Relation in the Ising model of the Lee-Yang branch point and critical behavior

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Abstract
In general for an Ising model, the end point of the distribution of zeros of the grand partition function along the unit circle in the activity plane is thought to be a branch point. In the case of the Bethe-lattice, this point is of the square-root type. We point out that for models which exactly satisfy scaling, the coefficients of the partial differential approximants introduced by Fisher are independent of the location and nature of this singularity. Thus, the use of these approximants effectively separates the analysis of the critical properties from those of the Lee-Yang branch point. This branch point is reflected in this approach through the boundary conditions for the partial differential equation. These boundary conditions determine the scaling function. We show this boundary condition problem to be equivalent to the monomer-dimer problem, and prove that there is no phase transition in that problem in any dimension.

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