Charge degrees in the quarter-filled checkerboard lattice

For a systematic study of charge degrees of freedom in lattices with geometric frustration, we consider spinless fermions on the checkerboard lattice with nearest-neighbor hopping $t$ and nearest-neighbor repulsion $V$ at quarter-filling. An effective Hamiltonian for the limit $|t|\ll V$ is given to lowest non-vanishing order by the ring exchange ($\sim t^{3}/V^{2}$). We show that the system can equivalently be described by hard-core bosons and map the model to a confining U(1) lattice gauge theory.