Written By: Qingyang Xu (website)

Date: July 5, 2025

We introduce the Lee-Yang Circle Theorem [1,2] and review the related research areas.

Lee and Yang at the Institute for Advanced Studies (IAS) Princeton [0]

Background

• Phase transitions occur when there is a discontinuity or singularity in the macroscopic thermodynamic functions $(F,G,Z)$ or their n-th order derivatives
  • Exception: topological phase transition has essential singularity $F \sim \exp(-\frac{C}{T-T_c})$
• However, in statistical mechanics we can only compute the partition function for finite systems (which is always finite and analytic) and take the thermodynamic limit $N \rightarrow \infty$
• Key question: how do we know if/when a phase transition occurs in this limit?
• Lee-Yang theorem provides a concrete method to identify phase transitions by analyzing the roots of partition function in the complex $\mathbb{C}$ plane

Lee-Yang Theorem [1]

In a ferromagnetic Ising model on an arbitrary lattice and dimension,

$$ \beta H_N[\sigma] =-\sum_{<ij>} J_{ij} \sigma_i \sigma_j + h \sum_{i=1}^N \sigma_i, \ \ z:= e^{-2h} $$

the roots of the partition function $Z_N(z)$ lie on the unit circle in $\mathbb{C}$ (i.e., $|z|=1$)

$$ Z_N(z) = \sum_{\{\sigma_i=\pm1\}} e^{-\beta H_N[\sigma]}=\sum_{\{\sigma_i=\pm1\}} z^{\frac{1}{2}\sum_{i=1}^N \sigma_i}e^{\sum_{<ij>} J_{ij} \sigma_i \sigma_j} $$

Implications

• Why are roots $z^*$ of $Z_N(z)$ important?
  • Because free energy $F=-\beta \ln Z$ has a singularity at $z=z^*$
• For finite system, $Z_N(z)$ is a polynomial of $z$ with positive coefficients. Therefore, $Z_N(z)$ does not have roots in positive real axis → finite systems do not have phase transition
  • Note: in actual experiments the transition is extremely sharp and close to a discontinuity
• Since $Z_N(z)$ is real, the distribution of complex roots is symmetric wrt the real axis $Re(z)$
• In thermodynamic limit $N \rightarrow \infty$, the system has a phase transition if the zeros of $Z_N(z)$ approach the positive real axis arbitrarily closely