Spin-orbital gap of multiorbital antiferromagnet with geometrical frustration

In order to discuss the spin-gap formation in a multiorbital system, we analyze an $e_g$-orbital Hubbard model on a geometrically frustrated zigzag chain by using a density-matrix renormalization group method. Due to the appearance of a ferro-orbital arrangement, the system is regarded as a one-orbital system, while the degree of spin frustration is controlled by the spatial anisotropy of the orbital. In the region of strong spin frustration, we observe a finite energy gap between ground and first-excited states, which should be called a spin-orbital gap. The physical meaning is clarified by an effective Heisenberg spin model including the correct effect of the orbital arrangement influenced by the spin excitation.

Figure 1

(a) Lattice arrangement and site numbering of $N$-site zigzag chain. (b) Schematic view of level splitting between $e_g$ orbitals.

Figure 2

(a) Original zigzag chain including two orbitals at each site. (b) Superblock configuration, in which each orbital is regarded as a site. The dotted line denotes the path for the finite-system algorithm.

Figure 3

DMRG results for orbital states and spin correlations in the ground state. The site dependence of $\left\{{\theta }_i\right\}$ representing orbital arrangement for (a) $\Delta =0$ and (b) $\Delta =1$. Spin correlation function measured from the center site of the lower chain for (c) $\Delta =0$ and (d) $\Delta =1$. (e) Electron densities in $3z^2-r^2$, $x^2-y^2$, and optimal orbitals as a function of $\Delta$. (f) Angle ${\theta }_{\mathbf{i}}$ at the center site of the system as a function of $\Delta$.

Figure 4

(a) Spin structure factor for $\Delta =0$ and 1 in the ground state. (b) The $\Delta$ dependence of the peak position, estimated from the DMRG results with $N=64$.

Figure 5

(a) Schematic view of spin-orbital configuration. (b) Spin-orbital correlation function in the spin-excited state. (c) The optimal set of $\left\{{\theta }_{\mathbf{i}}\right\}$ in the ground and spin-excited states. (d) The system-size dependence of the difference in the energy shift between ground and spin-excited states.

Figure 6

(a) The system-size dependence of spin gap at $\Delta =1$. Error bars for $N=40$ and 64 are estimated from the truncation error. Note that error bar for $N=40$ is within the size of the symbol. (b) Spin gap as a function of $\Delta$ for $U^{\prime }=20$ and $N=24$. (c) Spin gap as a function of $U^{\prime }$ for $\Delta =1$ and $N=24$.