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

TL;DR

This paper develops a free-fermion formulation for 2D Ising models on various lattices under zero and imaginary fields, providing exact solutions and revealing residual entropy in frustrated cases.

Contribution

It introduces an exact free-fermion mapping for Ising models on honeycomb, triangular, and Kagomé lattices under both zero and imaginary fields, including the first solution for Kagomé lattice with imaginary field.

Findings

Exact free-fermion representation for these Ising models.

First exact solution for Kagomé lattice under imaginary field.

Residual entropy persists in frustrated models with imaginary field.

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.