Superconductivity pairing mechanism in the Holstein model

Superconductivity driven by electron-phonon coupling (EPC) has been widely studied, but the potential for unconventional pairing, such as d-wave, remains debated. We perform determinant quantum Monte Carlo simulations of the Holstein model to systematically explore the potential of d-wave superconductivity across a wide range of parameters including temperature, EPC strength, and phonon frequency. We find that s-wave superconductivity dominates across all parameters. While we observe a nonzero d-wave pair-field susceptibility in the simulations, it primarily stems from van Hove singularities in the non-interacting band structure. Furthermore, increasing the vertex correction suppresses, rather than enhances, the d-wave susceptibility, underscoring the robustness of s-wave pairing in this model.