Definition
The Lee–Yang theorem is a result in statistical mechanics and mathematical physics that characterizes the distribution of zeros of the grand partition function (or, equivalently, the Lee–Yang zeros) for certain ferromagnetic spin systems in the complex plane of an external magnetic field. It states that, for a broad class of ferromagnetic models with pairwise interactions satisfying specific positivity conditions, all zeros of the partition function lie on the imaginary axis of the magnetic field variable, implying that phase transitions can only occur when the magnetic field is real.
Overview
First proved by Tsung-Dao Lee and Chen-Ning Yang in 1952, the theorem provides a rigorous foundation for understanding phase transitions through the analytic properties of the partition function. By examining the limit of an infinite system, the accumulation points of the zeros on the real axis correspond to singularities of the free energy, which signal a phase transition. The original work focused on the Ising model, but subsequent extensions have applied the theorem to a variety of lattice models, quantum spin systems, and even certain quantum field theories.
Etymology/Origin
The theorem is named after its discoverers, physicists Tsung‑Dao Lee and Chen‑Ning Yang, whose 1952 papers “Statistical Theory of Equations of State and Phase Transitions. II. Lattice Gas and Ising Model” (Physical Review, 87, 410) and “Statistical Theory of Equations of State and Phase Transitions. I. Theory of Condensation” introduced the concept of Lee–Yang zeros. The hyphenated form “Lee–Yang” follows standard conventions for eponymous scientific terms.
Characteristics
| Aspect | Description |
|---|---|
| Scope of applicability | Ferromagnetic Ising‑type models with real, non‑negative pairwise couplings; later generalized to broader classes of spin systems and certain quantum models. |
| Mathematical statement | For a system of $N$ spins with Hamiltonian $H = -\sum_{⟨i,j⟩} J_{ij}\sigma_i\sigma_j - h\sum_i \sigma_i$ where $J_{ij}\ge0$ and $\sigma_i=\pm1$, the zeros of the grand partition function $Z_N(z)=\sum_{M} \Omega(M) z^M$ (with fugacity $z=e^{2\beta h}$) all lie on the unit circle $ |
| Implications for phase transitions | In the thermodynamic limit $N\to\infty$, zeros become dense, and a non‑analyticity of the free energy occurs when the accumulation set of zeros crosses the real‑axis value of the magnetic field, indicating a phase transition (e.g., spontaneous magnetization). |
| Extensions | Lee–Yang circle theorem (original formulation for the Ising model); Fisher zeros (zeros in the complex temperature plane); Yang–Lee edge singularity (critical behavior at the endpoint of the zero distribution). |
| Proof techniques | Uses properties of polynomial stability, reflection positivity, and the Griffiths–Kelly–Sherman (GKS) inequalities. Modern proofs employ the theory of Pólya‑frequency functions and Lee–Yang–type theorems in complex analysis. |
| Experimental relevance | Lee–Yang zeros have been indirectly observed in experiments on quantum simulators and nuclear magnetic resonance systems, providing empirical validation of the theorem’s predictions. |
Related Topics
- Ising model – a prototypical lattice spin system to which the original Lee–Yang theorem was applied.
- Partition function zeros – general framework (Lee–Yang zeros, Fisher zeros, Yang–Lee edge) for studying phase transitions via complex analysis.
- Griffiths inequalities – set of correlation inequalities that underpin the positivity conditions required by the theorem.
- Statistical mechanics – the branch of physics concerning the macroscopic behavior of systems from microscopic interactions, within which the theorem resides.
- Quantum phase transitions – extensions of Lee–Yang concepts to zero‑temperature transitions driven by quantum fluctuations.
- Complex analysis in physics – the broader mathematical discipline that includes the study of analytic properties of thermodynamic functions.