Ordering and excitation in orbital compass model on a checkerboard lattice

We study an orbital compass model on a checkerboard lattice where orbital degree of freedom is represented by the pseudospin operator. Competition arises from an Ising interaction for the z component of pseudospins along the vertical/horizontal bonds and an Ising interaction for the x component along diagonal bonds. Classical and quantum compass models are analyzed by utilizing several analytical methods and numerical simulations. At a fully frustrated point where the two Ising interactions compete with each other, a macroscopic number of orbital configurations are degenerate in a classical ground state. This degeneracy is lifted by thermal and quantum fluctuations, and a staggered long-range order of the z component of the pseudospin is realized. A tricritical point for this order appears due to competition between the bond dependent Ising interactions. Roles of geometrical frustration on excitation dynamics are also examined.