Partition function zeros are powerful tools in understanding critical behavior. In this paper we present new 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 iπkBT/2. 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.
Read more at Physical Review Research:
https://journals.aps.org/prresearch/abstract/10.1103/b6d1-6sk5
Photo caption:
The construction of the dimer lattice for the free-fermion model. (a) The dimer cluster for a single vertex site. The associated dimer weights are presented and the six points are numbered. (b) All possible dimer arrangements of a cluster for each of the vertex configurations. Each thick line represents a dimer, while each thin line represents an edge uncovered by a dimer. As stated in the beginning of Sec. 2a1, “→” and “↑” represent dimers on the horizontal and vertical edges, respectively, while “←” and “↓” indicate the absence of a dimer.
The Fisher loci in the complex z plane of the honeycomb lattice Ising model in the imaginary field. The phases divided by the loci are shown.
12 Dec 2025
