Lee–Yang theorem

Remarks

The factorization holds because $\sum_i \sigma_i = N - 2m$, where $m$ is the number of minus spins, so each term is $e^{Nh}$ times a power $z^m$ of $z = e^{-2h}$. One proves the stronger multivariable statement: with a separate variable $z_i$ for each site, the polynomial has no zeros when all $\abs{z_i} < 1$. This is shown by building the system one site or bond at a time, using the Asano contraction, which preserves the absence of zeros in the product of unit discs. The symmetry $\sigma \mapsto -\sigma$ gives $z^N P(1/z) = P(z)$, which rules out zeros with $\abs{z} > 1$ too. The hypothesis $J_{ij} \ge 0$ is needed: for antiferromagnetic couplings the zeros can leave the unit circle.

Example: for one site, $Z(h) = e^h + e^{-h} = e^h (1 + e^{-2h})$, and the only zero $z = -1$ lies on the unit circle.

Going deeper

For a student

The theorem is the start of the theory of Lee–Yang zeros, where phase transitions appear as zeros pinching the real axis in the limit. It extends to many other spin models, including some continuous spins, and was greatly generalized by the theory of stable polynomials. It also gives correlation inequalities and the analyticity of the magnetization in $h$ for $\operatorname{Re} h \ne 0$. The density of zeros on the circle determines the free energy, much as charges determine an electrostatic potential.