Exact numerical results on finite one- and two-dimensional Heisenberg systems

Results of exact numerical work on the spectra and thermodynamic properties are presented for finite clusters in one and two dimensions, for the isotropic Heisenberg Hamiltonian and the Heisenberg Hamiltonian with longitudinal anisotropy. Eigenvalues of the isotropic Heisenberg loops (that is, those with periodic boundary conditions) of $N\le 12, 8, 6, 5$, 5 particles, each of spin ½, 1, $\frac{3}{2}$, 2 and $\frac{5}{2}$, respectively, are found. It is shown that the spin-$S$ linear chain ($N\to \infty$) will have ground-state energy per spin $\frac{{\epsilon }_{\infty }^s}{2J}=-1.404\mathrm{}\pm 0.002, -2.85\mathrm{}\pm 0.03, -4.7\mathrm{}\pm 0.4 \mathrm{and} -7.1\mathrm{}\pm 0.5$, for $S=1, \frac{3}{2}, 2, \mathrm{and} \frac{5}{2}$, respectively. For antiferromagnetic interaction, the lowest excited states of the loops with even number of particles have spin 1 and the familiar double sine spectrum; while for the spin-½ case the eigenvalues are very different from the result of the spin-wave theory, they approach this limit as the spin increases, and for spin $\frac{5}{2}$, the infinite-spin results of the spin-wave theory should be quite accurate. For ferromagnetic loops, there exist spin-wave bound states for higher spins just like those familiar in the spin-½ linear chain. Effective magnetic moments for these loops are tabulated. We also present the moments for the $N=8\mathrm{}$ spin-½ chain (with open ends), and find the difference to be small. Results on the 3×4 two-dimensional Heisenberg antiferromagnet are presented and compared with experimental data. Finally, we show some examples of the violation of the nondegeneracy rule, similar to those found by Heilmann and Lieb; levels belonging to different representations of a symmetry group remain permanently degenerate when a parameter is varied.