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.

Figure 1

(a) A honeycomb-lattice structure and sublattices $A$ and $B$. Bold arrows represent vectors connecting NN two sites. (b) Brillouin zone and reciprocal-lattice vectors for a honeycomb lattice.

Figure 2

(Color online). Pseudospin operator $\mathbf{T}$, and its projection components ${\tau }^{\eta }$ along the three-bond directions.

Figure 3

Eigenvalues of the orbital interaction $\widehat{J}\left(\mathbf{k}\right)$ in the momentum space.

Figure 4

(Color online). (a) A pair of triangular planes termed the $W$ layer, and (b) three Fe–O bond directions in a triangular lattice in $R{\text{Fe}}_2{\text{O}}_4$.

Figure 5

Schematic of the charge and spin structures in $2{\text{Fe}}^{2+}{\text{-Fe}}^{3+}$ plane (right) and in ${\text{Fe}}^{2+}\text{−}2{\text{Fe}}^{3+}$ plane (left) for $R{\text{Fe}}_2{\text{O}}_4$. Filled and open circles represent ${\text{Fe}}^{3+}$ and ${\text{Fe}}^{2+}$, respectively. At sites surrounded by dotted circles, spin directions are not uniquely determined due to frustration.

Figure 6

(Color online). Pseudospin configurations in the ground state. Arrows represent directions of PS’s and bold bars are for the projection components ${\tau }_i^{\eta }$ along the bond direction. A PS configuration, obtained by a uniform rotation of all PS’s in right figure, is also in the ground state.

Figure 7

(Color online). Contour map of the function $f\left({\theta }^{\ast },\mathbf{k}\right)$ in the Brillouin zone for ${\theta }^{\ast }=0$ in (a), and that for ${\theta }^{\ast }=\pi /6$ in (b).

Figure 8

(Color online). Left: PS configuration for ${\theta }^{\ast }=0$. Right: configuration obtained by $\pm \delta \theta$ rotations of PS’s in each zigzag chain.

Figure 9

A part of the free energy $\widehat{F}\left({\theta }^{\ast }\right)$ as a function of the PS angle ${\theta }^{\ast }$ obtained in the spin-wave approximation.

Figure 10

(Color online). Specific heat calculated in several cluster sizes. Low temperature data are enlarged in (b). The inset in (b) shows a peak position $T_{\text{O}}$ in the specific heat as function of $1/N$.

Figure 11

(Color online). Correlation functions $S^{zz}\left(\mathbf{k}\right)$ for several momentum $\mathbf{q}$. Cluster sizes are $2\times 2\times 2$ in (a), $2\times 3\times 3$ in (b), and $2\times 6\times 6$ in (c). The inset in (c) shows correlation function $S\left(\mathbf{k}\right)=4N^{-2}{\sum }_{ij}⟨{\mathbf{T}}_i\cdot {\mathbf{T}}_j⟩e^{i\mathbf{k}\cdot \left({\mathbf{r}}_i-{\mathbf{r}}_j\right)}$ at $\mathbf{k}=\left(\pi ,\pi ,\pi \right)$ calculated in the $e_g$ orbital model. A cubic cluster with ${18}^3$ sites is adopted (Refs. 11, 12).

Figure 12

(Color online) (a) Correlation function of a variable $q$ for the PS angle. The insets show the $q=1$ and $-1$ PS configurations. (b) Scaling analyses for the correlation length of $q_i=\text{cos}3{\theta }_i$.

Figure 13

(Color online) Temperature dependence of correlation functions of $q_i$ and specific heat. Broken, dotted and dash-dotted lines are for the correlation functions between the NN, second NN and third NN sites, respectively.

Figure 14

(Color online). Schematic view of the $q=1$ and $q\ne 1$ states and their low-lying fluctuations.

Figure 15

Energy correction $\Delta E\left({\theta }^{\ast }\right)$ due to the zero-point vibration as a function of the orbital angle ${\theta }^{\ast }$.

Figure 16

(a) Ground state energy and (b) energy gap for several size clusters. For the $2\times 3\times 3$ cluster, energy difference between the doubly degenerate ground states and the first-excited states are plotted. Broken lines are obtained by the least square fittings.

Figure 17

Correlation functions $S^{lm}\left(\mathbf{k}\right)$ for several momenta $\mathbf{k}$. Cluster sizes are $2\times 2\times 2$ in (a), $2\times 3\times 3$ in (b), and $2\times 3\times 4$ in (c).

Figure 18

(Color online). Some of the PS configurations where the honeycomb lattice is covered by NN bonds with the minimum bond energy. One of the $q=1$ states in (a), and one of the $q=-1$ in (b). In NN bonds surrounded by ellipses, the bond energy is the lowest.

Figure 19

(Color online). One example for the two PS configurations where a resonance state is possible due to the off-diagonal matrix element.

Figure 20

(Color online). Overlap integrals between the ground-state wave function and the trial functions. Squares are for the results where the variational parameters $A_l$ are taken to be one, and circles are obtained by optimizing $A_l$.

Figure 21

(Color online). A MC snapshot for PS configurations. Numbers in each hexagons denotes a number of the minimum-energy bonds $n_{\text{min}}$.

Figure 22

(Color online). Zero-dimensional fluctuation in the configuration II. Left: PS configuration. Right: a configuration obtained by $\pm \delta \theta$ rotation of PS’s in each hexagon with $n_{\text{min}}=0$.