Lattice-coupled antiferromagnet on frustrated lattices

The lattice-coupled antiferromagnetic spin model is analyzed for a number of frustrated lattices: triangular, Kagom\'e, and pyrochlore. We show, within the mean-field theory, that the classical ground state is free of lattice deformation in the triangular, Kagom\'e, and the tetrahedron-based, q=0, Eg phonon mode of the pyrochlore lattice. The q=0, Eᵤ phonon mode of pyrochlore is shown to have an energy gain of -2α² per spin over the undistorted ground state, where α is the spin-lattice interaction strength. Motivated by the picture of the hexagon spin cluster proposed in the recent experiment on ZnCr₂O₄ [S. H. Lee et al., Nature (London) 418, 856 (2002)], we also analyze the hexagon-based distortion within our model. Hexagon distortions give rise to mutually orthogonal arrangement of spins for nearby hexagons, and has an energy gain of -α²∕2 per spin. A general criterion for the lattice instability in the spin-lattice model is discussed.