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Phase diagram of 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 non-adiabatic limit ($t/ω_0 \leq 1$), by using a non-perturbative 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 $2 \leq g \leq 3$ considered. At higher values of $U/t$ and moderately strong e-ph interactions ($2 \leq g \leq 2.6$), 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.