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arXiv:math-ph/0111030 · Journal of Physics A Mathematical and General

Y(so(5)) symmetry of the nonlinear Schrödinger model with four-components

Hong-Biao Zhang, Mo-Lin Ge, Kang Xue

Published Jan 18, 2001• 1 citation
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Abstract

The quantum nonlinear Schrödinger model with four-component fermions exhibits a Y ( so (5)) symmetry when considered on an infinite interval. The constructed generators of the Yangian are proved to satisfy the Drinfel'd formula and, furthermore, the RTT relation with the general form of a rational R -matrix given by Yang-Baxterization associated with so (5) algebraic structure.

DOI
10.1088/0305-4470/34/4/316

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The quantum nonlinear Schrödinger model with four-component fermions exhibits a Y ( so (5)) symmetry when considered on an infinite interval. The constructed generators of the Yangian are proved to satisfy the Drinfel'd formula and, furthermore, the RTT relation with the general form of a rational R -matrix given by Yang-Baxterization associated with so (5) algebraic structure.
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 study certain non-symmetric wavefunctions associated with the quantum nonlinear Schrödinger model, introduced by Komori and Hikami using Gutkin’s propagation operator, which involves representations of the degenerate affine Hecke algebra. We highlight how these functions can be generated using a vertex-type operator formalism similar to the recursion defining the symmetric (Bethe) wavefunction in the quantum inverse scattering method. Furthermore, some of the commutation relations encoded in the Yang–Baxter equation for the relevant monodromy matrix are generalized to the non-symmetric case.
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.