Orbital-controlled magnetic transition between gapful and gapless phases in the Haldane system with $t_{2g}$-orbital degeneracy

In order to clarify a key role of the orbital degree of freedom in the spin $S=1$ Haldane system, we investigate ground-state properties of the $t_{2g}$-orbital degenerate Hubbard model on the linear chain by using numerical techniques. Increasing the Hund’s rule coupling in multiorbital systems, in general, there occurs a transition from an antiferromagnetic to a ferromagnetic phase. We find that the antiferromagnetic phase is described as the Haldane system with spin gap, while in the ferromagnetic phase, there exists the gapless excitation with respect to the orbital degree of freedom. Possible relevance of the present results to actual systems is also discussed.

Figure 1

Ground-state phase diagram for the four-site chain. The insets indicate schematic views for electron configurations in the AFM and FM phases. The oval enclosures represent the spin singlet pair for the VBS picture.

Figure 2

Ground-state energy per site as a function of $U_{\mathrm{eff}}$ for $J=6$ in the thermodynamic limit. Note that the origin of the energy is shifted (see the main text). The dashed and solid curves denote exact ground-state energies for the $S=1$ AFM and $T=1∕2$ AFO Heisenberg chains, respectively.

Figure 3

(a) Spin, string, and orbital correlation functions measured from the center of the chain for the AFM state. Here $r$ denotes the distance from the center position. The solid diamonds denote the spin-correlation function of $H_{\mathrm{AFM}}$. (b) Correlation functions for $U_{\mathrm{eff}}=34$. (c) The linear-log plot of the spin-correlation function in Fig. 3a and the two-orbital model. (d) The log-log plot of the orbital-correlation function for the FM state.