Partition function zeroes of a self-dual Ising model

Abstract

We consider the Ising model on an $M\times N$ rectangular lattice with an asymmetric self-dual boundary condition, and derive a closed-form expression for its partition function. We show that zeroes of the partition function are given by the roots of a polynomial equation of degree $2M-1$, which trace out certain loci in the complex temperature plane. Particularly, it is shown that (a) real solutions of the polynomial equations always lead to zeroes on the unit circle and a segment of the negative real axis, and (b) all temperature zeroes lie on two circles in the limit of $M\to\infty$ for any $N$. Closed-form expressions of the loci as well as the density of zero distributions in the limit of $N\to\infty$ are derived for M=1 and 2. In addition, we explain the reason of, and establish the criterion for, partition function zeroes of any self-dual spin model to reside precisely on the unit circle. This elucidates a recent finding in the case of the self-dual Potts model.