  Looking for superconductors with higher transition temperature requires a
guiding principle. In conventional superconductors, electrons pair up into
Cooper pairs via the retarded attraction mediated by electron-phonon coupling.
Higher-frequency phonon (or smaller atomic mass) leads to higher
superconducting transition temperature, known as the isotope effect.
Furthermore, superconductivity is the only instability channel of the metallic
normal state. In correlated systems, the above simple scenario could be easily
violated. The strong local interaction is poorly screened, and this conspires
with a featured Fermi surface to promote various competing electronic orders,
such as spin-density-wave, charge-density-wave and unconventional
superconductivity. On top of the various phases, the effect of electron-phonon
coupling is an intriguing issue. Using the functional renormalization group,
here we investigated the interplay between the electron correlation and
electron-phonon coupling in a prototype Hubbard-Holstein model on a square
lattice. At half-filling, we found spin-density-wave and charge-density-wave
phases and the transition between them, while no superconducting phase arises.
Upon finite doping, d-wave/s-wave superconductivity emerges in proximity to
spin-density-wave/charge-density-wave phases. Surprisingly, lower-frequency
Holstein-phonons are either less destructive, or even beneficial, to the
various phases, resulting in a negative isotope effect. We discuss the
underlying mechanism behind and the implications of such anomalous effects.
