Dimerization versus Orbital-Moment Ordering in a Mott Insulator ${\mathrm{Y}\mathrm{V}\mathrm{O}}_3$

We use exact diagonalization combined with mean-field theory to investigate the phase diagram of the spin-orbital model for cubic vanadates. The spin-orbit coupling competes with Hund’s exchange and triggers a novel phase, with the ordering of $t_{2g}$ orbital magnetic moments stabilized by the tilting of ${\mathrm{V}\mathrm{O}}_6$ octahedra. It explains qualitatively spin canting and reduction of magnetization observed in ${\mathrm{Y}\mathrm{V}\mathrm{O}}_3$. At finite temperature, an orbital instability in the $C$-type antiferromagnetic phase induces modulation of magnetic exchange constants even in the absence of lattice distortions. The calculated spin structure factor shows a magnon splitting at $\overrightarrow{q}=(0,0,\frac{\pi }{2})$ due to the orbital dimerization.

Figure 1

Phase diagram of the spin-orbital model at $T=0$, reflecting the competition between orbital valence bond (OVB), staggered orbitals ($C$-AF), and orbital moment ordering (OM), as obtained by the ED of a four-site embedded chain for $\varphi =11°$ (circles), and in the MFA (dashed lines). Orbital moments in the OM phase (open arrows) induce spin canting (solid arrows) with angle $\psi$. (Estimated parameters for ${\mathrm{Y}\mathrm{V}\mathrm{O}}_3$ are $\eta \simeq 0.12$, $\lambda \sim 0.3–0.4$ [15]).

Figure 2

Pair correlation functions along the $c$ axis: (a) orbital $⟨{\overrightarrow{\tau }}_i·{\overrightarrow{\tau }}_{i+1}⟩$, (b) spin $⟨{\overrightarrow{S}}_i·{\overrightarrow{S}}_{i+1}⟩$, and (c) spin components $⟨S_i^z⟩$ (full lines) and $⟨S_i^x⟩$ (dashed, long-dashed lines), as functions of $\lambda$ for the OVB ($\lambda <{\lambda }_c\simeq 0.37$) and the OM ($\lambda >{\lambda }_c$) phase, found by the ED at $\eta =0.07$. FM/AF bonds $(ij)$ in the OVB phase are shown by solid/dashed lines in (a) and (b).

Figure 3

Crossover from the $C$-AF to the OM phase for increasing $\lambda$ at $\eta =0.12$: (a) orbital order parameters: ${\tau }^z$ (dashed line), and $l^z$ (solid lines) at $\varphi =11°$; (b) spin canting angle $\psi =\arctan (S^x/S^z)$ (see Fig. 1), as obtained for the tilting angle: $\varphi =5°$, $11°$, and $20°$ (dashed, solid, and dotted lines).

Figure 4

Orbital Peierls dimerization in the $C$-AF phase at $T>0$: (a) spin-spin correlations $⟨{\overrightarrow{S}}_i·{\overrightarrow{S}}_{i+1}⟩$ on strong/weak FM bonds (solid/dashed lines); (b) spin response $S(\overrightarrow{q},\omega )$ in the dimerized $C$-AF phase for $\overrightarrow{q}=(0,0,q)$ with $q=0,\frac{\pi }{2},\pi$. Inset: filled (open) circles indicate strong (weak) features in $S(\overrightarrow{q},\omega )$; lines show the fitted spin-wave dispersion for $\eta =0.12$ and $\lambda =0.4$.