On cluster properties of classical ferromagnets in an external magnetic field

TL;DR

This paper proves that classical ferromagnetic spin systems with Lee-Yang property exhibit exponential decay of correlations in a non-zero external magnetic field, establishing a positive inverse correlation length and extending results to some quantum systems.

Contribution

The paper provides a short, transparent proof that correlation functions decay exponentially in classical ferromagnets with Lee-Yang property under a magnetic field, including complex fields, and extends to quantum systems.

Findings

Connected correlation functions decay exponentially with distance in a magnetic field.

Inverse correlation length is strictly positive for non-zero magnetic field.

The results include a mean-field lower bound on the inverse correlation length as the field approaches zero.

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.