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arXiv:2512.06310

Integrable construction of a two-dimensional lattice model with anisotropic Hubbard couplings

Ze Tao, Fujun Liu

nlin.SImath-phquant-phSubmitted Dec 6, 2025• Announced Dec 9, 2025• 0 citations
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Abstract

By defining a graded global R-operator $\mathbb{R}_{ab}^{(2D,2S)}$ 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.

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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.
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A class of recently introduced su (n) "free-fermion" models has recently been used to construct generalized Hubbard models. I derive an algebra defining the "free-fermion" models and give new classes of solutions. I then introduce a conjugation matrix and give a new and simple proof of the corresponding decorated Yang–Baxter equation. This provides the algebraic tools required to couple in an integrable way two copies of free-fermion models. Complete integrability of the resulting Hubbard-like models is shown by exhibiting their L and R matrices. Local symmetries of the models are discussed. The diagonalization of the free-fermion models is carried out using the algebraic Bethe Ansatz.
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.
In this paper we study the exact solution of a one-dimensional model of spin-12 electrons composed by a nearest-neighbor triplet pairing term and the on-site Hubbard interaction. We argue that this model admits a Bethe ansatz solution through a mapping to a Hubbard chain with imaginary kinetic hopping terms. The Bethe equations are similar to that found by Lieb and Wu [3] but with additional twist phases which are dependent on the ring size. We have studied the spectrum of the model with repulsive interaction by exact diagonalization and through the Bethe equations for large lattice sizes. One feature of the model is that it is possible to define the charge gap for even and odd lattice sites and both converge to the same value in the infinite size limit. We analyze the finite-size corrections to the low-lying spin excitations and argue that they are equivalent to that of the spin-12 isotropic Heisenberg model with a boundary twist depending on the lattice parity. We present the classical statistical mechanics model whose transfer matrix commutes with the model Hamiltonian. To this end we have used the construction employed by Shastry [6], [7] for the Hubbard model. In our case, however, the building block is a free-fermion eight-vertex model with a particular null weight.
Bootstrapping Fermionic RRR-operators with Decorated Yang--Baxter Equations
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