Distribution of zeros of the partition function for the finite two-dimensional Heisenberg model

Abstract

The exact excitation spectrum of the 3 × 3 square and the 3 × 4 rectangular lattices interacting via the nearest-neighbor anisotropic Heisenberg interaction H=JΣ(ij)[S_i^zSj^z+y(S_i^xS_j^x+S_i^yS_j^y)] has been obtained for various values of the anisotropy constant γ. For 0≤γ≤1, the zeros of the partition function in the complex µ=exp(-mH_e/k_BT) plane obey the generalized Lee-Yang theorem for ferromagnetic coupling. The usual ferromagnetic transition seems to persist up to γ≃0.6. For 0.6≲γ≤1, nothing definite about a ferromagnetic transition can be inferred from the behavior of the zeros. When γ>1, the ferromagnetic zeros do not obey the generalized Lee-Yang theorem at sufficiently low temperatures. For antiferromagnetic coupling, the zeros lie on the negative real axis for all values of γ and of the temperature of the studied lattices.