Orbital Compass Model in a Checkerboard Lattice

The orbital compass model in a checkerboard lattice is studied. There is a competition between the interaction for the z component of the orbital pseudo-spin along the vertical/horizontal direction and the interaction for the x component along the diagonal direction in a checkerboard lattice. In a frustration point where the two interactions compete with each other, a macroscopic number of the orbital pseudo-spin configurations are degenerated in the classical grounds state. This degeneracy is lifted by the thermal and quantum fluctuations and the z component long range order is realized. The tricritical point appears due to the coexistence of the orbital frustration and the geometrical frustration.