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 examine the nature, number, and interrelation of conservation laws in the one-dimensional Hubbard model. In previous work by Shastry [Phys. Rev. Lett. 56, 1529 (1986); 56, 2334 (1986); 56, 2453 (1986); J. Stat. Phys. 50, 57 (1988)], who studied the model on a large but finite number of lattice sites
(Na), only
Na+1 conservation laws, corresponding to
Na+1 operators that commute with themselves and the Hamiltonian, were explicitly identified, rather than the $\ensuremath{\sim}{2N}_{a}$ conservation laws expected from the solvability and integrability of the model. Using a pseudoparticle approach related to the thermodynamic Bethe ansatz, we discover an additional
Na+1 independent conservation laws corresponding to nonlocal, mututally commuting operators, which we call transfer-matrix currents. Further, for the model defined in the whole Hilbert space, we find there are two other independent commuting operators (the squares of the \ensuremath{\eta}-spin and spin operators) so that the total number of local plus nonlocal commuting conservation laws for the one-dimensional Hubbard model is
2Na+4. Finally, we introduce an alternative set of
2Na+4 conservation laws which assume particularly simple forms in terms of the pseudoparticle and Yang-particle operators. This set of mutually commuting operators lends itself more readily to calculations of physically relevant correlation functions at finite energy or frequency than the previous set.
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.