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We derive an effective electronic Hamiltonian for square latticeHubbard-Holstein model (HHM) in the strong electron-electron (e-e) andelectron-phonon (e-ph) coupling regime and under non-adiabatic conditions($t/\omega_0 \leq 1$), $t$ and $\omega_0$ being the electron hopping and phononfrequency respectively. Using Density Matrix Renormalization Group method, wesimulate this effective electronic model on $4-$Leg cylinder system atquarter-filling and present a phase diagram in $g-U$ plane where $g$ and $U$are being the e-ph coupling constant and Hubbard on-site interactionrespectively. For larger $g$, we find cluster of spins i.e. phase separation(PS) gives way to a charge density wave (CDW) phase made of NN singlets whichabruptly goes to another CDW phase as we increase $U$. But for smaller $g$, wefind a metallic phase sandwiched between PS and singlet CDW phase. This phaseis characterized by vanishing charge gap but finite spin gap -- suggesting asinglet superconducting phase.

Phase diagram of Hubbard-Holstein model on 4-leg tube system at quarter-filling