Frustrated classical Heisenberg model with biquadratic interactions in a rhombic lattice: exact ground-state phase diagram

The model cited has nearest-neighbor (nn) ferromagnetic and nnn Heisenberg interactions plus nn biquadratic interactions. The rhombic symmetry comes from assuming the nnn interaction only between sites on a square lattice connected by (1,1) (not (1,-1)) diagonals, as done for various multiferroic manganites{\footnote{T. Kimura et al., Phys. Rev. B~\textbf{68}, 060403(R) (2003)}$^,$\footnote{M. Mochizuki and N. Furukawa, J. Phys. Soc. Japan~\textbf{78}, 053704 (2009)}}. The biquadratic interactions replace the much smaller anisotropic terms usually used$^2$. The ground state problem in the thermodynamic limit is reduced, exactly, to a 3-spin problem, enabled by the LK cluster method\footnote{D. H. Lyons and T. A. Kaplan, J. Phys. Chem. Solids~\textbf{25}, 645 (1964)}, leading to the phase diagram. We find 4 phases: (1) ferromagnetic, (2) general-wave-vector ($\mathbf{Q}$) spiral, (3) up-up-down- down or ``E-type", degenerate with $\mathbf{Q}=(\pi,0)$, and (4) disordered. The uudd- $(\pi,0)$ degeneracy is removed in favor of uudd by a small ferromagnetic nnn interaction connecting sites along the (1,-1) diagonal (such an interaction was in fact found in ref. 1, where the observed uudd state was discussed). It is argued that the present model is probably realistic for these materials.