Doubly degenerate orbital system in honeycomb lattice: Implication of orbital state in layered iron oxide

We study a doubly degenerate orbital model on a honeycomb lattice. This is a model for orbital states in multiferroic layered iron oxides. The classical and quantum models are analyzed by spin-wave approximation, Monte Carlo simulation, and Lanczos method. A macroscopic number of degeneracy exists in the classical ground state. In the classical model, a peak in the specific heat appears at a temperature which is much lower than the mean-field ordering one. Below this temperature, the angle of orbital pseudospin is fixed, but conventional orbital orders are not suggested. The degeneracy in the ground state is partially lifted by thermal fluctuation. We suggest a role of zero-dimensional fluctuation in hexagons on a low-temperature orbital structure. Lifting of the degeneracy also occurs at zero temperature due to the quantum zero-point fluctuation. We show that the ground-state wave function is well represented by a linear combination of the states where a honeycomb lattice is covered by nearest-neighboring pairs of orbitals with the minimum bond energy.