The s=1/2 Heisenberg Antiferromagnet on the Triangular Lattice: Exact Results and Spin-Wave Theory for Finite Cells

We study the ground state properties of the S=$\frac{1}{2}$ Heisenberg antiferromagnet (HAF) on the triangular lattice with nearest-neighbour ($J$) and next-nearest neighbour ($\alpha J$) couplings. Classically, this system is known to be ordered in a $120^\circ$ N\'eel type state for values $-\infty<\alpha\le 1/8$ of the ratio $\alpha$ of these couplings and in a collinear state for $1/8<\alpha<1$. The order parameter ${\cal M}$ and the helicity $\chi$ of the $120^\circ$ structure are obtained by numerical diagonalisation of finite periodic systems of up to $N=30$ sites and by applying the spin-wave (SW) approximation to the same finite systems. We find a surprisingly good agreement between the exact and the SW results in the entire region $-\infty<\alpha< 1/8$. It appears that the SW theory is still valid for the simple triangular HAF ($\alpha=0$) although the sublattice magnetisation ${\cal M}$ is substantially reduced from its classical value by quantum fluctuations. Our numerical results for the order parameter ${\cal N}$ of the collinear order support the previous conjecture of a first order transition between the $120^\circ$ and the collinear order at $\alpha \simeq 1/8$.