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

Abstract

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 $120^{\circ }$ Néel 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 $\mathcal{M}$ 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 $\mathcal{M}$ is substantially reduced from its classical value by quantum fluctuations. Our numerical results for the order parameter $\mathcal{N}$ of the collinear order support the previous conjecture of a first order transition between the $120^{\circ }$ and the collinear order at $\alpha \simeq 1/8$.

27 pages, LaTeX, figures on request, ITP-UH-19/93

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