On the Lee-Yang property of some ferromagnets

Abstract

According to the Lieb-Sokal theorem, the partition function, $Z$, of a ferromagnetic spin model has the Lee-Yang property if the single-spin partition function has it. In this note, it is shown that for some spin models a ferromagnetic interaction can induce the Lee-Yang property of $Z$ even if the single-spin partition function fails to have it. In particular, this holds for the Blume-Capel model and for the annealed states of the $s=\pm 1$ site dilute Ising model with a neares-neighbor interaction on $\mathds{Z}^d$, as well as with interactions defined by a hierarchical structure similar to that of Dyson's hierarchical model.