Phonon Fluctuations Enhanced Superconductivity in a Three-Band Cuprate Model

The microscopic origin of high-$T_c$ superconductivity in cuprates remains an outstanding puzzle due to the interplay of intertwined orders. Experimental observations and theoretical studies have suggested crucial roles of phonons, with the $B_{1g}$ oxygen buckling mode emerging as a promising candidate. Here, we investigate the microscopic influence of this phonon in a three-band cuprate model. We find that the $B_{1g}$ phonon plays opposite roles in superconductivity depending on whether it lies in the adiabatic or antiadiabatic regime at a given temperature. Thermally populated adiabatic phonons suppress $d$-wave pairing through the loss of quasiparticle coherence, whereas antiadiabatic phonons simultaneously enhance coherence and the attractive $d$-wave pairing vertex, strengthening superconductivity across multiple diagnostics. This enhancement also exhibits pronounced electron-hole asymmetry arising from the different orbital characters of doped carriers. Our results therefore establish a microscopic mechanism through which $B_{1g}$ phonons can cooperate with electronic correlations to substantially enhance $d$-wave superconducting pairing in cuprates.