Lee–Yang theorem

Lee–Yang theorem. In statistical mechanics and mathematical physics, the Lee–Yang theorem is a fundamental result concerning the zeros of the partition function for certain lattice models. Proved in 1952 by Tsung-Dao Lee and Chen-Ning Yang, it establishes that for a class of ferromagnetic systems with Ising-type interactions, all zeros of the partition function in the complex plane of the magnetic field variable lie on the unit circle. This theorem provided the first rigorous explanation for the absence of phase transitions in a finite magnetic field and has profoundly influenced the study of critical phenomena.

Statement of the theorem

The theorem applies to a system of spin-½ particles on a finite lattice, interacting via ferromagnetic Ising couplings. The Hamiltonian includes a term for an external magnetic field \(h\). The central object is the partition function \(Z\), considered as a polynomial in the variable \(z = e^{-2\beta h}\), where \(\beta\) is the inverse temperature. The Lee–Yang theorem states that for real, non-negative couplings, all roots of \(Z(z) = 0\) lie on the unit circle \(|z| = 1\) in the complex plane. This implies that the free energy is analytic for all real \(h \neq 0\), so a phase transition, signaled by a non-analyticity, can only occur at zero magnetic field.

Historical context and significance

The theorem was published by Tsung-Dao Lee and Chen-Ning Yang in 1952, during a period of intense activity in the theory of phase transitions. Their work was contemporaneous with the development of Onsager's exact solution of the two-dimensional Ising model and the emerging renormalization group theory. The result resolved a long-standing puzzle about the analyticity of thermodynamic functions and provided a rigorous foundation for the study of critical exponents. It also connected deeply to earlier work by J. E. Mayer on cluster expansions and anticipated later developments in the use of complex analysis in statistical field theory. The theorem's significance was recognized when Lee and Yang were awarded the Nobel Prize in Physics in 1957 for their work on parity violation, though their contributions to statistical mechanics remain highly influential.

Physical interpretation and applications

Physically, the locus of zeros on the unit circle means that for a real magnetic field, the partition function is never zero, ensuring the free energy is a smooth function. This explains why a ferromagnet exhibits a sharp phase transition only at zero field, where the zeros can pinch the real axis in the thermodynamic limit. The theorem has been applied to analyze the Yang–Lee edge singularity, a critical phenomenon associated with the density of these zeros. It has also been used in studies of lattice gauge theory, quantum spin chains, and in understanding the Lee–Yang–Parr correlation functional in quantum chemistry. The concept of partition function zeros, or Lee–Yang zeros, has become a standard tool for probing phase structure in systems ranging from Bose–Einstein condensates to quark–gluon plasma.

Mathematical formulation and proof

Consider a finite graph \(\Lambda\) with vertices \(V\) and edges \(E\). At each vertex \(i \in V\) is a spin-½ variable \(\sigma_i = \pm 1\). The Hamiltonian is \(H = -\sum_{\langle ij \rangle} J_{ij} \sigma_i \sigma_j - h \sum_{i} \sigma_i\), with \(J_{ij} \geq 0\). The partition function is \(Z = \sum_{\{\sigma\}} e^{-\beta H} = e^{\beta h N} \sum_{\{\sigma\}} \prod_{\langle ij \rangle} e^{\beta J_{ij} \sigma_i \sigma_j} \prod_{i} z^{\sigma_i}\), where \(z = e^{-2\beta h}\) and \(N = |V|\). Lee and Yang proved that \(Z\), as a polynomial in \(z\), has all its roots on \(|z|=1\). A key step in the original proof uses the Asano contraction method, which leverages the fact that for ferromagnetic couplings, the polynomial is Hurwitz-stable after a transformation. Alternative proofs have since been given using the Griffiths–Kelly–Sherman inequalities and methods from the theory of perfect matchings and the Heilmann–Lieb theorem.

Generalizations and related results

The theorem has been extended to many other models. Important generalizations include the Heisenberg model with a magnetic field in the z-direction, certain Potts models, and vertex models like the ice-type model. The circle theorem was generalized by David Ruelle and others to systems with multi-component spins and complex external fields. A quantum version applies to the transverse field Ising model. Closely related is the Sato–Miwa–Jimbo theory of holonomic quantum fields. The study of the distribution of zeros, particularly their density near the real axis, connects to random matrix theory and the Riemann zeta function. The mathematical framework underpinning these results involves Lee–Yang measures, the theory of stable polynomials, and the Pólya–Schur theory of multiplier sequences.

Category:Statistical mechanics Category:Mathematical physics Category:Theorems in statistical mechanics Category:Tsung-Dao Lee Category:Chen-Ning Yang