Deformation of SU(4) singlet spin-orbital state due to Hund’s rule coupling

We investigate the ground-state property and the excitation gap of a one-dimensional spin-orbital model with isotropic spin and anisotropic orbital exchange interactions, which represents the strong-coupling limit of a two-orbital Hubbard model including the Hund’s rule coupling $\left(J\right)$ at quarter filling, by using a density-matrix renormalization group method. At $J=0$, spin and orbital correlations coincide with each other with a peak at $q=\pi ∕2$, corresponding to the SU(4) singlet state. On the other hand, spin and orbital states change in a different way due to the Hund’s rule coupling. With increasing $J$, the peak position of orbital correlation changes to $q=\pi$, while that of spin correlation remains at $q=\pi ∕2$. In addition, orbital dimer correlation becomes robust in comparison with spin dimer correlation, suggesting that quantum orbital fluctuation is enhanced by the Hund’s rule coupling. Accordingly, a relatively large orbital gap opens in comparison with a spin gap, and the system is described by an effective spin system on the background of the orbital dimer state.

Figure 1

Spin-orbital configuration in (a) a one-electron state and (b) a two-electron state.

Figure 2

Exchange interactions in the spin-orbital model (10): (a) $J_S∕K$, (b) $J_T^{zx}∕K$, $J_T^y∕K$, (c) $J_{ST}^y∕K$, and (d) $K$ in units of $t^2∕U^{\prime }$, as a function of $J∕U^{\prime }$.

Figure 3

(a)–(c) Three classes of relevant spin-orbital configurations in the ground state of the four-site system. (d) Weight of each class of (a)–(c) in the ground state as a function of $J∕U^{\prime }$.

Figure 4

(a) Spin and orbital correlation functions at $J=0$. (b) Spin and (c) orbital correlation functions at $J∕U^{\prime }=0.2$.

Figure 5

(a) Spin and (b) orbital structure factors at several values of $J$. (c) Spin and (d) orbital structure factors at $q=\pi ∕2$ and $q=\pi$ as a function of $J∕U^{\prime }$.

Figure 6

Log-log plot of (a) spin and (b) orbital correlation functions at several values of $J$. The results are shown for distances of four multiples.

Figure 7

(a) Spin and (b) orbital dimer correlation functions, measured from the center of the chain at several values of $J$. (c) Spin-orbital configuration in the dimer state. Dotted and solid oval enclosures represent spin- and orbital-singlet pairs, respectively.

Figure 8

(a) Spin and (b) orbital correlation functions between nearest-neighbor sites at $J∕U^{\prime }=0.2$.

Figure 9

System size dependence of (a) spin and (b) orbital gaps at several values of $J$. Dotted curve denotes the analytical result at $J=0$ given by Eq. (30).