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 the turning of a fermionic trajectory at a lattice vertex is encoded by the corresponding Kac--Ward phase factor. Within this approach the partition function is expressed through the determinant of a finite-dimensional momentum-space matrix, whose zeros determine the excitation spectrum and the critical coupling. We apply the method to the regular square, honeycomb, triangular, kagom\'e, and dual kagom\'e (dice or $T_3$) lattices. In all cases the known exact critical couplings are reproduced. Particular attention is given to the anisotropic kagom\'e lattice, for which the fermionic determinant yields the complete critical surface and the low-energy spectral equation. We also construct the fermionic action for the dual kagom\'e lattice and derive its anisotropic critical condition. In the isotropic dice model the spectrum reduces at low energy and momentum to a relativistic massive form, with the mass vanishing at $\cosh(2J_c)=(1+\sqrt3)/2$. The results demonstrate that the same fermionic construction provides a compact description of criticality and low-energy excitations for Ising models on lattices with different local geometries and coordination numbers.