Free-Fermion Models and Two-Dimensional Ising Models Under Zero Field and Imaginary Field i(π/2)k_BT

The 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 honeycomb, triangular and Kagomé lattices. Each Ising model is studied in the cases of a zero field and of an imaginary field i(π/2)kBT. 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 Kagomé lattices are odd free-fermion models. We obtain the exact solution of the Kagomé lattice Ising model under the imaginary field i(π/2)kBT, a result not previously reported in the literature. We also show that the frustrated Ising models on the triangular and Kagomé lattices in the imaginary field still exhibit a non-zero residual entropy.