Frustrated Classical Heisenberg and XY Models in Two Dimensions with Nearest-Neighbor Biquadratic Exchange: Exact Solution for the Ground-State Phase Diagram

The ground-state phase diagram is determined exactly for the frustrated classical Heisenberg model plus nearest-neighbor biquadratic exchange interactions on a two-dimensional lattice. A square-and a rhombic-symmetry version are considered. There appear ferromagnetic, incommensurate-spiral, "up-up-own- down'' (UUDD), and canted ferromagnetic states, a nonspiral coplanar state that is an ordered vortex lattice, plus a noncoplanar ordered state (a "conical vortex lattice''). In the rhombic case, which adds biquadratic terms to the Heisenberg model used widely for insulating manganites, the UUDD state found is the E-type state observed; this, along with accounting essentially for the variety of ground states observed in these materials, shows that this model probably contains the long-sought mechanism behind the UUDD state.