  Spontaneous charge ordering occurring in correlated systems may be considered
as a possible route to generate effective lattice structures with
unconventional couplings. For this purpose we investigate the phase diagram of
doped extended Hubbard models on two lattices: (i) the honeycomb lattice with
on-site $U$ and nearest-neighbor $V$ Coulomb interactions at $3/4$ filling
($n=3/2$) and (ii) the triangular lattice with on-site $U$, nearest-neighbor
$V$, and next-nearest-neighbor $V'$ Coulomb interactions at $3/8$ filling
($n=3/4$). We consider various approaches including mean-field approximations,
perturbation theory, and variational Monte Carlo. For the honeycomb case (i),
charge order induces an effective triangular lattice at large values of $U/t$
and $V/t$, where $t$ is the nearest-neighbor hopping integral. The
nearest-neighbor spin exchange interactions on this effective triangular
lattice are antiferromagnetic in most of the phase diagram, while they become
ferromagnetic when $U$ is much larger than $V$. At $U/t\sim (V/t)^3$,
ferromagnetic and antiferromagnetic exchange interactions nearly cancel out,
leading to a system with four-spin ring-exchange interactions. On the other
hand, for the triangular case (ii) at large $U$ and finite $V'$, we find no
charge order for small $V$, an effective kagome lattice for intermediate $V$,
and one-dimensional charge order for large $V$. These results indicate that
Coulomb interactions induce [case (i)] or enhance [case(ii)] emergent
geometrical frustration of the spin degrees of freedom in the system, by
forming charge order.
