Frustrated classical Heisenberg model in 1 dimension with added nearest-neighbor biquadratic exchange interactions

The ground state phase diagram is determined for the frustrated classical Heisenberg chain with added nearest-neighbor biquadratic exchange interactions. There appear ferromagnetic, incommensurate-spiral, and up-up-down-down phases; a lock-in transition occurs at the spiral boundary. The model contains an isotropic version of the ANNNI model; it is also closely related to a model proposed for some manganites. The Luttinger-Tisza method is not obviously useful due to the non-linear weak-constraint problem; however the ground state is obtained analytically by the exact cluster method of Lyons and Kaplan. The results are compared to the model of Thorpe and Blume, where the Heisenberg part of the energy is not frustrated.