Computation of the Yang-Lee Edge Singularity in Ising Models

Abstract

In 1952, Yang and Lee [1] established the link between the occurence of phase transition, and the location of zeroes of the partition function in the complex plane of a suitable activity variable. Lee and Yang [2] analyzed the location of zeroes for the lattice gaz as well as for the ferromagnetic spin one half Ising model. In the thermodynamic (large volume) limit these zeroes will become dense and form a cut in the complex plane. When the temperature is larger than the critical one, this cut ends up into a pair of singularities, which move with the temperature. The critical singularity is generated by the meeting of these two singularities (and possibly others). Nothing is known exactly about the nature of the singularities for the well-known Ising model in more than one dimension. This lack of knowledge indicates that a global understanding of the critical point in the two variables, temperature and magnetic field (or order parameter) has not yet been achieved. Our purpose is to recall the various attempts which have been made in this direction, and also to show why the problem is particularly hard. Although we shall see that these singularities occur for a pure imaginary field and therefore look highly unphysical, we think that the understanding of these singularities remains a challenge for physicists working in statistical mechanics.