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arXiv:cond-mat/0008352 · Journal of Physics A Mathematical and General

Transfer matrix eigenvalues of the anisotropic multiparametricUmodel

Jon Links, Angela Foerster

Published Jul 16, 2001• 3 citations
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Abstract

A multiparametric extension of the anisotropic U model is discussed which maintains integrability. The R -matrix solving the Yang-Baxter equation is obtained through a twisting construction applied to the underlying U q ( sl (2|1)) superalgebraic structure which introduces the additional free parameters that arise in the model. Three forms of Bethe ansatz solution for the transfer matrix eigenvalues are given which we show to be equivalent.

DOI
10.1088/0305-4470/34/29/307

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A multiparametric extension of the anisotropic U model is discussed which maintains integrability. The R -matrix solving the Yang-Baxter equation is obtained through a twisting construction applied to the underlying U q ( sl (2|1)) superalgebraic structure which introduces the additional free parameters that arise in the model. Three forms of Bethe ansatz solution for the transfer matrix eigenvalues are given which we show to be equivalent.
We solve the gl(1|2) generalized model by means of the algebraic Bethe ansatz. The resulting eigenvalue of the transfer matrix and the Bethe ansatz equations depend on three complex functions, called the parameters of the generalized model. Specifying the parameters appropriately, we obtain the Bethe ansatz equations of the supersymmetric t-J model, the Hubbard model, or of Yang’s model of electrons with delta interaction. This means that the Bethe ansatz equations of these (and many other) models can be obtained from a common algebraic source, namely from the Yang-Baxter algebra generated by the gl(1|2) invariant R-matrix.
The Perk–Schultz model may be expressed in terms of the solution of the Yang–Baxter equation associated with the fundamental representation of the untwisted affine extension of the general linear quantum superalgebra U q [ gl ( m | n )], with a multiparametric coproduct action as given by Reshetikhin. Here, we present analogous explicit expressions for solutions of the Yang–Baxter equation associated with the fundamental representations of the twisted and untwisted affine extensions of the orthosymplectic quantum superalgebras U q [ osp ( m | n )]. In this manner, we obtain generalizations of the Perk–Schultz model.
The algebraic Bethe ansatz can be performed rather abstractly for whole classes of models sharing the same R -matrix, the only prerequisite being the existence of an appropriate pseudo vacuum state. Here we perform the algebraic Bethe ansatz for all models with 9 × 9, rational, gl(1|2) invariant R -matrix and all three possibilities of choosing the grading. Our Bethe ansatz solution applies, for instance, to the supersymmetric t – J model, the supersymmetric U model and a number of interesting impurity models. It may be extended to obtain the quantum transfer matrix spectrum for this class of models. The properties of a specific model enter the Bethe ansatz solution (i.e. the expression for the transfer matrix eigenvalue and the Bethe ansatz equations) through the three pseudo vacuum eigenvalues of the diagonal elements of the monodromy matrix which in this context are called the parameters of the model.
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.
In this paper I construct lattice models with an underlying U q osp(2,2) superalgebra symmetry. I find new solutions to the graded Yang-Baxter equation. These trigonometric R-matrices depend on three continuous parameters: the spectral parameter, the deformation parameter q and the U(1) parameter, b, of the superalgebra. It must be emphasized that the parameter q is generic and the parameter b does not correspond to the 'nilpotency' parameter of. The rational limits are given; they also depend on the U(1) parameter and this dependence cannot be rescaled away. I give the Bethe ansatz solution of the lattice models built from some of these R-matrices, while for other matrices, due to the particular nature of the representation theory of osp(2,2), I conjecture the result. The parameter b appears as a continuous generalized spin. Finally I briefly discuss the problem of finding the ground state of these models.