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Nuclear Physics B

Anisotropic ising model on 2D Kagomé lattice as an inhomogeneous XYZ integrable model

Shahane A. Khachatryan, Hrachia Babujian, A. Sedrakyan

Published Oct 5, 2026• 0 citations
Publisher

Abstract

We investigate a generalized inhomogeneous two-dimensional Ising model on the kagomé lattice with two alternating couplings J i and J i ′ , i = 1 , 2 , 3 , along the three lattice directions. The local Boltzmann weights are mapped onto a non-symmetric eight-vertex R -matrix satisfying the free-fermion condition for arbitrary values of the six couplings. We analyze the Yang–Baxter structure for the cases J i ′ = J i and J i ′ = − J i and derive the corresponding one-dimensional quantum spin chains in the anisotropic limit. The first case yields a transverse-field Ising-type Hamiltonian, while the second leads to a modified chain with a graded permutation structure and shifted partition-function zeros. For a partially anisotropic model, we also obtain the free energy and specific heat.

DOI
10.1016/j.nuclphysb.2026.117686

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We investigate a generalized inhomogeneous two-dimensional Ising model on the kagomé lattice with two alternating couplings J i and J i ′ , i = 1 , 2 , 3 , along the three lattice directions. The local Boltzmann weights are mapped onto a non-symmetric eight-vertex R -matrix satisfying the free-fermion condition for arbitrary values of the six couplings. We analyze the Yang–Baxter structure for the cases J i ′ = J i and J i ′ = − J i and derive the corresponding one-dimensional quantum spin chains in the anisotropic limit. The first case yields a transverse-field Ising-type Hamiltonian, while the second leads to a modified chain with a graded permutation structure and shifted partition-function zeros. For a partially anisotropic model, we also obtain the free energy and specific heat.
We investigate a generalized inhomogeneous two-dimensional Ising model on the kagomé lattice with two alternating couplings JiJ_iJi​ and Ji′J'_iJi′​, i=1,2,3i=1,2,3i=1,2,3, along the three lattice directions. The local Boltzmann weights are mapped onto a non-symmetric eight-vertex RRR-matrix satisfying the free-fermion condition for arbitrary values of the six couplings. We analyze the Yang--Baxter structure for the cases Ji′=JiJ'_i=J_iJi′​=Ji​ and Ji′=−JiJ'_i=-J_iJi′​=−Ji​ and derive the corresponding one-dimensional quantum spin chains in the anisotropic limit. The first case yields a transverse-field Ising-type Hamiltonian, while the second leads to a modified chain with a graded permutation structure and shifted partition-function zeros. For a partially anisotropic model, we also obtain the free energy and specific heat.
The 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 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)kBT. 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)kBT, 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.
By defining a graded global R-operator Rab(2D,2S)\mathbb{R}_{ab}^{(2D,2S)}Rab(2D,2S)​ that couples free-fermion structures and incorporates anisotropic Hubbard interactions while satisfying the Yang--Baxter equation, we construct a strictly solvable two-dimensional lattice model. We then build the layer-to-layer transfer matrix through a bidirectional-monodromy construction and prove the model's integrability via the associated global RTT relations. Using the nested algebraic Bethe ansatz, we obtain the exact eigenvalues of the transfer matrix and derive the corresponding first- and second-level Bethe equations. Finally, by taking the logarithmic derivative of the transfer matrix at the regular point, we recover explicitly a local Hamiltonian that features anisotropic hopping, an on-site Hubbard interaction, and orbital-coupling contributions.
This paper considers aspects of a Kagome lattice system with states classified by plane partitions. Using two sets of free fermions, we rewrite the lattice in terms of two families of spin chains. In this formalism, the quantum crystals Hamiltonian becomes more transparent, and we determine expressions for the first 3 levels of plane partition states. A classical statistical model associated to local configurations in the Kagome lattice is also defined, and we show that a reduction of this classical system gives two Yang-Baxter integrable subsystems analogous to a descendant of the 6-vertex model.
Abstract We classify all fundamental integrable spin chains with two-dimensional (2D) local Hilbert space which have regular R -matrices of difference form. This means that the R -matrix underlying the integrable structures is of the form and reduces to the permutation operator at some particular point. We find a total of 14 independent solutions, eight of which correspond to well-known eight or lower vertex models. The remaining six models appear to be new and some have peculiar properties such as not being diagonalizable or being nilpotent. Furthermore, for even R -matrices, we find a bijection between solutions of the Yang–Baxter equation (YBE) and the graded YBE which extends our results to the graded 2D case.
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 T3T_3T3​) 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 12×1212\times1212×12 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=1/8β=1/8β=1/8 follows.