Collective modes of SDW in 1D extended Hubbard model at quarter-filling

We have examined the collective mode of spin density wave (SDW) for a quarter-filled one-dimensional (1D) extended Hubbard model with next-nearest-neighbor repulsive interaction (V2). Within the random phase approximation (RPA), we obtain a new collective mode associated with 2kF-charge density wave (CDW), which appears with increasing V2 (kF being a Fermi wave number). Furthermore, the corresponding effective Hamiltonian is derived in terms of the velocity of the excitation and the spin susceptibility.