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

We derive an effective electronic Hamiltonian for square lattice Hubbard-Holstein model (HHM) in the strong electron-electron (e-e) and electron-phonon (e-ph) coupling regime and under non-adiabatic conditions ($t/\omega_0 \leq 1$), $t$ and $\omega_0$ being the electron hopping and phonon frequency respectively. Using Density Matrix Renormalization Group method, we simulate this effective electronic model on $4-$Leg cylinder system at quarter-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 interaction respectively. 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 which abruptly goes to another CDW phase as we increase $U$. But for smaller $g$, we find a metallic phase sandwiched between PS and singlet CDW phase. This phase is characterized by vanishing charge gap but finite spin gap -- suggesting a singlet superconducting phase.