Frustrated quantum-spin system on a triangle coupled with $e_g$ lattice vibrations - Correspondence to Longuet-Higgins et al.'s Jahn-Teller model -

Abstract

We investigate the quantum three spin model $({\mathbf{S}}_1,{\mathbf{S}}_2,{\mathbf{S}}_3)$ of spin$=1/2$ on a triangle, in which spins are coupled with lattice-vibrational modes through the exchange interaction depending on distances between spin sites. The present model corresponds to the dynamic Jahn-Teller system $E_g\otimes e_g$ proposed by Longuet-Higgins {\it et al.}, Proc.R.Soc.A.{\bf 244},1(1958). This correspondence is revealed by using the transformation to Nakamura-Bishop's bases proposed in Phys.Rev.Lett.{\bf 54},861(1985). Furthermore, we elucidate the relationship between the behavior of a chiral order parameter $\hat{\chi }={\mathbf{S}}_1\cdot ({\mathbf{S}}_2\times {\mathbf{S}}_3)$ and that of the electronic orbital angular momentum ${\hat{\ell }}_z$ in $E_g\otimes e_g$ vibronic model: The regular oscillatory behavior of the expectation value $<{\hat{\ell }}_z>$ for vibronic structures with increasing energy can also be found in that of $<\hat{\chi }>$. The increase of the additional anharmonicity(chaoticity) is found to yield a rapidly decaying irregular oscillation of $<\hat{\chi }>$.