Orbital and spin interplay in spin-gap formation in pyroxene titanium   oxides ATiSi2O6 (A=Na, Li)
  Interplay between orbital and spin degrees of freedom is theoretically studied for the phase transition to the spin-singlet state with lattice dimerization in pyroxene titanium oxides ATiSi2O6 (A=Na, Li). For the quasi one-dimensional spin-1/2 systems, we derive an effective spin-orbital-lattice coupled model in the strong correlation limit with explicitly taking account of the t_2g orbital degeneracy, and investigate the model by numerical simulation as well as the mean-field analysis. We find a nontrivial feedback effect between orbital and spin degrees of freedom; as temperature decreases, development of antiferromagnetic spin correlations changes the sign of orbital correlations from antiferro to ferro type, and finally the ferro-type orbital correlations induce the dimerization and the spin-singlet formation. As a result of this interplay, the system undergoes a finite-temperature transition to the spin-dimer and orbital-ferro ordered phase concomitant with the Jahn-Teller lattice distortion. The numerical results for the magnetic susceptibility show a deviation from the Curie-Weiss behavior, and well reproduce the experimental data. The results reveal that the Jahn-Teller energy scale is considerably small and the orbital and spin exchange interactions play a decisive role in the pyroxene titanium oxides. 
Toshiya Hikihara and Yukitoshi Motome