On the Lee-Yang property of some ferromagnets
Yuri Kozitsky
TL;DR
The paper addresses whether the Lee-Yang property for the full partition function $Z_N(h)$ can be induced by ferromagnetic couplings even when the single-spin law $Z_1(h)$ lacks it. It leverages the Lieb-Sokal Laguerre-entire function framework and analyzes the two-spin transform $F_{2N}$ built from a nonnegative interaction matrix $K$ and a Laguerre-type single-spin factor $\phi$, deriving explicit bounds on $\theta$ in terms of the minimum coupling $\varkappa$ to guarantee imaginary-axis zeros. The main result provides a theorem: under Assumption 1 with nonnegative $K$ and bounds $\theta \le \sqrt{\frac{e^{\varkappa}+e^{-\varkappa}}{2}}$ (and a stronger version with an additional condition) the function $F_{2N}$ is Laguerre and has imaginary zeros only; the particular regime $\theta \le 1$ yields Lee-Yang for the Blume-Capel and annealed diluted Ising models in specified parameter ranges. Corollaries show concrete parameter regimes for these models (e.g., $\Delta<\varkappa/2$ for Blume-Capel and thinning bounds for diluted Ising), and connect the results to standard lattices and Dyson hierarchical structures, thereby extending Lee-Yang-type behavior beyond single-spin measures.
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.