Lattice Distortion Driven by Spin-Lattice Coupling

We analyzed lattice-coupled antiferromagnetic spin models on a variety of frustrated lattices. Inspired by the picture of a hexagonal spin cluster proposed for the paramagnetic ZnCrO$_4$ (S. H. Lee \textit{\ et al}., Nature(2002).), we considered hexagon contractions in the \emph{pyrochlore} lattice. Hexagon distortions give rise to mutually orthogonal arrangements of spins for nearby hexagons, and has an energy gain of $-alpha^2/2$ per spin, where $\alpha$ is the spin-lattice interaction strength. However, due to the local rotational symmetry of the $\langle S_i\cdot S_j \rangle$, mean-field theory predicts a lack of lattice displacement in the \emph {triangular} and \emph{kagom\'{e}}lattices. In contrast to the valence-bond-solid(VBS) state of the Affleck-Kennedy-Lieb- Tesaki type, we argue that a type of VBS order (partial VBS, PVBS) with only four of the six bonds of the triangular lattice being filled by singlets can be stabilized through spin-lattice interactions and lead to lattice deformations as in the compound YMnO$_3$ (Seongsoo Lee \textit{\ et al.}, PRB(2005)). The ground state is derived as the direct product of states, one of which represents the conventional long-range ordered spins, and the other given by the $\sqrt{3}\times\sqrt{3}$ modulation of the valence bond amplitudes, $|GS \rangle = |LRO \rangle \otimes |PVBS \rangle$. The excitation spectrum for the modulated valence bond state is worked out within the single- mode approximation. The spectrum offers a new collective mode, which is distinct from the spin wave excitations of the magnetically ordered ground state, and in principle, observable by neutron scattering.