On the Distribution of Zeros for the Heisenberg Model

Abstract

The distribution of energy eigenvalues is discussed in the Heisenberg model. In parti­ cular, 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 t is presented. It yields the theorem which states if all the zeros of the partition function in the case of spin t 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 1 are on the unit circle of the complex field plane. A theorem is given also on the distribution of zeros in antifer­ romagnetic case. Finally, an idea of coarse-graining of energy eigenvalues is proposed on the complex temperature plane in the Heisenberg model.