Phase diagram of the one-dimensional Hubbard-Holstein model at quarter filling

We derive an effective Hamiltonian for the one-dimensional Hubbard-Holstein model, valid in a regime of both strong electron-electron (e-e) and electron-phonon (e-ph) interactions and in the nonadiabatic limit (t/omega(0) <= 1), by using a nonperturbative approach. We obtain the phase diagram at quarter filling by employing a modified Lanczos method and studying various density-density correlations. The spin-spin AF (antiferromagnetic) interactions and nearest-neighbor repulsion, resulting from the e-e and the e-ph interactions, respectively, are the dominant terms (compared to hopping) and compete to determine the various correlated phases. As e-e interaction (U/t) is increased, the system transits from an AF cluster to a correlated singlet phase through a discontinuous transition at all strong e-ph couplings 2 <= g <= 3 considered. At higher values of U/t and moderately strong e-ph interactions (2 <= g <= 2.6), the singlets break up to form an AF order and then to a paramagnetic order all in a single sublattice; whereas at larger values of g (>2.6), the system jumps directly to the spin disordered charge-density-wave (CDW) phase.