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Yangian Symmetry and Quantum Inverse Scattering Method for the One-Dimensional Hubbard Model

Murakami, S, Gohmann, F, 村上修一, Shuichi Murakami

Published Jan 1, 1996• 20 citations
OpenAlex

Abstract

We develop the quantum inverse scattering method for the one-dimensional Hubbard model on the infinite interval at zero density. R-matrix and monodromy matrix are obtained as limits from their known counterparts on the finite interval. The R-matrix greatly simplifies in the considered limit. The new R-matrix contains a submatrix which turns into the rational R-matrix of the XXX-chain by an appropriate reparametrization. The corresponding submatrix of the monodromy matrix thus provides a representation of the Y(su(2)) Yangian. From its quantum determinant we obtain an infinite series of mutually commuting Yangian invariant operators which includes the Hamiltonian. Keywords Hubbard model, quantum inverse scattering, Yangian z E-mail:murakami@appi.t.u-tokyo.ac.jp x E-mail:frank@monet.phys.s.u-tokyo.ac.jp -- Address from Oct. 1996: Physikalisches Institut der Universitat Bayreuth, TP1, 95440 Bayreuth, Germany. 1 Introduction Among the exactly solvable 1d quantum systems the Hubbard...

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We develop the quantum inverse scattering method for the one-dimensional Hubbard model on the infinite interval at zero density. R-matrix and monodromy matrix are obtained as limits from their known counterparts on the finite interval. The R-matrix greatly simplifies in the considered limit. The new R-matrix contains a submatrix which turns into the rational R-matrix of the XXX-chain by an appropriate reparametrization. The corresponding submatrix of the monodromy matrix thus provides a representation of the Y(su(2)) Yangian. From its quantum determinant we obtain an infinite series of mutually commuting Yangian invariant operators which includes the Hamiltonian. Keywords Hubbard model, quantum inverse scattering, Yangian z E-mail:murakami@appi.t.u-tokyo.ac.jp x E-mail:frank@monet.phys.s.u-tokyo.ac.jp -- Address from Oct. 1996: Physikalisches Institut der Universitat Bayreuth, TP1, 95440 Bayreuth, Germany. 1 Introduction Among the exactly solvable 1d quantum systems the Hubbard...
In this chapter we carry out the thermodynamic limit on the level of the monodromy matrix introduced in Chapter 12. This means to change the strategy as compared to the Bethe ansatz solutions put forward in Chapters 3 and 12 which depended crucially on the use of periodic boundary conditions. The discreteness of the quasi momenta k j in the Bethe ansatz wave function was due to the finite length L of the system. For infinite L there will be no Lieb-Wu equations which were our main tool for studying the Hubbard model in this book. Instead the commutation relations for the elements of the infinite interval monodromy matrix will utilized in the calculations shown below. We basically follow the articles [335, 336], where the quantum inverse scattering method, in the way as originally designed in [131, 404, 454], was applied to the Hubbard model. Our account will be restricted to the case of zero electron density. Quite generally, the quantum inverse scattering method, as originally conceived in the spirit of the ‘inverse scattering theory’ for classical non-linear evolution equations, is restricted to uncorrelated vacua (ground states) which limits the applicability of the method. Nevertheless, applying it to the empty band ground state of the Hubbard model we shall obtain valuable additional insights into its structure. We shall reveal the connection between Shastry's R -matrix and the Yangian symmetry discussed in Chapter 14.
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.