Lee–Yang theorem
In statistical mechanics, the Lee–Yang theorem states that if partition functions of certain models in statistical field theory with ferromagnetic interactions are considered as functions of an external field, then all zeros are purely imaginary (or on the unit circle after a change of variable). The first version was proved for the Ising model by . Their result was later extended to more general models by several people. Asano in 1970 extended the Lee–Yang theorem to the Heisenberg model and provided a simpler proof using Asano contractions. extended the Lee–Yang theorem to certain continuous probability distributions by approximating them by a superposition of Ising models. gave a general theorem stating roughly that the Lee–Yang theorem holds for a ferromagnetic interaction provided it holds for zero interaction. generalized Newman's result from measures on R to measures on higher-dimensional Euclidean space. There has been some speculation about a relationship between the Lee–Yang theorem and the Riemann hypothesis about the Riemann zeta function; see . Statement Preliminaries Along the formalization in the Hamiltonian is given by where Sj's are spin variables, zj external field. The system is said to be ferromagnetic if all the coefficients in the interaction term Jjk are non-negative reals.