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

Abstract

We derive an effective electronic Hamiltonian for the square lattice Hubbard-Holstein model (HHM) in the strong electron-electron $(e\text{\ensuremath{-}}e)$ and electron-phonon $(e\text{\ensuremath{-}}ph)$ coupling regime and under nonadiabatic conditions $(t/{\ensuremath{\omega}}_{0}\ensuremath{\le}1)$, $t$ and ${\ensuremath{\omega}}_{0}$ 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 $g\text{\ensuremath{-}}U$ plane where $g$ and $U$ are the $e\text{\ensuremath{-}}ph$ 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.