Lee–Yang circle theorem

Statement

Let $\Lambda$ be finite and consider the ferromagnetic Ising model with couplings $J_{ij}\ge 0$ and a uniform external magnetic field $h\in\mathbb{C}$:

Let $Z_\Lambda(h)$ be the associated partition function .

Introduce the complex fugacity variable

Up to a nonvanishing prefactor, $Z_\Lambda(h)$ becomes a polynomial in $z$ (of degree $|\Lambda|$).

Lee–Yang circle theorem.
All zeros of this polynomial lie on the unit circle:

Equivalently, $Z_\Lambda(h)$ has no zeros for $\mathrm{Re}(h)\ne 0$.

Key hypotheses and conclusions

Hypotheses

• Finite volume $\Lambda$.
• Ising spins $\sigma_i\in\{-1,+1\}$.
• Ferromagnetic couplings $J_{ij}\ge 0$.
• A single (uniform) external field $h$ treated as a complex parameter.
• Partition function context: partition function , pressure .

Conclusions

• Location of zeros: zeros in the fugacity variable $z=e^{-2h}$ lie on $|z|=1$.
• Analyticity for real nonzero field: for real $h\ne 0$, $z$ is positive real and not on the unit circle, hence $Z_\Lambda(h)\ne 0$. Consequently the finite-volume pressure $\frac{1}{|\Lambda|}\log Z_\Lambda(h)$ is analytic in $h$ for real $h\ne 0$.
• Constraint on phase transitions: any nonanalyticity in the thermodynamic limit as a function of real $h$ can only occur at $h=0$ (a key input into uniqueness/analyticity results; compare uniqueness/analyticity corollary ).

Proof idea / significance

The theorem is proved by showing that the Ising partition function defines a polynomial with a strong stability property inherited from ferromagnetism. One classical approach uses inductive “contraction” arguments (e.g. Asano-type contractions) that preserve the zero-free region and ultimately force all zeros onto the unit circle.

In applications, Lee–Yang zeros provide a powerful route from finite-volume polynomial structure to thermodynamic analyticity and the study of symmetry breaking at $h=0$ (see phase transitions ).