Magnetic properties of the spin-1 two-dimensional $J_1-J_3$ Heisenberg model on a triangular lattice

Motivated by the recent experiment in NiGa${}_2$S${}_4$, the spin-1 Heisenberg model on a triangular lattice with the ferromagnetic nearest- and antiferromagnetic third-nearest-neighbor exchange interactions, $J_1=-(1-p)J$ and $J_3=pJ,J>0$, is studied in the range of the parameter $0 \leq p \leq 1$. Mori’s projection operator technique is used as a method, which retains the rotation symmetry of spin components and does not anticipate any magnetic ordering. For zero temperature several phase transitions are observed. At $ p \approx 0.2$ the ground state is transformed from the ferromagnetic order into a disordered state, which in its turn is changed to an antiferromagnetic long-range ordered state with the incommensurate ordering vector at $p \approx 0.31$\mathrm{.}Withgrowing$p$ the ordering vector moves along the line to the commensurate point $Q_c=(2\pi \mathrm{/}3,0)$, which is reached at $p=1$. The final state with the antiferromagnetic long-range order can be conceived as four interpenetrating sublattices with the $120\mathrm{deg}$ spin structure on each of them. Obtained results offer a satisfactory explanation for the experimental data in NiGa${}_2$S${}_4$.