k B T = 0.002 eV.
Momentum-resolved Spin Susceptibility: The distribution of spin susceptibility in momentum space may not only disclose the spin fluctuation modes or the magnetic order, but also provide the superconductive pairing information. The RPA spin susceptibility of the effective minimal model for pristine CsV 3 Sb 5 shows six peaks near Q 1 ≈ 1 4 ΓK and other five sixfold-symmetric wavevectors, as seen the red points in Fig.  3(a). In the lines connecting these maxima, the magnitude of the spin susceptibility is also very large, as seen the yellow lines in Fig. 3(a). These indicate that there exist multi-mode antiferromagnetic or spin-density-wave fluctuations. We also find that these peaks are enhanced with the increase of the Coulomb correlation U, as seen in Fig. 3(b).
Note that the e-ph coupling mechanism could not account for the origin of the superconductivity in CsV 3 Sb 5 , the spin-fluctuation mechanism seems a plausible candidate. In the theory of spin-fluctuation mediated superconductivity, the singlet pair scattering Γ s i, j (k, k ′ ) of a Cooper pair at k to k ′ is proportional to the RPA spin susceptibility 44,46 , one expects that the gap function on different parts Fermi surface connected by q 1 ≈ 1 4 ΓK and other sixfold-symmetric wavevectors has opposite signs, according to the linearized gap equations.   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 CsV 3 Sb 5 . The d xy -wave-like pairing is in consistent with the results of finite linear term of thermal conductivity at ultra low temperature 21 and the V-shaped gap 16 , which suggest the nodal superconductivity in CsV 3 Sb 5 . However, the study of impurity effects 14 on superconductivity by STM and the appearance of Hebel-Slichter peak 20 in 1/T 1 T just below T C proposes S-wave pairing in CsV 3 Sb 5 . To uncover the structure of the superconducting gap in CsV 3 Sb 5 , 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 Γ 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 5p bands of CsV 3 Sb 5 almost do not vary with the disappearance of the charge-densitywave order 28 , while the superconducting properties change considerably with increasing pressure, suggesting that the present fitted effective minimal model is responsible for the unconventional superconductivity in CsV 3 Sb 5 .
Contrast to the present singlet d xy -like superconductive pairing symmetry in our effective minimal model for CsV 3 Sb 5 , very recently Wu et al. fitted another effective 6-orbital model 33 based on the electronic structures of KV 3 Sb 5 , 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 AV 3 Sb 5 . Our present singlet pairing results agree with recent experiments, suggesting the plausible of the present effective minimal model for the superconductivity in AV 3 Sb 5 .
In summary, we have obtained an effective low-energy sixband model for CsV 3 Sb 5 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 AV 3 Sb 5 . 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 AV 3 Sb 5 is worthy of further investigations.