Free-fermion approach to the partition function zeros: Special boundary conditions and product form of solution

Partition function zeros are powerful tools in understanding critical behavior. In this paper, we present results of the Fisher zeros of two-dimensional Ising models, in the framework of free-fermion eight-vertex model. First, we succeed in finding special boundary conditions for the free-fermion model, under which the partition function of a finite lattice can be expressed in a double product form. Using appropriate mappings, these boundary conditions are transformed into the corresponding versions of the square, triangular, and honeycomb lattice Ising models. Each Ising model is studied in the cases of a zero field and of an imaginary field <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>i</a:mi> <a:mrow> <a:mo>(</a:mo> <a:mi>π</a:mi> <a:mo>/</a:mo> <a:mn>2</a:mn> <a:mo>)</a:mo> </a:mrow> <a:msub> <a:mi>k</a:mi> <a:mi>B</a:mi> </a:msub> <a:mi>T</a:mi> </a:mrow> </a:math> . For the square lattice model, we rediscover the famous Brascamp-Kunz (B-K) boundary conditions. For the triangular and honeycomb lattice models, we obtain the B-K-type boundary conditions, and the Fisher zeros are conveniently solved from the product form of partition function. The advantage of B-K-type boundary conditions is that the Fisher zeros of any finite lattice exactly lie on certain loci, and the accumulation points of zeros can be easily determined in the thermodynamic limit. Our finding and method would be very helpful in studying the partition function zeros of vertex and Ising models.