  We explore the possibility of inducing a topological insulator phase in a
honeycomb lattice lacking spin-orbit interaction using a metallic (or Fermi
gas) environment. The lattice and the metallic environment interact through a
density-density interaction without particle tunneling, and integrating out the
metallic environment produces a honeycomb sheet with in-plane oscillating
long-ranged interactions. We find the ground state of the interacting system in
a variational mean-field method and show that the Fermi wave vector, kF, of the
metal determines which phase occurs in the honeycomb lattice sheet. This is
analogous to the Ruderman-Kittel-Kasuya-Yosida (RKKY) mechanism in which the
metal's kF determines the interaction profile as a function of the distance.
Tuning kF and the interaction strength may lead to a variety of ordered phases,
including a topological insulator and anomalous quantum-hall states with
complex next-nearest-neighbor hopping, as in the Haldane and the Kane-Mele
model. We estimate the required range of parameters needed for the topological
state and find that the Fermi vector of the metallic gate should be of the
order of 3Pi/8a (with a being the graphene lattice constant). The net coupling
between the layers, which includes screening in the metal, should be of the
order of the honeycomb lattice bandwidth. This configuration should be most
easily realized in a cold-atoms setting with two interacting Fermionic species.
