On cluster properties of classical ferromagnets in an external magnetic field

Abstract: Correlation functions of ferromagnetic spin systems satisfying a Lee-Yang property are studied. It is shown that, for classical systems in a non-vanishing uniform external magnetic field $h$, the connected correlation functions decay exponentially in the distances between the spins, i.e., the inverse correlation length ("mass gap"), $m(h)$, is strictly positive. Our proof is very short and transparent and is valid for complex values of the external magnetic field $h$, provided that $\text{Re} \, h \not= 0$. It implies a mean-field lower bound on $m(h)$, as $h \searrow 0$, first established by Lebowitz and Penrose for the Ising model. Our arguments also apply to some quantum spin systems.