Orbital compass model

Abstract In a solid with orbital degree of freedom, an orbital configuration does not minimize simultaneously bond energies in equivalent directions. This is a kind of frustration effect which exists intrinsically in orbital degenerate system. We review in this paper the intrinsic orbital frustration effects in Mott insulating systems. We introduce recent our theoretical studies in three orbital models, i.e. the cubic lattice orbital model, the two-dimensional orbital compass model and the honeycomb lattice orbital model. We show numerical results obtained by the Monte-Carlo simulations in finite size systems, and introduce some non-trivial orbital states due to the orbital frustration effect. 

In Sect. 2, the microscopic model which describes the inter-site orbital interactions are introduced. In Sect. 3, the numerical study in the orbital model in a cubic lattice is presented. The non-trivial orbital states in the two-dimensional orbital compass model and the honeycomb lattice orbital model are introduced in Sects. 4 and 5, respectively. The last section is devoted to the summary of this paper. 

Although the orbital model in (5) is derived in a simple cubic lattice, it is shown that, in general, the several orbital models are represented in similar forms of the Hamiltonian. This will be shown, in the following sections, for the orbital compass model in a two dimensional square lattice and the honeycomb lattice orbital model. 

As a different kind of orbital models with the intrinsic frustration effect, we introduce, in this section, the orbital compass model in the two-dimensional square lattice [16,17,20-22]. This model is given by... 

Fig. 6 Dispersion relation of the orbital interaction of the two-dimensional orbital compass model in (11) represented in the BriUouin zone for the square lattice...

We examine the orbital compass model by utilizing the quantum Monte-Carlo method is a finite-size cluster [4], The simulations have been performed on a square lattice of Lx L sites with periodic-boundary conditions. 

Using the orbit radii equation, calculate hydrogen s first seven electron orbit radii and then construct a scale model of those orbits. Use a compass and a metric ruler to draw your scale model on two sheets of paper that have been taped together. (Use caution when handling sharp objects.) Using the orbit energy equation, calculate the energy of each electron orbit and record the values on your model. 

---