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-down-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.

Figure 1

Phase diagram, ${\gamma }^{\prime }=0$ (rhombic symmetry). In the upper-right and lower-left regions there is large degeneracy that is lifted by ${\gamma }^{\prime }\ne 0$ in favor of the states given.

Figure 2

The ground states in the UUDD, $(\pi ,0)/(0,\pi )$ region of Fig. 1.

Figure 3

Spiral and canted ferromagnets, ${\mathrm{CF}}_n$ (for illustrative value $\pi /4$ of the turn-angle $q_0$.). Vortex lattice: ground state in regions VL of Fig. 4.

Figure 4

Phase diagrams, ${\gamma }^{\prime }=\gamma$ (square symmetry), for $XY$ and Heisenberg (HEIS) models, respectively.

Figure 5

Conical vortex lattice: ground state in the conical VL region of Fig. 4 HEIS. Open arrows indicate plaquette-spin patterns.