Coexistence of charge-density wave and spin-Peierls orders in quarter-filled quasi-one-dimensional correlated electron systems

Charge and spin-Peierls instabilities in quarter-filled $\mathrm{}(n=1/2\mathrm{})\mathrm{}$ compounds consisting of coupled ladders and/or zigzag chains are investigated. Hubbard and $t-J$ models including local Holstein and/or Peierls couplings to the lattice are studied by numerical techniques. Next nearest-neighbor hopping and magnetic exchange, and short-range Coulomb interactions are also considered. We show that, generically, these systems undergo instabilities towards the formation of charge-density waves, bond order waves, and (generalized) spin-Peierls modulated structures. Moderate electron-electron and electron-lattice couplings can lead to a coexistence of these three types of orders. In the ladder, a zigzag pattern is stabilized by the Holstein coupling and the nearest-neighbor Coulomb repulsion. In the case of an isolated chain, bond-centered and site-centered ${2k}_F$ and ${4k}_F$ modulations are induced by the local Holstein coupling. In addition, we show that, in contrast to the ladders, a small charge ordering in the chains strongly enhances the spin-Peierls instability. Our results are applied to the ${\mathrm{NaV}}_2{\mathrm{O}}_5$ compound (trellis lattice) and various phases with coexisting charge disproportionation and spin-Peierls order are proposed and discussed in the context of recent experiments. The role of the long-range Coulomb potential is also outlined.