A Property of Zeros of the Partition Function for Ising Spin Systems

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Abstract
Given an Ising antiferromagnet on a lattice with an AB substructure (bipartite lattice), one can consider the associated ferromagnet in which all the exchange constants are negated. Suppose the ferromagnet is above its critical temperature in the sense that there is an arc (−θ, θ) of the unit circle on which the partition function has no zeros in z = exp(2βH). We prove that the original antiferromagnet partition function will have no zeros in z in the disc orthogonal to the unit disc and passing through the two end points of the arc. In other words, the antiferromagnet free energy is analytic in the magnetic field for small fields.