Phase diagram of the Hubbard-Holstein model on a four-leg tube system at quarter filling

We derive an effective electronic Hamiltonian for the square lattice Hubbard-Holstein model (HHM) in the strong electron-electron (e-e) and electron-phonon (e-ph) coupling regime and under nonadiabatic conditions (t/omega(0) <= 1), t and omega(0) being the electron hopping and phonon frequency respectively. Using the density matrix renormalization-group method, we simulate this effective electronic model on a four-leg cylinder system at quarter filling and present a phase diagram in the g-U plane where g and U are the e-ph coupling constant and Hubbard on-site interaction respectively. For larger g, we find that a cluster of spins, i.e., phase separation (PS), gives way to a charge density wave (CDW) phase made of nearest-neighbor 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 the singlet CDW phase. This phase is characterized by a vanishing charge gap but a finite spin gap, suggesting a singlet superconducting phase.