Robustness of bipolaronic superconductivity to electron-density-phonon coupling

We study bipolaron formation and bipolaronic superconductivity on a square lattice, where electrons couple to both local Holstein phonons via on-site charge density and nonlocal bond Su-Schrieffer-Heeger phonons via modulation of hopping amplitudes. Using an unbiased Diagrammatic Monte Carlo method, we investigate how the interplay between these two types of electron-phonon coupling affects the bipolaron binding energy, effective mass, spatial extent (quantified by the mean-squared radius), and the superconducting transition temperature $T_c$. We find that, in some parameter space, the moderate Holstein coupling, though detrimental to $T_c$ when acting alone, can enhance superconductivity when combined with the bond SSH coupling by further compressing the bipolaron without significantly increasing its mass. Similarly, introducing bond SSH coupling into a Holstein bipolaron reduces its size while keeping the effective mass nearly unchanged, leading a higher $T_c$. These effects give rise to nonmonotonic behavior and reveal a cooperative regime in which both couplings work together to enhance superconductivity. We further examine phonon frequency asymmetry, particularly the case $\omega_H/t = 2\omega_B/t$, and show that in the deep adiabatic regime, adding Holstein coupling can even raise $T_c$ when combined with bond SSH coupling. These results highlight the distinct and complementary roles of local Holstein and non-local bond SSH electron-phonon couplings, and suggest strategies for optimizing high-$T_c$ superconductivity in systems with multiple phonon modes.