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The spin-1 two-dimensional J1-J2 Heisenberg antiferromagnet on a triangular lattice

The spin-1 Heisenberg antiferromagnet on a triangular lattice with the nearest- and next-nearest-neighbor couplings, $J_1=(1-p)J$ and $J_2=pJ$, $J>0$, is studied in the entire range of the parameter $p$. Mori&#39;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 four second-order phase transitions are observed. At $p\approx 0.038$ the ground state is transformed from the long-range ordered $120^\circ$ spin structure into a state with short-range ordering, which in its turn is changed to a long-range ordered state with the ordering vector ${\bf Q^\prime}=\left(0,-\frac{2π}{\sqrt{3}}\right)$ at $p\approx 0.2$. For $p\approx 0.5$ a new transition to a state with a short-range order occurs. This state has a large correlation length which continuously grows with $p$ until the establishment of a long-range order happens at $p \approx 0.65$. In the range $0.5<p<0.96$, the ordering vector is incommensurate. With growing $p$ it moves along the line ${\bf Q&#39;-Q}_1$ to the point ${\bf Q}_1=\left(0,-\frac{4π}{3\sqrt{3}}\right)$ which is reached at $p\approx 0.96$. The obtained state with a long-range order can be conceived as three interpenetrating sublattices with the $120^\circ$ spin structure on each of them.