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

TL;DR

This work provides exact solutions for anisotropic 2D Ising models on honeycomb, triangular, and Kagomé lattices in both zero and imaginary fields by mapping to free-fermion eight-vertex models. The authors develop and combine decorated-lattice, star-triangle, and weak-graph expansion methods to transform Ising Boltzmann factors into vertex weights that satisfy the free-fermion condition, yielding even or odd free-fermion formulations depending on the case. A key contribution is the first exact solution for the Kagomé lattice Ising model in the imaginary field, along with quantified residual entropies in frustrated regimes. The results illustrate the broad applicability of the free-fermion formulation to 2D lattice systems and set the stage for extending these techniques to other lattices or fields.

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 Kagomé lattices. Each Ising model is studied in the cases of a zero field and of an imaginary field $i(π/2){k_B}T$. 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 Kagomé lattices are odd free-fermion models. We obtain the exact solution of the Kagomé lattice Ising model under the imaginary field $i(π/2){k_B}T$, 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.

Figures (6)

• Figure 1: The vertex configurations of the sixteen-vertex model. The even subcase consists of vertices (1)--(8), while the odd subcase consists of vertices (9)--(16).
• Figure 2: The diagrammatic representation of the decorated lattice technique. A spin $\tilde{s}$ is added and the interaction $J$ is replaced by $\tilde{J}$.
• Figure 3: The diagrammatic representation of the star-triangle transformation. A spin $\tilde{s}$ is added and the interactions $J_1$, $J_2$ and $J_3$ are replaced by $J^{\prime}_1$, $J^{\prime}_2$ and $J^{\prime}_3$.
• Figure 4: The diagrammatic representation of applying the decorated lattice technique to the honeycomb lattice model. The region bounded by dashed lines in (a) forms a vertex site of the sixteen-vertex model. The details of the vertex unit are shown in (b).
• Figure 5: The diagrammatic representation of applying the star-triangle transformation to the triangular lattice model. The transformed system, which is on the honeycomb lattice as shown by the dash lines in (a), can be mapped into the sixteen-vertex model in the same way as that in Fig. 4(a). The details of the vertex unit are shown in (b).