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Coupling between spin and lattice degrees of freedom are important ingeometrically frustrated magnets where they can lead to degeneracy lifting andnovel orders. We show that moderate spin-lattice couplings in triangular andKagome antiferromagnets can induce complex colinear magnetic orders. Whenclassical Heisenberg spins on the triangular lattice are coupled to Einsteinphonons, a rich variety of phases emerge, including the experimentally observedfour sublattice state and the five sublattice 1/5th plateau state seen in themagneto-electric material CuFeO$_2$. In addition we predict magnetizationplateaus at 1/3, 3/7, 1/2, 3/5 and 5/7 at these couplings. Strong spin-latticecouplings induce a striped colinear state, seen in $\alpha$-NaFeO$_2$ andMnBr$_2$. On the Kagome lattice, moderate spin-lattice couplings inducecolinear order, but an extensive degeneracy remains.

Spin phonon induced colinear order and magnetization plateaus in triangular and kagome antiferromagnets. Applications to CuFeO_2