The eigenfunctions of a two-dimensional finite 3×3 lattice of Heisenberg model ( s =1/2) are obtained exactly with the aid of group theoretical technique and by using a computer. The eigenvalues of the anisotropic exchange Hamiltonian have been computed. The zeros of the partition function for the two-dimensional finite 3×3 lattice lie on the same unit circle in the complex magnetic field plane as those obtained in the three-dimensional finite 2×2×2 lattice. The zeros of the partition function obtained by coarse-graining of eigenvalues distribute on a line in the complex temperature plane. The thermodynamic quantities–specific heat, entropy, magnetization–has been numerically computed for ferro- and antiferromagnetic coupling. The critical index of the magnetization versus field is discussed on the basis of these numerical results.
The distribution of energy eigenvalues is discussed in the Heisenberg model. In particular, the first and second moments of energy eigenvalues are formulated. A method to reduce the partition function of the Heisenberg model with higher spin to that of spin ½ is presented. It yields the theorem which states if all the zeros of the partition function in the case of spin ½ are on the unit circle of the complex magnetic field plane, then it is true in the case of general spin. Furthermore, it is proved at low temperature that the zeros of the partition function of the ferromagnetic Heisenberg model with spin ½ are on the unit circle of the complex field plane. A theorem is given also on the distribution of zeros in antiferromagnetic case. Finally, an idea of coarse-graining of energy eigenvalues is proposed on the complex temperature plane in the Heisenberg model.