Crossover between electron-electron and electron-phonon mediated pairing on the kagome lattice

We study electron-electron and electron-phonon mediated pairing in the Holstein extended Hubbard model on the kagome lattice near the van Hove fillings, and we investigate their combined effects on electron pairing states. We find that their combination can promote exotic pairings in a crossover region, where the filling is close to a van Hove singularity. In particular, at the $p$-type van Hove filling, the ${E}_{1u}$ ($p$-wave) and ${B}_{2u}$ (${f}_{{y}^{3}\ensuremath{-}3y{x}^{2}}$-wave) pairings become leading, and at the $m$-type van Hove filling, the ${E}_{1u}$ and ${A}_{2g}$ ($i$-wave) pairings get promoted. Moreover, we show that the electron-phonon interaction acquires a significant momentum dependence, due to the sublattice texture of the Fermi surfaces, which can promote non-$s$-wave pairing. We present a detailed analysis of these pairing propensities, and we discuss implications for the vanadium-based kagome superconductors ${A\text{V}}_{3}{\mathrm{Sb}}_{5}$.