Critical couplings of two dimensional Ising model on various lattices

Abstract

We develop a unified fermionic-field formulation of the two-dimensional Ising model on several planar lattices using the Kac-Ward representation. Grassmann fields are associated with directed lattice links, while changes in the direction of fermionic trajectories are encoded by the corresponding Kac-Ward phase factors. The resulting quadratic action reduces, after Fourier transformation, to a finite-dimensional momentum-space fermionic matrix. Its determinant determines the excitation spectrum, while its zero modes define the critical surface. We apply this construction to the square, honeycomb, triangular, kagom\'e, and dual kagom\'e (dice or $T_3$) lattices. The known critical couplings of the homogeneous models are reproduced as consistency checks. For the fully anisotropic kagom\'e model with three independent couplings we recover the previously known fully anisotropic kagom\'e critical surface within the present Kac-Ward construction and the corresponding low-energy spectral equation. We construct the analogous $12\times12$ fermionic matrix for the anisotropic dice lattice and derive its critical surface, which is related to that of the kagom\'e lattice by the Ising duality transformation. The same fermionic formulation also provides direct access to physical correlation functions. Expressing the spin-spin correlators through fermionic Green functions and the corresponding Toeplitz/Pfaffian representation, we obtain the spontaneous magnetization of the anisotropic kagom\'e model and of the six-coordinated sublattice of the anisotropic dice model. In both cases the magnetization is controlled by the same spectral polynomial that determines the fermionic gap and vanishes on the corresponding critical surface. In the homogeneous limits the known results are recovered, and the standard two-dimensional Ising magnetization exponent $\beta=1/8$ follows.