Preliminaries

Along the formalization in Newman (1974) the Hamiltonian is given by

${\displaystyle H=-\sum J_{jk}S_{j}S_{k}-\sum z_{j}S_{j}}$

where S_j's are spin variables, z_j external field. The system is said to be ferromagnetic if all the coefficients in the interaction term J_jk are non-negative reals.

The partition function is given by

${\displaystyle Z=\int e^{-H}d\mu _{1}(S_{1})\cdots d\mu _{N}(S_{N})}$

where each dμ_j is an even measure on the reals R decreasing at infinity so fast that all Gaussian functions are integrable, i.e.

${\displaystyle \int e^{bS^{2}}d|\mu _{j}(S)|<\infty ,\,\forall b\in \mathbb {R} .}$

A rapidly decreasing measure on the reals is said to have the Lee-Yang property if all zeros of its Fourier transform are real as the following.

${\displaystyle \int e^{hS}d\mu _{j}(S)\neq 0,\,\forall h\in \mathbb {H} _{+}:=\{z\in \mathbb {C} \mid \Re (z)>0\}}$

Theorem

The Lee–Yang theorem states that if the Hamiltonian is ferromagnetic and all the measures dμ_j have the Lee-Yang property, and all the numbers z_j have positive real part, then the partition function is non-zero.

${\displaystyle Z(\{z_{j}\})\neq 0,\,\forall z_{j}\in \mathbb {H} _{+}}$

In particular if all the numbers z_j are equal to some number z, then all zeros of the partition function (considered as a function of z) are imaginary.

In the original Ising model case considered by Lee and Yang, the measures all have support on the 2 point set −1, 1, so the partition function can be considered a function of the variable ρ = e^πz. With this change of variable the Lee–Yang theorem says that all zeros ρ lie on the unit circle.