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arXiv:cond-mat/0012374 · Journal of the Physical Society of Japan

FermionicR-Operator and Algebraic Structure of 1D Hubbard Model: Its Application to Quantum Transfer Matrix

Yukiko Umeno

Published Sep 15, 2001• 10 citations
arXivHTMLPDF

Abstract

The algebraic structure of the 1D Hubbard model is studied by means of the fermionic R-operator approach. This approach treats the fermion models directly in the framework of the quantum inverse scattering method. Compared with the graded approach, this approach has several advantages. First, the global properties of the Hamiltonian are naturally reflected in the algebraic properties of the fermionic R-operator. We want to note that this operator is a local operator acting on fermion Fock spaces. In particular, SO(4) symmetry and the invariance under the partial particle hole transformation are discussed. Second, we can construct a genuinely fermionic quantum transfer transfer matrix (QTM) in terms of the fermionic R-operator. Using the algebraic Bethe Ansatz for the Hubbard model, we diagonalize the fermionic QTM and discuss its properties.

DOI
10.1143/jpsj.70.2531

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The algebraic structure of the 1D Hubbard model is studied by means of the fermionic R-operator approach. This approach treats the fermion models directly in the framework of the quantum inverse scattering method. Compared with the graded approach, this approach has several advantages. First, the global properties of the Hamiltonian are naturally reflected in the algebraic properties of the fermionic R-operator. We want to note that this operator is a local operator acting on fermion Fock spaces. In particular, SO(4) symmetry and the invariance under the partial particle hole transformation are discussed. Second, we can construct a genuinely fermionic quantum transfer transfer matrix (QTM) in terms of the fermionic R-operator. Using the algebraic Bethe Ansatz for the Hubbard model, we diagonalize the fermionic QTM and discuss its properties.
The SO (4) invariance of the transfer matrix for the one-dimensional Hubbard model is clarified from the viewpoint of the quantum inverse scattering method. We demonstrate the SO (4) symmetry by means of the fermionic L -operator and the fermionic R -matrix, which satisfy the graded Yang-Baxter relation. The transformation law of the fermionic L -operator under the SO (4) rotation is identified with a kind of gauge transformation, which determines the corresponding transformation of the fermionic creation and annihilation operators under the SO (4) rotation. The transfer matrix is confirmed to be invariant under the SO (4) rotation, which ensures the SO (4) invariance of the conserved currents including the Hamiltonian. Furthermore, we show that the representation of the higher conserved currents in terms of the Clifford algebra gives manifestly SO (4) invariant forms.
We discuss the integrable boundary conditions for the one-dimensional (1D) Hubbard Model in the framework of the Quantum Inverse Scattering Method (QISM). We use the fermionic R-matrix proposed by Olmedilla et al. to treat the twisted periodic boundary condition and the open boundary condition. We determine the most general form of the integrable twisted periodic boundary condition by considering the symmetry matrix of the fermionic R-matrix. To find the integrable open boundary condition, we shall solve the graded reflection equation, and find there are two diagonal solutions, which correspond to a) the boundary chemical potential and b) the boundary magnetic field. Non-diagonal solutions are obtained using the symmetry matrix of the fermionic R-matrix and the covariance property of the graded reflection equation. They can be interpreted as the SO(4) rotations of the diagonal solutions.
We reconsider the quantum inverse scattering approach to the one-dimensional Hubbard model and work out some of its basic features so far omitted in the literature. It is our aim to show that the R-matrix and monodromy matrix of the Hubbard model, which have now been known for ten years, have good elementary properties. We provide a meromorphic parametrization of the transfer matrix in terms of elliptic functions. We identify the momentum operator for lattice fermions in the expansion of the transfer matrix with respect to the spectral parameter and thereby show the locality and translational invariance of all higher conserved quantities. We work out the transformation properties of the monodromy matrix under the su(2) Lie algebra of rotations and under the -pairing su(2) Lie algebra. Our results imply invariance of the transfer matrix for the model on a chain with an even number of sites.
Exact integrability and algebraic Bethe ansatz of the small-polaron model with the open boundary condition are discussed in the framework of the quantum inverse scattering method (QISM). We employ a new approach where the fermionic R-operator which consists of fermion operators is a key object. It satisfies the Yang-Baxter equation and the reflection equation with its corresponding K-operator. Two kinds of 'super-transposition' for the fermion operators are defined and the dual reflection equation is obtained. These equations prove the integrability and the Bethe ansatz equation which agrees with the one obtained from the graded Yang-Baxter equation and the graded reflection equations.
We discuss the one-dimensional Hubbard model for a finite chain of atoms, that is within the single band approximation, in the context of the transfer matrix. We show two ways of constructing the finite Hilbert space, one with the basis of electron configurations, and the second created as the antisymmetric part of the orbital and spin function tensor product. We classify the set of all electron configurations according to the action of the symmetric group. We present the Bethe Ansatz (BA) solution for the example of three electrons moving on a ring with N = 6 nodes. After that, we concentrate on giving the motivation for exploring the properties of the transfer matrix. We discuss the scattering and the winding operators in the tensor product space, resulting from the physical and the auxiliary spaces. We consider the transfer matrix as the partial trace over the auxiliary space from the winding operator for the auxiliary particle. After that, we derive the explicit expressions for some elements of the transfer matrix in the basis of the spin electron configurations.
A new integrable model which is a variant of the one-dimensional Hubbard model is proposed. The integrability of the model is verified by presenting the associated quantum R-matrix which satisfies the Yang–Baxter equation. We argue that the new model possesses the SO(4) algebra symmetry, which contains a representation of the η-pairing SU(2) algebra and a spin SU(2) algebra. Additionally, the algebraic Bethe ansatz is studied by means of the quantum inverse scattering method. The spectrum of the Hamiltonian, eigenvectors, as well as the Bethe ansatz equations, are discussed.