发表机构
Alikhanyan National Science Laboratory（阿里哈尼扬国家科学实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究利用Kac–Ward表示构建二维伊辛模型的统一费米子场形式，复现多种晶格的精确临界耦合，还推导了各向异性kagomé与对偶kagomé晶格的临界条件，为不同晶格伊辛模型的临界性提供了紧凑描述。
AI 中文摘要
我们利用Kac–Ward表示，针对多种平面晶格上的二维伊辛模型，构建了统一的费米子场形式化表述。格拉肖场与有向晶格边关联，而费米子轨迹在晶格顶点处的转向则由对应的Kac–Ward相位因子编码。在该方法框架下，配分函数可通过有限维动量空间矩阵的行列式表示，其零点决定了激发谱与临界耦合。我们将该方法应用于规则正方形晶格、蜂窝晶格、三角晶格、 kagomé晶格以及对偶kagomé晶格（又称骰子晶格或$T_3$晶格），所有情况均复现了已知的精确临界耦合。研究特别关注各向异性kagomé晶格，费米子行列式可给出其完整的临界曲面与低能谱方程；我们还构建了对偶kagomé晶格的费米子作用量并推导了其各向异性临界条件。在各向同性骰子模型中，低能低动量下的谱简化为相对论性质量形式，当$\text{cosh}(2J_c)=(1+\text{sqrt}3)/2$时质量消失。上述结果表明，该统一费米子构造可为不同局域几何与配位数的晶格伊辛模型提供临界性与低能激发的紧凑描述。
英文摘要
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 $12\times12$ 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.
Comments20 pages, 6 figures
