Cooperative orbital ordering and Peierls instability in the checkerboard lattice with doubly degenerate orbitals

It has been suggested that the metal-insulator transitions in a number of spinel materials with partially filled t_2g d orbitals can be explained as orbitally driven Peierls instabilities. Motivated by these suggestions, we examine theoretically the possibility of formation of such orbitally driven states within a simplified theoretical model, a two-dimensional checkerboard lattice with two-directional metal orbitals per atomic site. We include orbital ordering and interatom electron-phonon interactions self-consistently within a semiclassical approximation, and onsite intraorbital and interorbital electron-electron interactions at the Hartree-Fock level. We find a stable, orbitally induced Peierls bond-dimerized state for carrier concentration of one electron per atom. The Peierls bond distortion pattern continues to be period two bond dimerization even when the charge density in the orbitals forming the one-dimensional band is significantly smaller than 1. In contrast, for carrier density of half an electron per atom the Peierls instability is absent within one-electron theory as well as mean-field theory of electron-electron interactions, even for nearly complete orbital ordering. We discuss the implications of our results in relation to complex charge, bond, and orbital ordering found in spinels.