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Spontaneous charge ordering occurring in correlated systems may be consideredas a possible route to generate effective lattice structures withunconventional couplings. For this purpose we investigate the phase diagram ofdoped extended Hubbard models on two lattices: (i) the honeycomb lattice withon-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. Thenearest-neighbor spin exchange interactions on this effective triangularlattice are antiferromagnetic in most of the phase diagram, while they becomeferromagnetic 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 otherhand, for the triangular case (ii) at large $U$ and finite $V'$, we find nocharge order for small $V$, an effective kagome lattice for intermediate $V$,and one-dimensional charge order for large $V$. These results indicate thatCoulomb interactions induce [case (i)] or enhance [case(ii)] emergentgeometrical frustration of the spin degrees of freedom in the system, byforming charge order.

Emergent lattices with geometrical frustration in doped extended Hubbard models