Location of zeros in the complex temperature plane: absence of Lee-Yang theorem

In the Yang-Lee theory (1952) the occurrence of a phase transition at real (physical) values of the magnetic field is related to the behaviour of the density of zeros of the partition function in the complex magnetic field plane. As shown by Lee and Yang, these zeros fall on lines for a wide class of ferromagnetic models. In extensions of the Yang-Lee theory to the complex temperature plane, it has often been assumed that the zeros of the partition function would again fall on lines rather than in areas. Though this happens to be true for the isotropic Ising model, the authors point out that there is in general no basis for this assumption. They discuss the location of the zeros of the partition function of the anisotropic Ising model in the complex temperature plane and show that they indeed always fall in areas.