Charge ordering in quarter-filled ladder systems coupled to the lattice

We investigate charge ordering in the presence of electron-phonon coupling for quarter-filled ladder systems by using exact diagonalization. As an example we consider NaV2O5 using model parameters obtained from first-principles band-structure calculations. The relevant Holstein coupling to the lattice considerably reduces the critical value of the nearest-neighbor Coulomb repulsion at which formation of the zig-zag charge-ordered state occurs, which is then accompanied by a static lattice distortion. Energy and length of a kink-like excitation on the background of the distorted lattice are calculated. Spin and charge spectra on ladders with and without static distortion are obtained, and the charge gap and the effective spin-spin exchange parameter J are extracted. J agrees well with experimental results. Analysis of the dynamical Holstein model, restricted to a small number of phonons, shows that low frequency lattice vibrations have a strong influence on the charge ordering, particularly in the vicinity of the phase transition point. By investigating the charge order parameter we conclude that phonons produce dynamical zig-zag lattice distortions. A model with only static distortions gives a good description of the system well away from the transition point while overestimating the amount of charge ordering in the vicinity of the phase transition.