Free-fermion models and the two-dimensional Ising models under the zero field and imaginary field $i(\pi/2){k_B}T$

Abstract

Ising model is famous in condensed matter and statistical physics. In this work we present a free-fermion formulation of the two-dimensional classical Ising models on the honeycomb, triangular and Kagom\'e lattices. Each Ising model is studied in the cases of a zero field and of an imaginary field $i(\pi/2){k_B}T$. We employ the decorated lattice technique, star-triangle transformation and weak-graph expansion method to exactly map each Ising model in both cases into an eight-vertex model on the square lattice. The resulting vertex weights are shown to satisfy the free-fermion condition. In the zero field case, each Ising model is an even free-fermion model. In the case of the imaginary field, the Ising model on the honeycomb lattice is an even free-fermion model while the models on the triangular and Kagom\'e lattices are odd free-fermion models. We obtain the exact solution of the Kagom\'e lattice Ising model under the imaginary field $i(\pi/2){k_B}T$, a result not previously reported in the literature. We also show that the frustrated Ising models on the triangular and Kagom\'e lattices in the imaginary field still exhibit a non-zero residual entropy.