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, kagomé, and dual kagomé (dice or T3) lattices. The known critical couplings of the homogeneous models are reproduced as consistency checks. For the fully anisotropic kagomé model with three independent couplings we recover the previously known fully anisotropic kagomé critical surface within the present Kac-Ward construction and the corresponding low-energy spectral equation. We construct the analogous 12×12 fermionic matrix for the anisotropic dice lattice and derive its critical surface, which is related to that of the kagomé 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 kagomé 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 β=1/8 follows.
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BibTeX
@misc{khachatryan2026critical,
title = {{Critical couplings of two dimensional Ising model on various lattices}},
author = {Sh. Khachatryan and A. Sedrakyan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.16949},
eprint = {2608.16949},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.16949},
}
RIS
TY - UNPB
TI - Critical couplings of two dimensional Ising model on various lattices
AU - Sh. Khachatryan
AU - A. Sedrakyan
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2608.16949
UR - https://arxiv.org/abs/2608.16949
AB - 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, kagomé, and dual kagomé (dice or T_3) lattices. The known critical couplings of the homogeneous models are reproduced as consistency checks. For the fully anisotropic kagomé model with three independent couplings we recover the previously known fully anisotropic kagomé critical surface within the present Kac-Ward construction and the corresponding low-energy spectral equation. We construct the analogous 12×12 fermionic matrix for the anisotropic dice lattice and derive its critical surface, which is related to that of the kagomé 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 kagomé 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 β=1/8 follows.
ER -