Lee-Yang Circle Theorem

• Lee-Yang Circle Theorem is a fundamental result in statistical mechanics stating that all zeros of the partition function for ferromagnetic Ising models lie on the unit circle in the complex plane.
• It employs ferromagnetic symmetry, reciprocal root pairing, and maximum-modulus principles to connect the analytic properties of the partition function with the onset of phase transitions.
• The theorem has extensive applications and generalizations, influencing the study of diluted models, spin glasses, quantum systems, and even complex networks.

The Lee-Yang Circle Theorem establishes a precise geometric constraint on the locations of zeros of the partition function in ferromagnetic spin systems, notably the Ising model. It asserts that, under ferromagnetic interactions and suitable conditions, all zeros of the partition function as a function of a complexified external field (often parametrized via a fugacity variable) lie exactly on the unit circle in the complex plane. The theorem structurally connects partition-function analyticity to the onset of phase transitions, with zeros "pinching" the real axis at criticality in the thermodynamic limit, and provides a foundation for analyzing nonanalytic behavior in statistical mechanics and allied fields.

1. Formal Statement and Proof Structure

Consider the ferromagnetic Ising model on a finite graph $\Gamma=(V,E)$ with spins $\sigma(v)\in\{\pm1\}$, coupling strength $J>0$, and external field $h\in\mathbb{R}$, at temperature $T>0$. The Hamiltonian is

$$H(\sigma) = -J \sum_{\{v,w\}\in E} \sigma(v)\sigma(w) - h \sum_{v\in V} \sigma(v),$$

and the partition function reads

$Z(J,h,T) = \sum_{\sigma} e^{-H(\sigma)/T}.$

Introducing the complex fugacity variable $z = e^{-2h/T}$ (and $t = e^{-2J/T} \in [0,1]$), the partition function (up to normalization) becomes a polynomial in $z$: $Z(z,t) = \sum_{k=0}^{|V|} a_k\,z^k,$ where $a_k$ are real, non-negative coefficients.

Theorem (Lee-Yang, 1952): For any finite graph $\Gamma$ and any $0\leq t\leq1$, all zeros of $Z(z,t)$ lie on the unit circle $|z|=1$.

The proof uses ferromagnetic symmetry to show the polynomial is "self-reciprocal" and invokes a factorization argument in pairs of reciprocal roots, together with a maximum-modulus principle, to exclude zeros off $|z|=1$ (Chio et al., 2018, Ghosh, 2012).

2. Extensions, Generalizations, and Variants

The classical theorem generalizes to a broad array of models and circumstances:

• Multivariate Lee-Yang property: For partition functions $Z(z_1,\dots,z_N)$, the property holds if $Z(z_1,\dots,z_N)=0$ implies $|z_i|=1$ for all $i$.
• Dilute and hierarchical ferromagnets: Ferromagnetic interactions can restore the Lee-Yang property even when the single-site partition function does not have it, e.g., Blume-Capel or annealed dilute Ising models with sufficient pairwise coupling (Kozitsky, 5 Mar 2025).
• Spin glasses: In 2D $\pm J$ Ising spin glasses, Song (2024) proves that the zeros of the squared partition function are densely packed on the unit circle for all realizations of disorder, extending the circle law to quenched disorder and revealing a finite-temperature spectral-gap crossover (Song, 12 Mar 2025).
• Self-avoiding walks: Generating functions for adsorbing self-avoiding walks exhibit Lee-Yang zeros asymptotically accumulating on a circle of radius determined by the free energy (Rensburg, 2016).
• Quantum extensions: For the quantum ferromagnetic Heisenberg model, analogous circle theorems have been established (Fröhlich et al., 2012).

Failures and modifications are observed:

• Complex networks: On annealed scale-free networks with degree distribution $p(K)\sim K^{-\lambda}$, the circle theorem holds only for $\lambda>5$; for $3<\lambda<5$, zeros depart from $|z|=1$ (Krasnytska et al., 2015).
• Bounded-degree graphs: For graphs with bounded maximum degree, there exists a critical gap around $z=1$ (the real axis) where zeros do not occur above a temperature-dependent threshold (Peters et al., 2018).

3. Dynamics of Lee-Yang Zeros and Phase Transitions

In the thermodynamic limit, zeros accumulate to form continuous distributions:

• Ferromagnets: For $T<T_c$, zeros densely cover the unit circle; as $T$ approaches $T_c$, the zeros "pinch" the real axis—signaling criticality.
• Cayley tree: Zeros correspond to iterates of expanding Blaschke products; their statistical distribution yields singular measures, with fractal local dimension determined by Lyapunov exponents, and governs nonanalytic free-energy exponents (Chio et al., 2018).
• Spin glasses: In the 2D $\pm J$ model, a temperature-dependent angular gap in zero density emerges at finite $T<T^*$, interpreted as a precursor to $T=0$ spin-glass criticality (Song, 12 Mar 2025).

4. Measure-Theoretic, Dynamical, and Fractal Aspects

The limiting zero distribution is described by a singular measure $\mu$ supported on the unit circle. For the Cayley tree, this measure is not absolutely continuous with respect to Lebesgue measure and has a Hausdorff dimension strictly less than 1 except possibly at critical points (Chio et al., 2018). The local (pointwise) dimension $d_\mu(\phi)$ is related to dynamical invariants (Lyapunov exponents), and the singular part of the free energy near a zero at angle $\phi$ behaves as $|r-1|^{d_\mu(\phi)}$ (Chio et al., 2018, Garcia-Saez et al., 2015).

5. Numerical, Experimental, and Applied Verification

Recent computational and experimental work connects the theorem to physical observables:

• Tensor network methods: Direct calculation of the zero density $g(\theta)$ in 2D/3D Ising models confirms the unit circle law, elucidates critical and edge singularities, and determines exponents (Garcia-Saez et al., 2015).
• Multipoint Padé approximants: Extraction of Lee-Yang zeros from rational approximations to magnetization yields precise numerical confirmation and finite-size scaling results for exponents $\beta$ and $\delta$ (Singh et al., 2023).
• Quantum probe experiments: The first physical observation of Lee-Yang zeros was achieved using a central-spin coherence protocol, effectively realizing imaginary magnetic fields and mapping measured zero-times to the unit circle (Peng et al., 2014).

6. Rigorous Applications and Analytical Consequences

The theorem underpins key results:

• Analyticity of correlations: For systems satisfying the Lee-Yang property, connected correlations are analytic functions of $h$ in the half-plane $\mathrm{Re}\,h>0$, enabling proofs of independence of boundary conditions, monotonicity, concavity, mass gap, and scaling bounds for critical exponents (Fröhlich et al., 2012).
• Non-perturbative control: In extended models (Blume-Capel, diluted Ising), analytic continuation and Lieb-Sokal-type arguments rigorously constrain the location of phase boundaries (Kozitsky, 5 Mar 2025).

7. Connections to Advanced Topics

• Orthogonal polynomial characterizations: Lee-Yang measures (those whose Fourier transforms have only real zeros) relate to the Laguerre-Pólya class and admit explicit approximation via symmetrized Slater determinants of orthogonal polynomials, with applications across statistical mechanics, quantum theory, and even links to the Riemann Hypothesis (1311.0596).
• Superconductivity and the "semicircle theorem": In BCS superconductivity, partition function zeros form a semicircle in the complex interaction plane rather than a full circle, reflecting a fundamentally different universality and the onset of essential singularity at the Fermi surface (Li et al., 2022).

Table: Lee-Yang Circle Theorem—Model Scope and Observed Behavior

Model Class	Zeros Location	Notable Features
Ising ferromagnet (regular graph)	$|z|=1$ (unit circle)	Circle theorem holds, zeros pinch real axis
Cayley tree, rooted/full	$|z|=1$ (unit circle)	Fractal singular measure, dynamical connection
Ising spin glass (2D, $\pm J$)	$|z|=1$ (unit circle)	Spectral gap at finite $T$, law for $Z^2$ only
Bounded-degree graphs	Partial circle, zero-free arc	Gap appears above $T_c^*$
Scale-free networks ($\lambda>5$)	$|z|=1$ (unit circle)	Circle theorem holds in mean-field regime
Scale-free networks ($3<\lambda<5$)	Off unit circle	Theorem fails, zeros scatter
BCS superconductors	Semicircle in complex $U$	RG separatrix, essential singularity

In summary, the Lee-Yang Circle Theorem provides a foundational geometric criterion for the singularities and analytic structure of phase transitions in ferromagnetic systems, with deep connections to complex dynamics, fractal geometry, and physical observables in statistical, quantum, and disordered models. Its generalizations, limitations, and extensions continue to inform the rigorous analysis of many-body critical phenomena.