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The su(n) Hubbard model

Z. Maassarani

Published Jul 26, 2013• 21 citations
OpenAlex

Abstract

The one-dimensional Hubbard model is known to possess an extended su(2) symmetry and to be integrable. I indroduce an integrable model with an extended su(n) symmetry. This model contains the usual su(2) Hubbard model and has a set of features that make it the su(n) generalization of the Hubbard model. Complete integrability is shown by introducing the L-matrix and showing that the transfer matrix commutes with the hamiltonian. While the model is integrable in one dimension, it provides a generalization of the Hubbard hamiltonian in any dimension.

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The Yangian symmetry of the Hubbard model

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The one-dimensional Hubbard model is known to possess an extended su(2) symmetry and to be integrable. I indroduce an integrable model with an extended su(n) symmetry. This model contains the usual su(2) Hubbard model and has a set of features that make it the su(n) generalization of the Hubbard model. Complete integrability is shown by introducing the L-matrix and showing that the transfer matrix commutes with the hamiltonian. While the model is integrable in one dimension, it provides a generalization of the Hubbard hamiltonian in any dimension.
Abstract. 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. PACS numbers: 71.10.Fd, 71.10.Pm. Submitted to: J. Phys. A: Math. Gen. Integrable variant of the one-dimensional Hubbard model 2
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.
New integrable variant of the one-dimensional Hubbard model with variable-range correlated hopping is studied. The Hamiltonian is constructed by applying the quantum inverse scattering method on the infinite interval at zero density to the one-parameter deformation of the L-matrix of the Hubbard model. By construction, this model has Y(su(2))\\PhiY(su(2)) symmetry in the infinite chain limit. Multiparticle eigenstates of the model are investigated through this method. z E-mail:murakami@appi.t.u-tokyo.ac.jp 2 1 Introduction As a model of strongly correlated electrons, the Hubbard model has been attracting much interest in solid state physics. Especially, in one dimension, the model is exactly solvable [1] and its thermodynamic properties can be calculated out, which give a good testing ground for theories of strongly correlated electron systems. From the point of view of the integrability of the one-dimensional Hubbard model, there have been many works, including the pioneering work o...
In this chapter we will reveal another piece of the algebraic structure of the Hubbard model. As was first observed by Uglov and Korepin [462], the Hubbard Hamiltonian on the infinite line is invariant under the action of the direct sum of two so-called Yangian quantum groups, extending the rotational and the η-pairing su(2) symmetries we encountered earlier. Following [172] we shall address the issue in a more general context. We present two pairs of fermionic representations of the Y(su(2)) Yangian quantum group which commute with the trigonometric [162] and hyperbolic [40, 41] versions of a Hubbard Hamiltonian with non-nearest-neighbour hopping. In both cases the two representations are also mutually commuting, hence can be combined into a representation of Y(su(2))⊕Y(su(2)). The generators of the Yangian symmetry of the ordinary Hubbard model (with nearest-neighbour hopping) and of a number of other interesting models like the Haldane-Shastry spin-chain [194, 394] are obtained as special cases of our general result. Introduction Quantum groups were introduced by Drinfeld [107, 109]. His original intention was to put what we called the Yang-Baxter algebra into the mathematically more conventional context of Hopf algebras. The Yangians are special quantum groups. Their representation theory [80, 81] is intimately related to the classification of integrable quantum systems with rational R -matrices.
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.