The extended Hubbard Hamiltonian is a widely accepted model for uncovering the effects of strong correlations on the phase diagram of low-dimensional systems, and a variety of theoretical techniques have been applied to it. In this paper the world-line quantum Monte Carlo method is used to study spin, charge, and bond order correlations of the one-dimensional extended Hubbard model in the presence of coupling to the lattice. A static alternating lattice distortion (the ionic Hubbard model) leads to enhanced charge density wave correlations at the expense of antiferromagnetic order. When the lattice degrees of freedom are dynamic (the Hubbard-Holstein model), we show that a similar effect occurs even though the charge asymmetry must arise spontaneously. Although the evolution of the total energy with lattice coupling is smooth, the individual components exhibit sharp crossovers at the phase boundaries. Finally, we observe a tendency for bond order in the region between the charge and spin density wave phases.
Theoretical and computational studies of the quantum phase diagram of the one-dimensional half-filled extended Hubbard model (EHM) indicate a narrow bond-order wave (BOW) phase with finite magnetic gap
Em for on-site repulsion $U<{U}^{\ensuremath{\ast}}$, the critical point, and nearest-neighbor interaction ${V}_{c}\ensuremath{\approx}U/2$ near the boundary of the charge-density wave (CDW) phase. Potentials with more extended interactions that retain the EHM symmetry are shown to have a less cooperative CDW transition with higher ${U}^{\ensuremath{\ast}}$ and wider BOW phase. Density-matrix renormalization group is used to obtain
Em directly as the singlet-triplet gap, with finite
Em marking the BOW boundary
Vs(U). The BOW/CDW boundary
Vc(U) is obtained from exact finite-size calculations that are consistent with previous EHM determinations. The kinetic energy or bond order provides a convenient new estimate of ${U}^{\ensuremath{\ast}}$ based on a metallic point at
Vc(U) for $U<{U}^{\ensuremath{\ast}}$. Tuning the BOW phase of half-filled Hubbard models with different intersite potentials indicates a ground state with large charge fluctuations and magnetic frustration. The possibility of physical realizations of a BOW phase is raised for Coulomb interactions.