Competition between charge and spin order in the $t-U-V$ extended   Hubbard model on the triangular lattice
  Several new classes of compounds can be modeled in first approximation by electrons on the triangular lattice that interact through on-site repulsion $U$ as well as nearest-neighbor repulsion $V$. This extended Hubbard model on a triangular lattice has been studied mostly in the strong coupling limit for only a few types of instabilities. Using the extended two-particle self consistent approach (ETPSC), that is valid at weak to intermediate coupling, we present an unbiased study of the density and interaction dependent crossover diagram for spin and charge density wave instabilities of the normal state at arbitrary wave vector. When $U$ dominates over $V$ and electron filling is large, instabilities are chiefly in the spin sector and are controlled mostly by Fermi surface properties. Increasing $V$ eventually leads to charge instabilities. In the latter case, it is mostly the wave vector dependence of the vertex that determines the wave vector of the instability rather than Fermi surface properties. At small filling, non-trivial instabilities appear only beyond the weak coupling limit. There again, charge density wave instabilities are favored over a wide range of dopings by large $V$ at wave vectors corresponding to $\sqrt(3) \times \sqrt(3)$ superlattice in real space. Commensurate fillings do not play a special role for this instability. Increasing $U$ leads to competition with ferromagnetism. At negative values of $U$ or $V$, neglecting superconducting fluctuations, one finds that charge instabilities are favored. In general, the crossover diagram presents a rich variety of instabilities. We also show that thermal charge-density wave fluctuations in the renormalized classical regime can open a pseudogap in the single-particle spectral weight, just as spin or superconducting fluctuations. 
B. Davoudi, S. R. Hassan and A.-M. S. Tremblay