Yang–Lee theory

Yang–Lee theory
Name	Yang–Lee theory
Field	Statistical physics
Introduced	1952
Founders	C. N. Yang; T. D. Lee
Notable concepts	Lee–Yang zeros; partition function zeros; phase transitions

Yang–Lee theory is a foundational framework in statistical physics that characterizes phase transitions via the distribution of zeros of the partition function in the complex plane. Developed to explain singularities in thermodynamic quantities, the theory links analytic properties of partition functions to critical phenomena and has influenced work across mathematical physics, condensed matter physics, and complex systems.

Introduction

Yang–Lee theory was introduced to study the analytic structure of the partition function for models such as the Ising model, offering a criterion for the emergence of phase transitions in the thermodynamic limit. The original work analyzes zeros in the complex fugacity plane and relates their accumulation to nonanalytic behavior studied in contexts like the Van der Waals equation, Ising model on a lattice, and models solved by the Bethe ansatz. Key contributors include Chen Ning Yang and Tsung-Dao Lee, whose 1952 papers initiated a broad literature connecting rigorous results from rigorous statistical mechanics to applications in critical phenomena and universality class classification schemes related to work by Kenneth G. Wilson and Leo Kadanoff.

Historical background

The theory arose from mid-20th-century investigations into analytic properties of thermodynamic functions by Yang and Lee, following earlier statistical mechanics studies by Ludwig Boltzmann, James Clerk Maxwell, and Josiah Willard Gibbs. Their approach paralleled rigorous developments by Otfried E. Lanford III and later formalizations by Barry Simon and Elliott H. Lieb. Subsequent milestones include connections to the Yang–Baxter equation discovered in work with R.J. Baxter and relationships to transfer-matrix methods employed by Lars Onsager in solving the two-dimensional Ising model. The Lee–Yang circle theorem and extensions influenced research by Michael Fisher, Robert Griffiths, Tom Kennedy, and David Ruelle, and informed modern numerical studies by groups at institutions like Institute for Advanced Study and Princeton University.

Mathematical formulation

Yang–Lee theory frames the partition function Z of a finite system as a polynomial (or entire function) in a complex variable such as fugacity or magnetic field; its zeros z_i determine analytic structure. The Lee–Yang circle theorem proves that for certain ferromagnetic Ising-type Hamiltonians the zeros lie on the unit circle in the complex fugacity plane; this proof uses methods reminiscent of Perron–Frobenius theory and combinatorial identities exploited by Paul Erdős and George Pólya. In the thermodynamic limit the accumulation points of zeros correspond to singularities of free energy, a viewpoint that links to singularity classification in Renormalization Group theory developed by Kenneth G. Wilson and fixed-point analyses influenced by Michael E. Fisher. Mathematical tools include complex analysis techniques from Bernhard Riemann and Hermann Weyl, combinatorial expansions akin to work by Gian-Carlo Rota, and rigorous bounds related to studies by Eugene Wigner.

Applications and implications

Yang–Lee ideas apply to equilibrium and non-equilibrium systems, influencing studies of the Ising model, Potts model, lattice gas representations used by László Tisza, and quantum phase transitions analyzed in frameworks by Philip W. Anderson and Subrahmanyan Chandrasekhar. The theory underpins modern analyses of critical points in materials investigated at facilities like CERN and Brookhaven National Laboratory, and informs algorithmic methods in computational physics developed by researchers at Los Alamos National Laboratory and Argonne National Laboratory. It also impacts mathematical studies of zeros in polynomials explored by Gábor Szegő and Issai Schur, and connects to integrable models studied by Rodney Baxter and Alexander Zamolodchikov. Practical implications extend to interpreting experiments by groups at Bell Labs and IBM Research on magnetism, as well as to theoretical work on topological phases discussed by Xiao-Gang Wen.

Experimental tests and observations

Experimental probes of Lee–Yang zeros involve measurements of magnetization, susceptibility, and complex-field responses. Techniques leveraging neutron scattering at facilities such as Oak Ridge National Laboratory and synchrotron experiments at European Synchrotron Radiation Facility have tested predictions for magnetic systems like iron alloys and rare-earth compounds studied at Argonne. Recent experimental reconstructions of partition-function zeros used quantum simulators developed by teams at Harvard University, MIT, and University of California, Berkeley employing cold atoms and trapped ions—platforms pioneered by researchers including Immanuel Bloch and Rainer Blatt. Observations relate to finite-size scaling analyses similar to approaches by Kurt Binder and to numerical techniques refined by Mark Newman and Michael E. J. Newman.

Extensions and generalizations

Generalizations of Yang–Lee theory encompass non-ferromagnetic interactions, non-Hermitian Hamiltonians, and dynamical phase transitions studied in contexts related to the Kardar–Parisi–Zhang equation and driven-dissipative systems explored at Max Planck Institute for the Physics of Complex Systems. Extensions include Fisher zeros in the complex temperature plane, studies of zeros for the Potts model and XY model, and applications in quantum information and entanglement spectra analyzed by researchers at Perimeter Institute and Institute for Quantum Information. Mathematical expansions involve connections to random matrix theory advanced by Freeman Dyson and to Lee–Yang distributions in complex networks researched by groups at Santa Fe Institute.

Category:Statistical mechanics